THERY OF QUADRATIC EQUATIONS
SIP-1
1) Find the roots of the equation x²+ 9x -10= 0. -10,1
2) Find the roots of the equation 4x² -17x +4 = 0. 1/4,4
3) Find the nature of the roots of the equation 9x² -3x +1= 0. Complex
4) Find the nature of the roots of the equation 5x² - x -4= 0. Rational and unequal
5) If the sum of the roots of the equation kx² - 52x +24= 0 is 13/6, find the product of its roots. 24
6) If the roots of the equation 6x² - 7x + b= 0 are reciprocals of each other, find b. 6
7) The roots of a quadratic equation are a and - a. The product of its roots is -9. Form the equation in variable x. x²-9=0
8) The roots of the equation x² - 12x + k = 0 are in the ratio 1:2. Find k. 32
9) A quadratic equation has rational coefficients. One of the roots is 2+√2. Find its other root. 2-√2
10) I can buy 9 books less for Rs 1050 if the price of each book goes up by Rs 15. Find the original price and the number of books I could buy at that price. 35
11) P and Q are the roots of the equation x² - 22x +120 = 0. Find the value of
a) P²+ Q². 244
b) 1/P + 1/Q. 11/60
c) difference of P and Q. 2
12) If √(x +9) + √(x + 29)= 10. Find x. 7
13) 4ˣ⁺² + 4²ˣ⁺¹= 1280, find x. 2
14) The minimum value of 2x² + bx + c is known to be 15/2 and occurs at x= -5/2. Find the value of b and c. 10, 20
15) Find the number of positive and negative roots of the equation x² - ax + b = 0 where a> 0 , b> 0. 1 negative roots, 2 or 0 positive roots
16) If -1 and 2 are two of the roots of the equation x⁴+ 3x³+ 2x² +2x -4= 0. Then find the other two roots.
SIP-2
1) Find a quadratic equation whose roots are 3,4.
a) x² +7x +12= 0.
b) x² -7x +12= 0.
c) x² +7x -12= 0.
d) x² +7x -12= 0.
2) Find the roots of the equation x² -12x +13= 0.
a) 1,13 b) -1,-13 c) 6+√23, 6-√23 d) none
3) If the sum of the roots and the product of the roots of a equation are 13 and 30 respectively, find its roots.
1) 10,3 b) -10,-3 c) 10,-3 d) -10,3
4) Find the value of the discriminant of the equation 3x² +7x +2= 0.
a) 6.25 b) 25 c) 43 d) 5
5) Find the nature of the roots of the equation 2x² +6x -5= 0.
a) complex conjugate
b) real and equal
c) conjugate surds
d) unequal and rational.
6) Find the degree of the equation (x³- 3)² -6x⁵= 0.
a) 5 b) 6 c) 9 d) none
7) How many many roots (both real and complex) does (xⁿ - a)²= 0 have?
a) 2 b) n+1 c) 2n d) n
8) Find the signs of the roots of the equation x² +x -420= 0.
a) both are positive
b) both are negative
c) the roots are of opposite signs with the numerically larger root being positive
d) The roots are of opposite signs with the numerically root being negative.
9) Construct a quadratic equation whose roots are 2 more than the roots of the equation x² +9x + 10= 0.
a) x² +5x -4 = 0.
b) x² +13x +32= 0.
c) x² - 5x -4 = 0.
b) x² -13x +32= 0
10) Construct a quadratic equation whose roots are reciprocal of the roots of the equation 2x² +8x + 5= 0.
a) 5x² +8x +2= 0.
b) 8x² +5x +2= 0
c) 2x² +5x +8 = 0.
b) 8x² +2x +5= 0
11) The square of the sum of the roots of a quadratic equation E is 8 times the product of its roots. Find the value of the square of the sum of the roots divided by the product of the roots of the equation whose roots are reciprocals of these of E.
a) 8 b) 1/8 c) 1 d) 4
12) Construct a quadratic equation whose roots are one third of the roots of the equation x² +6x + 10= 0.
a) x² + 18x +90= 0.
b) x² +16x +80= 0
c) 9x² +18x +10= 0.
b) x² +17x +90= 0
13) Find the maximum value of the equation -3x² +4x +5.
a) 19/3 b) 31/12 c) 3/19 d) -19/3
14) The Quadratic expression ax² +bx +c has its maximum/ minimum value at
a) -B/2a b) b/2a c) -2b/a d) 2b/a
15) The expression (4ac - b²)/4a represents the maximum/minimum value of the expression ax²+ bx + c. Which of the following is true?
a) it represents the maximum value when a > 0.
b) it represents the minimum value when a < 0.
c) both a and b
d) neither a or b
16)
PROGRESSION
Sap- 1
1) The sum of n terms of two arithmetic series are in the ratio of (7n +1): (4n +27). Find the ratio of their nth term.
2) In an AP of which 'a' is the last term, if the sum of the 1st p terms is equal to zero, show that the sum of the next q terms is -{aq(p+ q)/(p -1)}
3) a) The interior angles of a polygon are in AP. The smallest angle is 120 & the common difference is 5. Find the number of sides of the polygon.
b) The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygon it its largest interior angle is 172.
4) There are n AM's between 1 & 31 such that 7th mean: (n -1)th mean= 5:9, then find the value of n.
5) Prove that the average of the numbers n sin n°, n= 2,4,6,....,180, is cot 1°.
6) Find the value of the sum ¹⁵⁹ₖ₌₀∑ k. cos k°.
7) The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369. The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.
8) In a set of four numbers, the first three are in GP and the last three are in AP with common difference 6. If the first number is the same as the fourth, find the four numbers.
9) The 1st, 2nd, 3rd terms of an arithmetic series are a, b and a² where a is negative. The 1st, 2nd , 3rd terms of a geometric series are a, a² and b find the
a) value of a and b.
b) sum of infinite geometric series if it exists. Otherwise find the sum to n terms of the GP
c) sum of 40 term of the AP series.
10) Let 'X' denotes the value of the product.
(1+ a+ a²+ a²+....∞)(1+ b+ b²+ b³+.....∞)
Where a and b are the roots of the equation 11x²- 4x -2= 0 and Y denotes the numerical value of the infinite series.
(Logᵥ.
11) Find three numbers a,b,c between 2 and 18 such that;
a) their sum is 25.
b) the numbers 2, a, b are consecutive terms of an AP
c) the number b, c, 18 are consecutive terms of a GP.
12) If one AM 'a' and two GM's p and q be inserted between any two given numbers then show that p³+ q³= 2apq.
13) In the Quadratic equation A(√3 - √2)x²+ B/(√3+ √2) x + c = 0 with m,n as its roots. If A= ⁴√(49+ 20√6) ; B= sum of the infinite GP as 8√3 + 8√6/√3 + 16/√3+ ....∞ and|m - n|= (6√6)ᵏ where k= log₆10 - 2 log₆√5 + log₆√(log₆18+ log₆72), then find the value of C.
14) If S₁, S₂, S₃,.....Sₙ,.... are the sums of infinite geometric series whose first terms are 1,2,3,....n,.... and whose common ratio are 1/2, 1/3, 1/4, ....1/(n +1),..... respectively, then find the value of ²ⁿ⁻¹ᵣ₌∑Sᵣ².
15) Find the sum of the first n terms of the sequence: 1+ 2(1+ 1/n)+ 3(1+ 1/n)²+ 4(1+ 1/n)³+ ......
16) Find the nth term and the sum to n terms of the sequence:
i) 1+5+13+29+61+......
ii) 6+13+22+13+....
17) Find the sum of the n terms of the sequence 1/(1+ 1²+ 1⁴) + 2/(2+ 2²+ 2⁴) + 3/(1+ 3²+ 3⁴)+ ......
18) Let σ denotes the sum of the infinite series
∞ₙ₌₁∑ (n²+ 2n +3)/2ⁿ compute the value of (1³+2³+3³+.....σ³).
19) If the sum √(1+ 1/1²+ 1/2²) + √(1+ 1/2²+ 1/3²) + √(1+ 1/2²+ 1/4²)+ .....+ √(1+ 1/1999²+ 1/2000²) equal to n - 1/n where n ∈ N. Find n.
20) If the 10th term of an HP is 21 and 21st term of the same HP is 10, then find 210th term.
21) The pth term Tₚ of HP is Q(p+ q) and qᵗʰ term Tq is p(p+ q) when p>2, q> 2. Show that
a) Tₚ₊q = pq
b) Tₚq = p+ q
c) Tₚ₊q > Tₚq
22) a) The harmonic mean of two numbers is 4. The arithmetic mean A & the geometric mean G satisfy the relation 2A+ G²= 27. Find the two numbers.
b) The AM of two numbers exceeds their GM by 15 & HM by 27. Find the numbers.
23) If a,b,c,d,e be 5 numbers such that a,b,c are in AP; b,c,d are in GP & c,d,e are in HP then:
a) show that a,c,e are in GP
b) show that e= (2b - a)⅖/a.
c) If a= 2 and e= 18, find all possible values of b,c,d.
24) If A₁, A₂, A₃, .....A₅₁ are arithmetic means inserted between the numbers a and b, then find the value of {(b + A₅₁)/(b - A₆₁)} - {(A₁ + a)/(A₁ - a).
25) Sum of the following series to n terms and to infinity:
a) 1/(1.4.7) + 1/(4.7.10)+ 1/(7.10.13) +....
b) ⁿᵣ₌₁ ∑ r(r+1)(r+2)(r+3)
c) ⁿᵣ₌₁ ∑ 1/(4r²-1).
d) 1/4 + (1.3)/(4.6) + (1.3.5)/(4.6.8) + .......,
Answer
1) (14n -6)(8n +23)
3)a) 9 b) 12
4) 14
6) -180
7)27
8) (8,-4,2,8)
9) a) -1/2, -1/8 b) -1/3 c) 545/2
10) 11/15
11) 5,8,12
13) 128
14) n(2n +1)(4n +1)/3
15) n²
16)i) 2ⁿ⁺¹-3; 2ⁿ⁺²- 4 - 3n ii) n²+ 4n +1; (1/6)n (n +1)(2n +13)+ n
17) n(n +1)/2(n²+ n +1)
18) 8281
19) 2000
20) 1
22) a) 6,3 b) 120,30
23) 4,6,9 or -2,-6,-18
24) 102
25) (1/24) - [(1/6)(3n +1)(3n +4)]: 1/24 ii) (1/5) n(n +1)(n +2)(n +3)(n +4) iii) n/(2n +1) iv) 2[1/2 - (1.3.5.....(2n -1)(2n +1))/(2.4.6....(2n)(2n +2)), 1
PROGRESSION
Sap-2
1) The first three consecutive terms of a geometrical progression are the real roots of the equation 2x³- 19x²+ 57x - 54=0 find the sum to infinite number of the terms of GP.
2) if sin x, sin2x and cosx, sin4x form an increasing geometric sequence, find the numerical value of Cos 2x. Also find the common ratio of geometric sequence.
3) Find the condition on α and β if x₁, x₂, x₃ satisfying the cubic x³ - x² +αx+ and β =0 are AP.
4) Find the sum of the infinite series (1. 3)/2+ (3.5)/2³ + (5.7)/2³ + (7.9)/2⁴+ ... to ∞.
5) Two distinct, real infinite geometric series each have a sum of 1 and have the same second term. The third term of one of the series is 1/8. If the second term of both the series can be written in the form (√m - n)/p, where m,n and p are positive integer and m is not divisible by the square of any prime, find the value of 100m + 10n + p.
6) One of the roots of the equation 2000x⁶+ 100x⁵+ 10x³+ x- 2=0 is of the form (m + √n)/r, when n is non zero integer and n and r are relatively prime natural numbers . Find the value of m+ n + r.
7) Let S= ⁹⁹ ₙ₌₁∑ 5¹⁰⁰/{(25)ⁿ+ (5)¹⁰⁰}. Find [S]
Where [y] denotes largest integer less than or equals to y.
8) A computer solved several problems in succession. The time it took the computer to solve each successful problem was the same number of times smaller than the time to to solve that preceding problem. How many problems were suggested to the computer if it spent 63.5 minutes to solve all the problems except for the first, 127 minutes to solve all the problems except for the last one and 31.5 minutes to solve all the problems except for the first two?
9) if n is a root of the equation x²(1- ac) - x(a²+ c²) - (1+ ac)= 0 & if n HM's are inserted between a and c, show that the difference between the first and the last mean is equals to ac(a - c)
10) If 2²ˢᶦⁿˣ ⁻¹ , 14, 3⁴ ⁻ ²ˢᶦⁿ²ˣ form first three terms of an AP., then find the sum 1+ sin2x + sin²2x + ..... ∞.
11) Given that the cubic ax³- ax²+ 9bx - b= 0 (a≠0) has all three positive roots. Find the harmonic mean of the roots independent of a and b, hence deduce that the roots are all equal. Find also the minimum value of (a+ b) if a and b∈N.
12) If tan(π/12 -x), tan(π/12), tan(π/12 +x), in order are three consecutive terms of a GP, then the sum of all the solutions in [0, 314] is kπ. Find the value of k.
13) The sequence a₁, a₂, a₃, ....a₉₈ satisfies the relation aₙ₊₁ = aₙ +1 for n= 1,2,3,....97 and has the sum equal to 4949. Evaluate ⁴⁹ₖ₌₁ ∑ a₂ₖ.
14) a) The value of x+ y+ z is 15 if a, x, y, z, b are in AP while the value of (1/x) + (1/y) + (1/z) is 5/3 if a, x, y, z, b are in HP. Find a, b.
b) The value of xyz is 15/2 or 18/5 according as the series a, x, y, z, b is an AP or HP.
Find the values of a and b assuming them to be positive integer.
15) A cricket player n(n > 1) matches during his career and made a total of
{(n²-12n + 39)(4.6ⁿ - 5.3ⁿ+1)}/5 runs. If Tᵣ represent the runs made by the player in the rth match such that T₁ = 6 and Tᵣ = 3Tᵣ₋₁ + 6ʳ, 2≤ r ≤ n then find n.
16) If the roots of 10x³- cx²- 54x - 27=0 are in harmonic progression, then find c and all the roots.
17) If a,b,c be in GP and log꜀a, logᵥc, logₐv be an AP, then show that the common difference of the AP must be 3/2.
18) In a GP the ratio of the sum of the first 11 terms of the sum of the last eleven terms is 1/8 and the ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is 2. Find the number of terms in the GP.
19) Given a 3 digit number whose digits are three successive terms of a GP. if we subtract 792 from it, we get a number written by the same digits in the reverse order. Now if we subtract four from the hundred's digit of the initial number and leave the other digits unchanged , we get a number whose digits are successive terms of an AP. Find the number.
20) let a ₙ be a sequence such that a₁ = 3, aₙ₊₁ = 3aₙ +1 (n= 1,2,3...). If the value of ∞ₙ₌₁∑ aₙ/5ⁿ = p/q (where p and q are their lowest form), then find the value of (p + q).
Answer
1) 27/2 2) (√5-1)/2 , √2 3) α ≤ 1/3; β≥ -1/27 4) 23 5) 518 6) 200 7) 49 8) 8 problems, 127.5 minutes 10) 2 11) 28 12) 4950 13) 2499 14)a) a=1, b= 9 or b= 1, a=9 b) a= 1; b= 3 or vice versa 15) n= 6 16) C= 9; (3,-3/2,-3/5) 18) n=38 19) 931. 20) 21
Sap-3
1) The sum of roots of the equation ax⅖+ bx + c=0 is equals to the sum of squares of their reciprocals. Find whether bc², ca² and ab² in AP, GP or HP?
2) Solve the following equations for x and y
log₂x + log₄x + log₁₆x + .....= y.
(5+9+13+....+(4y+1)/(1+3+5+....+(2y-1)= 4 log₄x.
3) Let α, β are the roots of x²- x + p and γ, δ be the roots of x²- 4x + q= 0. If α, β, γ, δ are in GP, then the integral values of p and q respectively, are
a) (-2, 32) b) (-2,3) c) (-6,3) d) (- 6,-32)
4) If the sum of the first 2n terms of the AP 2, 5, 8,.... is equals to the sum of the first n terms of the AP 57, 59, 61,....., then n equals
a) 10 b) 12 c) 11 d) 13
5) Let the positive numbers a,b,c,d be in AP. Then abc, acd, bcd are.,
a) not in AP/ GP/ HP b) in AP c) in GP d) HP
6) Let a₁, a₂, .... be positive real numbers in GP. For each n,
Let Aₙ, Gₙ, Hₙ, be respectively, the arithmetic mean, geometric mean and harmonic mean of a₁, a₂, a₃, .....aₙ. Find an expression for the GM of G₁, G₂, .....Gₙ in terms of A₁, A₂, .....Aₙ, H₁, H₂, .....Hₙ.
7) suppose a,b,c are in AP and a², b², c² are in GP. If a< b < c and a+ b+ c = 3/2, then the value of a is
a) 1/2√2 b) 1/2√3 c) 1/2 - 1/√3 d) 1/2 - 1/√2
8) Let a,b,c be positive real numbers. If a, A₁, A₂, b are in AP; a G₁, G₂, b are in GP and a, H₁, H₂, b are HP, prove that (G₁ G₂)/(H₁H₂) = (A₁+ A₂)/(H₁+ H₂)= {(2a+ b)(a+ 2b)/9ab.
9) If a, b, c are in AP, a², b², c² are in HP, then show that either a= b= c or a, b, -Cl/2 form a GP
10) The first term of an infinite geometric progression is x and its sum is 5. Then
a) 0≤x≤ 10 b) 0< x < 10 c) -10< x < 0 d) x> 10
11) If a, b, c are positive real numbers, then show that {(1+ a)(1+ b)(1+c)}⁷> 7⁷ a³b⁴c⁴.
12) In the Quadratic equation ax²+ bx + c= 0, if ∆= b²- 4ac and α + β, α²+ β², α³+ β³ are in GP, where α, β are the roots of ax²+ bx+ c= 0, then
a) ∆≠ 0 b) b∆= 0 c) c∆= 0 d) ∆= 0
13) If total number of a runs scored in n matches is {(n+1)/4} (2ⁿ⁺¹ - n -2) where n> 1, and the runs scored in the kᵗʰ match are given by k.2ⁿ⁺¹⁻ᵏ, where 1≤ k ≤ n. Find n.
14) For n= 1,2,3,...., let Aₙ = (3/4) - (3/4)²+ (3/4)³+ ....+(-1)ⁿ⁻¹(3/4)ⁿ and Bₙ = 1- Aₙ. Find the smallest natural number n₀ such that Bₙ > Aₙ for all n ≥ n₀.
Comprehension(3 questions)
Let Vᵣ denote the sum of the first 'r' terms of an arithmetic r(AP) whose first term is 'r' and the common difference is (2r -1).
Lrt Tᵣ = Vᵣ₊₁ - Vᵣ -2 and Qᵣ = Tᵣ₊₁ - Tᵣ for r= 1,2...
15) The sum V₁ + V₂+ ....+ Vₙ is
a) (n/12)(n+1)(3n²- n +1)
b) (n/12)(n+1)(3n²+ n +2)
c) (n/12)(2n²- n +1)
d) (1/3)(2n²- 2n +3)
16) Tᵣ is always
a) an odd number b) an even number c) a prime number d) a composite number
17) Which of the following is a correct statement?
a) Q₁, Q₂, Q₃, ......are in AP with common difference 5.
b) Q₁, Q₂, Q₃, ......are in AP with common difference 6.
c) Q₁, Q₂, Q₃, ......are in AP with common difference 11.
d) Q₁= Q₂=Q₃, = .....
Comprehension (3 questions):
Let A₁, G₁, H₁, denote the arithmatic, geometric and harmonic mens respectively, of two distinct positive members. For n≥ 2, let Aₙ₋₁ and Hₙ₋₁ have arithmetic, geometric and harmonic mean as Aₙ, Gₙ, Hₙ respectively.
18) Which one of the following statement is correct?
a) G₁> G₂> G₃> ....
b) G₁< F₂< G₃ < ....
c) G₁ = G₂ = G₃= ....
d) G₁ < G₃ < G₅ < .....and G₂> G₄> G₆ > ....
19) Which one of the following statements is correct?
a) A₁> A₂> A₃> ....
b) A₁ < A₂ < A₃ < ....
c) A₁ > A₃ > A₅ > ..... and A₂ < A₄< A₆ < ....
d) A₁< A₃< A₅ < ....and A₂> A₄> A₆> ...
20) Which of the following statements is correct?
a) H₁> H₂> H₃> ...
b) H₁< H₂< H₃< .....
c) H₁> H₃> H₅ > .... and H₂< H₄< H₆< ....
d) H₁< H₃< H< .... and H₂> H₄ > H₆> ....
21) A straight line through the vertex P of a triangle PQR intersect the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then
a) 1/PS + 1/ST < 2/√(QS x SR)
b) 1/PS + 1/ST > 2/√(QS x SR)
c) 1/PS + 1/ST < 4/QR
d) 1/PS + 1/ST > 4/QR
Assertion & Reason:
22) Suppose four distinct positive numbers a₁, a₂, a₃,a₄ are in GP. Let b₁ = a₁ , b₂ = b₁ + a₂ , b₃ = b₂ + a₃ and b₄ = b₃ + a₄
Statement -1: The number b₁, b₂, b₃, b₄ are neither in AP nor GP.
And
Statement -2: The number b₁, b₂, b₃, b₄ are in HP.
A) Statement -1 is true, Statement -2 is true; statement -2 is a correct explanation for statement -1.
B) Statement -1 is True, Statement -2 is True; statement -2, is NOT a correct explanation for statement -1
C) Statement -1 is True, Statement -2 is False
D) Statement -1 is False, Statement -2 is True
23) If the sum of first n terms of an AP is cn², then the sum of squares of these n terms is
a) n(4n²-1)c²/6
b) n(4n²+1)c²/3
c) n(4n½-1)c²/3
d) n(4n²+1)c²/6
24) Let a₁, a₂, a₃, .....a₁₁ be real numbers satisfying
a₁ = 15, 27 - 2a₂ > 0 and aₖ = 2aₖ₋₁ - aₖ₋₂ for k= 3,4, ....11.
If (a₁²+ a₂²+ ....+ a₁₁²)/11= 90, then the value of (a₁ + a₂ + ...a₁₁) is equal to
25) The minimum value of the sum of real numbers a⁻⁵ , a⁻⁴, 3a⁻³, 1, a⁸, and a¹⁰ with a> 0 is
26) Let a₁, a₂, a₃, ...., a₁₀₀ be an arithmetic progression with a₁ = 3 and Sₚ = ᵖᵢ₌₀∑ aᵢ, 1≤ p≤ 100. For any integer n with 1≤ n ≤ 20, let m= 5n. If Sₘ/Sₙ does not depend on n, then a₂ is
Answer
1) AP 2) 2√2, 3. 3) A 4C 5D 6) [(A₁, A₂, .....Aₙ)(H₁, H₂, ....., Hₙ)]¹⁾²ⁿ 7) d 10b 12c 13) 7 14) n₀= 6 15b 16d 17b 18c 19a 20b 21b,d 22c 23c 24) 0 25) 8 26) 9 or 3.
Sap-1 (SEQUENCE & SERIES, BINOMIAL THEOREM, SOLUTION OF TRIANGLE)
1) (1+ x)(1+ x+ x²)(1+ x + x² + x³)....(1+ x+ x² +....x¹⁰⁰) When written in the ascending power of x then the highest exponents of x is
a) 4950 b) 5050 c) 5250 d) none
2) The sum ¹⁰ₖ₌₁∑ k.k! equals
a) 10! b) 11! c) 10!+1 d) 11!-1
3) If a,b,c are distinct positive real in HP., then the value of the expression, (b + a)/(b - a) + (b+ c)/(b - c) is
a) 1 b) 2 c) 3 d) 4
4) In a triangle ABC, R(b + c)= a√(bc) where R is the circumradius of the triangle. Then the triangle is
a) isosceles but not right b) rifr ut not isosct c) right isosceles d) equilateral
5) If the coefficient of x⁷ & x⁸ in the expansion of (2+ x/3)ⁿ are equal, then the value of n is
a) 15 b) 45 c) 55 d) 56
6) Let (5+ 2√6)ⁿ = p+ f where n ∈N and p ∈ N and 0< f < 1 then the value of, f² - f + pf - p is
a) a natural number
b) a negative integer
c) a prime number
d) are irrational number
7) Consider the triangle pictured as shown. If 0 < α< π/2 then the number of integral values of c is
a) 35 b) 23 c) 24 d) 25
8) The sum of infinity of the series 1/1+ 1/(1+2) + 1/(1+2+3) + .....is equal to
a) 2 b) 5/2 c) 3 d) none
9) In the expansion of {(x +1)/(x²⁾³ - x¹⁾³ +1) - (x -1)/(x - x¹⁾²)}¹⁰, the term which does not contain x is
a) ¹⁰C₀ b) ¹⁰C₇ c) ¹⁰C₄ d) none
10) In an acute angled triangle ABC, point D,E,F are the feet of the perpendiculars from A, B and C on to BC, AC and AB respectively. H is the intersection of AD and BE. If sinA= 3/5 and BC= 39, the length of AH is
a) 44 b) 48 c) 52 d) 54
11) A triangle has sides 6,7,8. The line through its incentre parallel to the shortest side is drawn to meet the other two sides at P and Q. The length of the segment PQ is
a) 12/5 b) 15/4 c) 30/7 d) 33/9
12) In the expansion of (1+ x + x²+ ......+ x²⁷)(1+ x + x² +....x¹⁴)², the coefficient of x²⁸ is
a) 195 b) 224 c) 378 d) 405
13) Triangle ABC has BC= 1 and AC= 2. The maximum possible value of the angle A is
a) π/6 b) π/4 c) π/3 d)π/2
14) If the constant term of the binomial expansion (2x - 1/x)ⁿ is -160, then n is
a) 4 b) 6 c) 8 d) 10
15) Along a road lies and odd number of stones placed at interval of 10m. These stone have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stone in succession, thereby covering a distance of 4.8 km. Then the number of stones is
a) 15 b) 29 c) 31 d) 35
16) Triangle ABC is right angled at A. The points P and Q are on the hypotenuse BC such that BP= PQ= QC. If AP= 3 and AQ= 4 then the length BC is equal to
a) √27 b) √36 c) √45 d) √54
17) If (1+ x - 3x²)²¹⁴⁵ = a₀ + a₁x + a₂x² + ..... then a₀ - a₁ + a₂ - a₃ + .....ends with
a) 1 b) 3 c) 7 d) 9
18) In the expansion of {√(p/q) + ¹⁰√(p⁷/q³)}ⁿ, there is a term similar to pq, then that term is equal to
a) 210pq b) 252pq c) 120pq d) 45pq
19) If S= 1² + 3² + 5² + ....(99)² then the value of the sum 2²+ 4²+ 6² + ....+ (100)² is
a) S+ 2550 b) 2S c) 4S d) S+ 5050
20) The coefficient of x⁴⁹ in the expansion of (x -1)(x - 1/2)(x - 1/2²) .......(x - 1/2⁴⁹) is equal to
a) -2(1- 1/2⁵⁰) b) +ve coefficient of xbc) -ve coefficient of x d) -2(1- 1/2⁴⁹)
21) In an isosceles triangle ABC, AB= AC, angrBAC= 108° and AD trisects angle BAC and BD> DC. The ratio BD/DC is
a) 3/2 b) (√5+1)/2 c) √5-1 d) 2
22) The sum S= ²⁰C₂ + 2. ²⁰C₃ + 3. ²⁰C₄ + .....+ 19.²⁰C₂₀ is equal to
a) 1+ 5.2²⁰ b) 1+ 2²¹ c) 1+ 9.2²⁰ d) 2²⁰
23) In an AP with first term a and common difference d (a,d≠0) , the ratio 'ρ' of the sum of the first n terms to sum of n terms succeeding them does not depend on n. Then the ratio 'ρ', respectively are
a) 1/2,1/4 b) 2,1/3 c) 1/2,1/3 d) 1/2,2
24) In ∆ ABC if a= 8, b= 9, c= 10, Then the value of tanC/sinB is
a) 32/9 b) 24/7 c) 21/4 d) 18/5
25) The number of values of r satisfying the equation ³⁹C₃ᵣ₋₁ - ³⁹Cᵣ² = ³⁹Cᵣ²₋₁ - ³⁹C₃ᵣ is
a) 1 b) 2 c) 3 d) 4
26) In a triangle abcr, CD is the bisector of the angle C. If cos(C/2) has the value 1/3 and l(CD)= 6, then (1/a + 1/b) has the value equal to
a) 1/9 b) 1/12 c) 1/6 d) none
27) Let a,b,c be the three sides of a triangle then the equation b²x²+ (b²+ c²- a²)+ c²= 0 has
a) both imaginary roots
b) both positive and one negative roots
c) both negative roots
d) one positive and one negative roots.
28) If 6⁸³ + 8⁸³ is divided by 49, then the remainder is
a) 35 b) 5 c) 1 d) 0
29) The term independent of x in the expansion of+9x - 1/3√x)¹⁸, x> 0, is α times the corresponding binomial coefficient. Then α is
a) 3 b) 1/3 c) -1/3 d) 1
30) The arithmetic mean of the nine numbers in the given set {9, 98, 999, .....999999999) is a 9 digit number N, all whose digits are distinct. The number N does not contain the digit
a) 0 b) 2 c) 5 d) 9
31) With usual notations, in a triangle ABC, a cos(B- C)+ b cos(C - A)+ c cos(A - B) is equal to
a) abc/R² b) abc/4R² c) 4abc/R² d) abc/2R².
32) Greatest term in the binomial expansion of (a + 2x)⁹ when a= 1 & x= 1/3 is
a) 3ʳᵈ & 4ᵗʰ b) 4ᵗʰ & 5ᵗʰ c) only 4ᵗʰ d) only 5ᵗʰ
33) The sum of the binomial coefficients of (2x + 1/x)ⁿ is equal to 256. The constant term in the expansion is
a) 1120 b) 2110 c) 1210 d) none
34) If for an AP a₁ , a₂, a₃, .....aₙ, .....a₁ + a₃ + a₅ = -12 and a₁a₂a₃= 8, then the value of a₂ + a₄ + a₆ equal to
a) -12 b) -16 c) -18 d) -21
35) With usual notations in a triangle ABC, (II₁). (II₂). (II₃) has the value equal to
a) R²r b) 2R²r c) 4R²r d) 16R²r
36) Given (1- 2x + 5x² - 10x³)ⁿ= 1+ a₁x + a₂x² +.... and that a₁² = 2a₂ then the value of n is
a) 6 b) 2 c) 5 d) 3
37) A sector OABO of central angle θ is constructed in a circle with centre O and of radius 6. The radius of the circle that is circumscribed about the triangle OAB, is
a) 6 cos(θ/2) b) 6 sec(θ/2) c) 3(cos(θ/2+2) d) 3 sec(θ/2)
38) The expansion of (1+ x)ⁿ has 3 consecutive terms with coefficients in the ratio 1:2:3 and can be written in the form ⁿCₖ : ⁿCₖ₊₁: ⁿCₖ₊₂, the sum of all possible values of (n + k) is
a) 18 b) 21 c) 28 d) 32
39) Number of rational terms in the expansion of (√2+ ⁴√3)¹⁰⁰ is
a) 25 b) 26 c) 27 d) 28
40) Let a≤ b ≤ c be the length of the sides of a triangle T. If a² + b²< c² then which one of the following must be true?
a) All 3 angles of T are acute.
b) some angle of T is obtuse
c) one angle of T is a right angle.
d) no such triangle can exist.
41) The coefficient of the middle term in the binomial expansion in powers of x of (1+ αx)⁴ and of (1- αx)⁶ is the same if α equals
a) -5/3 b) 10/2 c) -3/10 d) 3/5
42) ³⁶⁰ₖ₌₁∑[1/{k √(k+1)+ (k+1)√k} is the ratio of two relative prime positive integers m and n. The value of (m + n) is equal to
a) 43 b) 41 c) 39 d) 37
43) If x∈ R, the numbers (5¹⁺ˣ + 5¹⁻ˣ), a/2, (25ˣ + 25⁻ˣ) form an AP. Then 'a' must lie in the interval
a) [1,5] b) [2,5] c) [5,12] d) [12,∞]
44) Id C₀, C₁, C₂....denotes the combinatiorial coefficients in the expansion of (1+ x)¹⁰, then the value of C₀/. + C₁/2 + C₂/3 + ......+C₁₀/11 is equal to
a) 2¹¹/11 b) (2¹¹ -1)/11 c) 3¹¹/11 d) (3¹¹-1)/11
45) Let triangle ABC be an isosceles triangle with AB= AC. Suppose that the angle bisectors of its angle B meets the side AC at a point D and that BC= BD+ AD. Measure of the angle A in degree, is
a) 80 b) 100 c) 110 d) 130
46) If the sum of the first 11 terms of an arithmetical progression equals that of the first 19 terms, then the sum of its 30 terms is
a) equals to 0 b) equals to -1 c) equal to 1 d) non unique
47) The remainder, when (15²³ + 23²³) is divided by 19, is
a) 4 b) 15 c) 0 d) 18
48) In a ∆ ABC, the value of (a cosA + b cosB + c cosC)/(a+ b+ c) is equal to
a) r/R b) R/2r c) R/r d) 2r/R
49) With usual notations in a ∆ ABC, if R= k(r₁ + r₂)+ +r₂+ r₃)(r₃ + r₁)/(r₁r₂ + r₂r₃ + r₃r₁) where k has the value equal to
a) 1 b) 2 c) 1/4 d) 4
50) If n ∈N and n is even, then 1/1.(n -1)! + 1/3!(n -3)! + 1/5!(n -5)!+......+ 1/(n -1)!1!=
a) 2ⁿ b) 2ⁿ⁻¹/n! c) 2ⁿn! d) none
51) The remainder, if 1+2+2²+ 2³ + ....+2¹⁹⁹⁹ is divided by 5 is
a) 0 b) 1 c) 2 d) 3
52) Let s₁, s₂, s₃ ..... and t₁, t₂, t₃....are two arithmetic sequence such that s₁ = t₁ ≠ 0; s₂ = 2t₂ and ¹⁰ᵢ₌₁∑ sᵢ = ¹⁵ᵢ₌₁∑ tᵢ. Then the value of (s₂ - s₁)/(t₂ - t₁) is
a) 8/3 b) 3/2 c) 19/8 d) 2
53) In the expansion of (3⁻ˣ⁾⁴ + 3⁵ˣ⁾⁴)ⁿ the sum of the binomial coefficients is 64 and the term with the greatest binomial coefficients exceeds the third term by (n -1), then the value of x must be
a) 1 b) 2 c) 0 d) -1
54) If the incircle of the ∆ ABC touches its sides respectively at L, M, N and if x,y,z be the circumradius of the triangles MN, NIL, LIM where I is the incentre then the product xyz is equal to:
a) Rr² b) rR² c) (1/2) Rr² d) (1/2) rR²
55) Sum of all the rational terms in the expansion of (3¹⁾⁴ + 4¹⁾³)¹², is
a) 27 b) 256 c) 283 d) none
56) ABC is an acute angled triangle with circumcentre O orthocentre H. If AO= AH then the measure of the angle A is
a) π/6 b) π/4 c) π/3 d) 4π/12
57) Let L and M be the respective intersection of the internal and external angle bisectors of the triangle ABC at C and the side AB produced. If CL= CM, then the value of (a²+ b²) is (where a and b have their usual meanings)
a) 2R² b) 2√2R² c) 4R² d) 4√2R².
58) α, β, γ, δ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k. The value of 4 sin(α/2)+ 3 sin(β/2)+ 2 sin(γ/2)+ sin(δ/2) is equal to
a) 2√(1- k) b) 2√(1+ k) c) 2√k d) 2k
59) Coefficient of αᵗ in the expansion of, (α+ p)ᵐ⁻¹ + (α + p)ᵐ⁻² (α+ q)+ (α+ p)ᵐ⁻³ (α+ q)²+ .....(α+ q)ᵐ⁻¹, where α= - q and p≠ s is
a) ᵐCₜ(pᵗ - qᵗ)/(p - q)
b) ᵐCₜ(pᵐ⁻ᵗ - qᵐ⁻ᵗ)/(p- q)
c) ᵐCₜ(pᵗ + qᵗ)/(p - q)
d) ᵐCₜ(oᵐ⁻ᵗ + qᵐ⁻ᵗ)/(p - q).
60 Let aₙ, n ∈N is an AP with common difference d and all whose terms are nonzero. If n approaches infinity, then the sum 1/a₁a₂ + 1/a₂a₃ + .....+ 1/aₙaₙ₊₁ will approach
a) 1/a₁d b) 2/a₁d c) 1/2a₁d d) a₁d
61) If (11)²⁷ + (21)²⁷ when divided by 16 leaves the remainder
a) 0 b) 1 c) 2 d) 14
62) In a ∆ ABC if a+ b + c= 3a then cot(B/2). cot(C/2) has the value equal to
a) 4 b) 3 c) 2 d) 1
63) The last two digits of the number 3⁴⁰⁰ are:
a) 81 b) 43 c) 29 d) 01
64) Let f,g,h be the lengths of the perpendiculars from the circumcentre of the ∆ ABC on the sides a, b and c respectively. If a/f+ b/g + c/h = λ (abc)/(fhh) then the value of λ is
a) 1/4 b) 1/2 c) 1 d) 2
65) If (1+ x + x²)²⁵ = a₀ + a₁x + a₂x²+.....a₅₀x⁵⁰ then a₀+ a₂+ a₄ +....+ a₅₀ is
a) even b) odd and of the form 3n c) odd and of the form (3n -1) d) odd and of the form (3n +1)
66) The largest real value for x such that ⁴ₖ₌₀∑ {5 ⁴⁻ᵏ/(4- k)! (xᵏ/k!)}= 8/3 is
a) 2√2-5 b) 2√2+ 5 c) -2√2- 5 d) - 2√2+ 5
67) The sum of the first three terms of an increasing GP is 21 and the sum of their squares is 189. Then the sum of its first n terms is
a) 3(2ⁿ-1) b) 12(1- 1/2ⁿ) c) 6(1- 1/2ⁿ) d) 6(2ⁿ-1)
68) In a ∆ abc if b= a(√3-1) and angle C=30° then the measure of the angle A is
a) 15 b) 45 c) 75 d) 105
69) In a ∆ ABC, a= a₁ , b= a₂, c= a₃ such that aₚ₊₁ = 5ᵖ/3²⁻ᵖ . aₚ+2²⁻ᵖ - (4p -2)/5ᵖ. aₚ) where p= 1,2 then
a) r₁= r₂ b) r₃ = 2r₁ c) r₂= 2r₁ d) r₂ = 3r₁
70) (2n +1)(2n +3)(2n +5).....(4n -1) is equal to
a) (4n)!/(2ⁿ. (2n)!(2n)!)
b) ((4n)!n!)/(2ⁿ. (2n)!(2n)!)
c) ((4n)!n!)/((2n)!(2n)!)
d) ((4n)!n!)/(2ⁿ. (2n)!)
71) The sum of the roots (real or complex) of the equation x²⁰⁰¹ + (1/2 - x)²⁰⁰¹= 0 is
a) 2000 b) 2001 c) 1000 d) 500
72) If 'O' is the circumcentre of the ∆ ABC and R₁, R₂ and R₃ are the radii of the circumcircle of triangles OBC, OCA, OAB respectively then a/R₁ + b/R₂ + c/R₃ has the value equal to
a) abc/2R³ b) R³/abc c) 4∆/R² d) ∆/4R².
73) The sum of the series (1²+1)!+ (2²+1)2! +(3²+1).3! + ....(n²+1).n! I
is
a) (n+1)(n +2)! b) n(n+1)! c) (n+1)(n+1)! d) none
74) The expression 1/√(4x +1) [[{1+√(4x+1)}/2]⁷ - [{1- √(4x+1)}/2]⁷ is a polynomial in x of degree
a) 7 b) 6 c) 4 d) 3
75) The sum ∞ₙ₌₁∑{n/(n⁴+4)} is equal to
a) 1/41/33/81/2
76) If a≠ 1 and ln a² + (ln a²)²+ (ln a²)³+......= 3(ln a + (ln a)² + (ln a)³+ (ln a)⁴+...) then 'a' is equal to
a) e¹⁾⁵ b) √e c) ³√e d) ⁴√e
77) The median of a ∆ ABC are 9cm, 12cm, and 15cm respectively. Then the area of the traingle
a) 96 sq cm b) 84 sq cm c) 72 square cm d) 60 square cm
78) If the second term of the expansion [a¹⁾¹³ + a/√a⁻¹]ⁿ is 14a⁵⁾² then the value of ⁿC₃/ⁿC₂ is
a) 4 b) 3 c) 12 d) 6
79) If r₁, r₂, r₃ be the radii of excircles of the triangle ABC, then ∑r₁/√∑r₁r₂ is equal to
a) ∑cot(A/2) b) ∑cot(A/2 cot(B/2) c) ∑ tan(A/2) d) Π tan(A/2)
80) The value of (4. ⁿC₁ + 4². ⁿC₂ + 4³. ⁿC₃ +.....4ⁿ) is
a) 0 b) 5ⁿ +1 c) 5ⁿ d) 5ⁿ - 1
81) 1/(2.4) + 1.3/(2.4.6) + (1.3.5)/(2.4.6.8) + (1.3.5.7)/(2.4.6.8.10)+.......∞ is equal to
a) 1/4 b) 1/3 c) 1/2 d) 1
82) The sum of the coefficients of all the even powers of x in the expansion of (2x²- 3x +1)¹¹ is
a) 2.6¹⁰ b) 3.6¹⁰ c) 6¹¹ d) none
83) The sum ¹⁰⁰ₖ₌₁∑ k/(k⁴ + k² +1) is equal to
a) 4950/10101 b) 5050/10101 c) 5151/10101 d) none
84) If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then abc/xyz is equal to
a) Π tan(A/2) b) ∑ cot(A/2) c) ∑ tan(A/2) d) ∑sin(A/2)
85) Last three digits of the number N= 7¹⁰⁰ - 3¹⁰⁰ are
a) 100 b) 300 c) 500 d) 000
86) A circle of radius r is inscribed in a square. The midpoint of sides of the squares have been connected by line segment and a new square resulted. The sides of the resulting square were also connected by segments so that a new square was obtained and so on, then the radius of the circle inscribed in the nth square is
a) (2⁽¹⁻ⁿ⁾/²)r
b) (2⁽³⁻³ⁿ⁾/²)r
c) (2⁻ⁿ⁾²)r
d) (2⁽⁵⁻³ⁿ⁾/²)r
87) The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equals to{where ∆ is the area of the triangle ABC)
a) ∆ b) 2∆ c) 3∆ d) 4∆
88) In a triangle ABC, angle ABC=120, AB=3 and BC= 4. If perpendicular constructed on the side AB at A and to the side BC at C meets at D then CD is equal to
a) 3 b) 8√3/3 c) 5 d) 10√3
89) If abcd= 1 where a,b,c,d are positive reals then the minimum value of a² + b² + c² + d² + ab+ ac+ ad+ bc+ bd+ cd is
a) 6 b) 10 c) 12 d) 20
90) A triangle has base 10cm long and the base angle of 50° and 70°. If the perimeter of the triangle is x+ y cos x° where z ∈(0,90) then the value of x+ y+ z equals
a) 60 b) 55 c) 50 d) 40
91) The positive value of a so that the coefficient of x⁵ is equal to that in the x¹⁵ in the expansion of (x² + a/x³)¹⁰ is
a) 1/2√3 b) 1/√3 c) 1 d) 2√5
92) A sequence of equilateral triangle is drawn . The altitude of each is √3 times the altitude of the proceeding triangle, the difference between the area of the first triangle and the sixth triangle is 968√3 square unit. The perimeter of the first triangle is
a) 10 b) 12 c) 16 d) 18
93) Let ABC be a triangle with angle BAC=2π/3 and AB= x such that (AB)(AC)=1. If x varies then the longest possible lengths of the angle bisector AD equal to
a) 1/31/22/33/2
94) If a, b, c are three consecutive positive terms of a GP, then the graph of y= ax²+ bx + c is
a) A curve that intersect the x-axis at two distinct points.
b) entirely below the x-axis.
c) entirely above the x-axis.
d) tangent to the x-axis.
95) Set of value of r for which, ¹⁸Cᵣ₋₂ + 2. ¹⁸Cᵣ₋₁+ ¹⁸Cᵣ ≥ ²⁰C₁₃ contains:
a) 4 elements b) 5 elements c) 7 elements d) 10 elements
96) For which positive integers n is the ratio ⁿₖ₌₁₌∑k²/ ⁿₖ₌₁∑k an integer
a) odd n only b) even n only c) n= 1+ 6k only, where k≥ 0 and k∈I d) n= 1+ 3k, integer k≥ 0
Comprehensive type paragraph question number 97 to 99 analtitude and bisector are drawn in the triangle from the vertex it is known that the length of the side in the magnitude of the angles form an arithmetic progression the area of the circle let be the circumcenter of the radius of the circle inscribed in let me the image of the point to respect to side then the length is equal paragraph for question number 100 to 100 to consider the binomial expansion
ⁿ ᵗʰᵗʰₘᵣᵖₘ₌₁Πₘₙⁿᵣ₌₁∑ᵣₙ∞ᵣ₌₅ᵣ⁻¹⁻¹⁻¹⁻¹₁₂₃₁₂₃₁₁₁₀₀₁₀₀ₙₙ₁₁₂₂₃₃¹⁰⁰ᵣ₌₁ᵣᵣ³⁻ˡᵒ¹¹²²⁻³³⁻³ˡᵒᵍ⁵⁶⁻⁵⁾²₂₃₁²²²²₁²²ⁿ₀₁₂²₀₁₂ⁿ²∑Π²²²²ⁿⁿ₁₂²₃₄²₁₂₃₄₁₃₂₄
σ ∈
αβ ₁₂₃³² αβ ∞ ⁹⁹ ₙ₌₁∑¹⁰⁰ₙⁿ¹⁰⁰ ˢᶦⁿˣ ⁻¹ ⁴ ⁻ ²ˢᶦⁿˣ. ₁₂₃₉₈ₙ₊₁ₙ⁴⁹ₖ₌₁ ∑₂ₖⁿⁿᵗʰ₁ ᵣᵣ₋₁ʳ ꜀ᵥₐ ₙ₁ₙ₊₁ₙ∞ₙ₌₁∑ₙₙ
₂₄₁₆₄αβγδαβγδ₁₂ₙₙₙ₁₂₃ₙ₁₂ₙ₁₂ₙ₁₂ₙ²²²₁₂₁₂₁₂₁₂₁₂₁₂₁₂²²²²⁷⁷⁴⁴⁴⁴αβαβαβαβⁿ⁺¹ᵗʰⁿ⁺¹⁻ᵏₙⁿ⁻¹ⁿₙₙ₀ₙₙ₀ᵣᵣᵣ₊₁ᵣᵣᵣ₊₁ᵣ₁₂ₙᵣ₁₂₃₁₂₃₁₂₃₁₂₃₁₁₁ₙ₋₁ₙ₋₁ₙₙₙ₁₂₃₁₂₃₁₂₃₁₃₅₂₄₆₁₂₃₁₂₃₁₃₅₂₄₆₁₃₅₂₄₆₁₂₃₁₂₃₁₂₃₁₃₅₂₄₆₁₃₅₂₄₆ ₁₂₃₄₁₁₂₁₂₃₂₃₄₃₄₁₂₃₄₁₂₃₄₁₂₃₁₁₁₂ₖₖ₋₁ₖ₋₂₁₂₁₁₁₂₁₁⁻⁵⁻⁴⁻³⁸¹⁰₁₂₃₁₀₀₁ₚᵖᵢ₌₀ᵢₘₙ₂
COORDINATE GEOMETRY
Equation of Straight line
Sap-1
1) The number of points on x-axis which are at a distance c(c< 3) from the point (2,3) is
a) 2 b) 1 c) infinite d) no point d
2) The distance between the points P(a cosα, a sinα) and Q(a cosβ, a sinβ) is
a) 4a sin{(α-β)/2} b) 2a sin{(α + β)/2} c) 2a sin{(α-β)/2} d) 2a cos{(α-β)/2}. c
3) Determine the ratio in which y - x + 2 divides the line joining (3,-1) and (8,9). 2:3
4) If (1,4) is the centroid of a triangle and its two vertices are (4,-3) and (-9,7) then third vertices is
a) (7,8) b) (8,8) c) (8,7) d) (6,8). b
5) The vertices of a triangle are A(0.-6), B(-6,0) and C(1,1), respectively, then coordinates of the excentre opposite to vertex A is.
a) (-3/2,-3/2) b) (-4,3/2) c) (-3/2,3/2) d) (-4,6). d
a) (-3/2,-3/2) b) (-4,3/2) c) (-3/2,3/2) d) (-4,6). d
6) If the vertices of a triangle are (1,2),(4,-6) and (3,5) then the area is
a) 25/2 b) 12 c) 5 d) 25. a
7) The point A divides the join of the points (-5.1) and (3,5) in the ratio k: 1 and coordinates of points B and C are (1,5) and (7,-2) respectively. If the area of ∆ ABC be 2 units, then k equals to
a) (7,9) b) (6,7) c) 7,31/9 d) 9,31/9. c
8) Show that the coordinates of the vertices of an equilateral triangle can not be rational.
9) The ends of the rod of length l moves on two mutually perpendicular lines, find the locus of the point on the rod which divides it in the ratio m₁: m₂
a) m₁²x²+ m₂²y²= l²/(m₁ + m₂)²
b) (m₂x)²+ (m₁y)²= {(m₁m₂l)/(m₁ + m₂)}²
c) (m₁x)²+ (m₂y)²= {(m₁m₂l)/(m₁ + m₂)}²
d) none. C
10) If A(a,0) and B(-a,0) are two fixed points of ∆ ABC. If its vertex C moves in such way that cotA + cotB= λ, where λ is a constant, then the locus of the point C is
a) yλ = 2a b) y= λa c) ya = 2λ d) none a
11) The equation of the lines which passes through the point (3,4) and the sum of its intercept on the axes is 14 is
a) 4x - 3y= 24, x - y= 7
b) 4x + 3y= 24, x + y= 7
c) 4x + 3y=- 24, x + y=- 7
d) 4x - 3y= -24, x - y=- 7. b
12) Two points A and B move on the positive direction of x-axis and y-axis respectively, such that OA+ OB= K. Show that the locus of the foot of the perpendicular from the origin O on the line AB is (x + y)(x²+ y²)= Kxy.
13) Find the equation of the straight line on which the perpendicular from origin makes an angle 30° with x-axis and which forms a triangle of area (50/√3) square. units with the coordinates axes. x√3+ y= 10
14) Equation of a line which passes through point A(2,3) and makes an angle of 45° with x-axis. If this line meet the line x+ y+1=0 at point P then distance AP is
a) 2√3 b) 3√2 c) 5√2 d) 2√5. b
15) A variable line is drawn through O, to cut two fixed straight lines L₁ and L₂ in A₁ and A₂ respectively. A point A is taken on the variable line such that (m+ n)/OA = m/OA₁ + n/OA₂.
Show that the locus of A is a straight line passing through the point of intersection of L₁ and L₂ where O is being the origin.
16) A straight line through P(-2,-3) cuts the pair of straight line x²+ 3y²+4xy - 8x - 6y - 9= 0 in Q and R. Find the equation of the line if PQ. PE = 20. 3x - y + 3=0
17) If the line y - √3 x +3=0 cuts the parabola y²= x + 2 at A and B, then find the value of PA. PB (where P=(√3,0). N 4(2+√3)/3
18) If x + 4y -5=0 and 4x + ky +7=0 are two perpendicular lines then k is
a) 3 b) 4 c) -1 d) -4. C
19) A line L passes through the points (1,1) and (2,0) and another line M which is perpendicular to L passes through the point (1/2,0). The area of the triangle formed by these lines with y-axis is
a) 25/8 b) 25/16 c) 25/4 d) 25/32. b
20) If the straight line 3x + 4y+ 5 - k(x + y +3)= 0 is parallel to y-axis, then the value of k is
a) 1 b) 2 c) 3 d) 4
21) If the algebraic sum of perpendiculars from n given points on a variable straight line is zero then show that the variable straight line passes through a fixed point.
22) Show that no line can be drawn through the point (4,-5) so that its distance from (-2,3) will be equal to 12.
23) Three lines x+ 2y+3=0, x + y= 7, 2x - y= 4 form 3 sides of two squares. Find the equation of remaining sides of these squares. 2x - y= -6, 2x - y= 14.
24) Find the equation to the sides of an isosceles right angled triangle, the equation of whose hypotenuse is 3x + 4y= 4 and the opposite vertex is the point (2,2). -x +7y= 12, 7x + y= 16
25) Let P(sinθ, cosθ)(0≤θ≤2π) be a point and let OAB be a triangle with vertices (0,0), ((√3/2),0) and (0,√(3/2)). Find θ if P lies inside the ∆ OAB. 0<θ<π/12 or 5π/12< θ< 3π/4
26) Through what angles should the axes be rotated so that the equation 9x² - 2√3xy = 10 may be changed to, 3x² + 5y²= 5? 60°
27) For the straight lines 4x + 3y= 6, 5x +12y +9= 0, find the equation of the
a) bisector of the obtuse angle between them.
b) bisector of the acute angle between them.
c) bisector of the angle which contains origin. 9x - 7y= 41, 7x +9y= 3, 7x + 9y= 3
28) Show that each member of the family of straight lines
(3sinθ + 4 cosθ)x + (2 sinθ - 7 cosθ)y + (sinθ + 2 cosθ)= 0 (θ is a parameter) passes through a fixed point.
29) λx¹- 10xy + 12y²+ 5x - 16y -3=0 represents a pair of straight lines, then λ is equal to
a) 4 b) 3 c) 2 d) 1. c
30) Show that the two straight lines x¹(tan²θ+ cos²θ) - 2xy tanθ + y² sin²θ = 0 represented by the equation are such that the difference of their slopes is 2.
31) If pair of straight lines x¹- 2pxy - y²= 0 and, x² - 2qxy - y²= 0 be such that each pair bisects the angle between the other pair, show that pq= -1.
32) The chord √6y = √8 px + √2 of the curve py²+ 1= 4x subtends a right angle at origin then find the value of p. (-9±√33)/8.
Sap-2
1) If (3,-4) and (-6,5) are the extremities of the diagonal of a parallelogram and (-2,1) is its third vertex, then its fourth vertex is
a) (-1,0) b) (-1,1) c) (0,-1) d) none
2) The ratio in which the line joining the points (3,-4) and (-5,6) is divided by x-axis.
a) 2:3 b) 6:4 c) 3:2 d) none
3) The circumcentre of the triangle with vertices (0,0),(3,0),(0,4) is
a) (1,1) b) (2,3/2) c) (3/2,2) d) none
4) the midpoints of the sides of a triangle are (5,0),(5,12),(0,12) then orthocentre of this triangle is
a) (0,0) b) (0,24) c) (10,0) d) (13/3,8)
5) Area of a triangle whose vertices are (a cosθ, b sinθ),(-a sinθ, b cosθ),(- a cosθ, - b sinθ) is
a) ab sinθ cosθ b) a sinθ cosθ c) ab/2 d) ab
6) The point A divides the join of the points (-5,1) and (3,5) in the ratio k: 1 and coordinates of points B and C are (1,5),(7,-2) respectively. If the area of ∆ ABC be 2 units, then k equals
a) 7.9 b) 6,7 c) 7,31/9 d) 9,31/9
7) If A(cosα, sinα), B(sinα, - cosα), C(1,2) are the vertices of a ∆ ABC, then as α varies, the locus of its centroid is
a) x² + y² - 2x - 4y +3=0
b) x² + y² - 2x - 4y +1=0
c) 3(x² + y²) - 2x - 4y + 1=0 d) none
8) The points with the coordinates (2a, a), (3b,b) & (c,c) are collinear
a) for no value of a,b,c
b) for all values of a, b, c
c) if a, c/5, b are in HP
d) if a, 2c/5, b are in HP
9) A stick of length 10 units rests against the floor and a wall of a room. If the stick begins to slide on the floor then the locus of its middle point is
a) x²+ y²= 2.5 b) x²+ y²= 25 c) x²+ y²= 100 d) none
10) The equation of the line cutting an intercept of 3 on negative y-axis and inclined at an angle tan⁻¹(3/5) to the x-axis is
a) 5y - 3x +15= 0
b) 5y - 3x -15= 0
c) 3y - 5x +15= 0 d) none
11) The equation of a straight line which passes through the point (-3,5) such that the portion of it between the axes is divided by the point in the ratio 5:3 (reckoning from x-axis) will be
a) x+ y -2=0 b) 2x+ y +1=0 c) x+ 2y - 7=0 d) x - y + 8 =0
12) The coordinates of the vertices P, Q, R and S of square PQRS inscribed in the triangle ABC with vertices A≡ (0,0), B(3,0 and C≡ (2,1) given that two of its vertices P, Q are on the side AB are respectively:
a) (1/4,0),(3/8,0),(3/8,1/8),(1/4,1/8)
b) (1/2,0),(3/4,0),(3/4,1/4),(1/2,1/4)
c) (1,0),(3/2,0),(3/2,1/2),(1,1/2)
d) (3/2,0),(9/4,0),(9/4, 3/4),(3/2, 3/4)
13) The equation of perpendicular bisector of the line segment joining the points (1,2) and (-2,0) is
a) 5x + 2y= 1 b) 4x + 6y= 1 c) 6x + 4y= 1 d) none
14) The number of possible straight lines, passing through (2,3) and forming a triangle with coordinates axes, whose area is 12 square.units, is
a) one b) two c) three d) four
15) Points A and B are in the first quadrant; point O is the origin. If the slope of OA is 1, slope of OB is 7 and OA= OB, then the slope of AB is
a) -1/5 b) -1/4 c) -1/3 d) -1/2
16) Coordinates of a point which is at 3 distance from point (1,-3) of the line 2x + 3y= -7 is
a) (1+ 9/√13, 3- 6/√13)
b) (1- 9/√13, - 3+ 6/√13)
c) (1+ 9/√13, -3+ 6/√13)
d) (1- 9/√13, 3- 6/√13)
17) The angle between the lines y- x +5=0 and √3 x - y +7=0
a) 15° b) 60° c) 45° d) 75°
18) A line is perpendicular to 3x + y= 3 and passes through a point (2,2). Its y intercept is
a) 2/3 b) 1/3 c) 1 d) 4/3
19) The equation of the line passing through the point (c,d) and parallel to the line ax+ by + c=0 is
a) a(x + c)+ b(y + d)= 0
b) a(x + c)- b(y + d)= 0
c) a(x +p- c)+ b(y - d)= 0 d) none
20) The position of the point (8,-9) with respect to the lines 2x + 3y-4=0 and 6x + 9y + 8 =0 is
a) point lies on the same side of the lines.
b) point lies on one of the lines.
c) point lies on the different sides of the line.
d) none
21) If origin and (3,2) are contained in the same angle of the lines 2x + y- a=0 , x - 3y+ a=0 , then 'a' must lie in the interval
a) (-∞,0) U(8,∞) b) (-∞,0) U(3,∞) c) (0,3) d) (3,8)
22) The line 2x + 2y-6 =0 will divide the quadrilateral formed by the lines x + y- 5=0 , - 2x + y- 8=0 , 2x + 3y=0 and y- x =0
a) two quadrilateral
b) one pentagon and one triangle
c) two triangle d) none
23) If the point (a,2) lies between the lines x - y- 1=0 and 2(x -y) - 5 =0 , then the set of values of a is
a) (-∞,3) U(9/2,∞) b) (3,9/2) c) -∞,3) d) (9/2,∞)
24) A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are three non collinear points in cartesian plane. Number of paralellograms that can be drawn with these three points as vertices are
a) one b) two c) three d) four
25) If P=(1,0); Q= (-1,0) and R= (2,0) are three given points, then the locus of the points S satisfying the relation, SQ²+ SR²= 2 SP¹ is
a) a straight line parallel to x-axis
b) a circle passing through the origin
c) a circle with the centre at the origin
d) a straight line parallel to y-axis.
26) The area of triangle formed by the lines x + y- 3 =0, x - 3y +9 =0 and 3x - 2y + 1=0
a) 16/7 b) 10/7 c) 49
27) The coordinates of foot of the perpendicular drawn on the line 3x - 4y -5=0 from the point (0,5) is
a) (1,3) b) (2,3) c) (3,2)) d) (3,1)
28) Distance of the point (2,5) from the line 3x + y +4 =0 measured parallel to the line 3x - 4y +8 =0 is
) 15/2 b) 9/2 c) 5 d) none
29) Three vertices of triangle ABC are A(-1,11), B(-9,-8), C(15,-2). The equation of angle bisectors of angle A is
a) 4x - y- 7=0 b) 4x + y - 7 =0 c) x + 4y- 7=0 d) x - 4y- 7=0
30) If line y - x +2 =0 is shifted parallel to itself towards the positive direction of the x-axis by a perpendicular distance of 3√2 units, then the equation of the new line is
a) y= x -4 b)!y= x +1 c) y= x -(2+ 3√2) d) y= x -8
31) The coordinates of the point of reflection of the origin (0,0) in the line 4x - 2y- 5 =0 is
a) (1,-2) b) (2,-1) c) ) (4/5,-2/5) d) (2,5)
32) If the axes are rotated through an angle of 30° in the anticlockwise direction, the coordinates of point (4,-2√3) with respect to new axes are
a) (2,√3) b) (√3,-5) c) (2,3) d) (√3,2)
33) Keeping the origin constant axes are rotated at an angle 30° in clockwise direction then the new coordinates of (2,1) with respect to old axes is
a) ((2+√3)/2, √3/2)
b) ((2√3+1)/2, (-2+ √3)/2)
c) ((2√3+1)/2, (2- √3)/2) d) none
34) If one diagonal of a square is along the line x= 2y and one of its vertex is (3,0), then its sides through this vertex are given by the equations
a) y- 3x +9=0, x - 3y -3=0
b) y+ 3x +9=0, x - 3y -3=0
c) y +3x -9=0, x +3y -3=0
d) y- 3x +9=0, x + 3y -3=0
35) The line (p+ 2q)x + (p- 3q) = p - q for different values of p and q passes through a fixed point whose coordinates are
a) (3/2,5/2) b) (2/5,2/5) c) (3/5,3/5) d) (2/5,3/5)
36) Given the family of lines, a(3x +4y +6)+ b(x + y +2)=0, The line of the family situated at the greatest distance from the point P(2,3) has equation
a) 4x +3y+ 8= 0 b) 5x +3y+ 10= 0 c) 15x +8y+ 30= 0 d) none
37) The base BC of a triangle ABC is bisected at the point (p,q) and the equation to the side AB & AC are px + qy = 1 & qx + py= 1. The equation of the median through A is
a) (p - 2q)x +(q - 2p) y+ 1= 0
b) (p +q)x +(q - 2p) y -2 = 0
c) (2pq - 1)(px + qy -1)= (p²+ q²-1)(qx + py -1) d) none
38) The equation 2x⅖+ 4xy - py²+ 4x + qy+1= 0 will represent two mutually perpendicular straight lines, if
a) p= 1 and q=2 or 6
b) p= -2 and q=-2 or 8
c) p= 2 and q= 0 or 8
d) p= 2 and q=0 or 6
39) Equation of the pair of straight lines through origin and perpendicular to the pair of straight lines 5x²- 7xy - 3y²= 0 is
a) 5x²- 7xy - 5y²= 0
b) 3x²+ 7xy +5y²= 0
c) 2x²- 7xy - 5y²= 0
d) 3x²+ 7xy - 5y²= 0
40) If the straight line joining the origin and the points of intersection of the curve 5x² + 11xy - 6y²+ 4x - 2y +3 = 0 and x + ky -1= 0 are equally inclined to the coordinate axes then the value of k
a) is equal to 1
b) is equal to -1
c) is equal to 2
d) does not exist in the set of real numbers
1a 2a 3c 4a 5d 6c 7c 8d 9b 10a 11d 12d 13c 14c 15d 16b 17a 18d 19c 20a 21a 22a 23b 24c 25d 26b 27d 28c 29b 30d 31b 32b 33b 34d 35d 36a 37c 38c 39a 40b
SAP- 3
1) If the lines x sin²A + y sinA +1= 0
x sin²B y sinB +1=0
x sin²C + y sin²C +1= 0 are concurrent where A, B, C are angles of triangle then ∆ ABC must be
a) equilateral b) isosceles c) right angle d) no such triangle exist
2) The coordinates of a point on the line 2x - y+5=0 such that |PA - PB| is maximum where A is (4,-2) and B is (2,-4) will be
a) (11, 27) b) (-11,- 17) c) (- 11, 17) d) (0,5)
3) The line x+ y = p meets the axis of x and y at A and B respectively. A triangle APQ is inscribed in the triangle OAB , O being the origin, with right angle at Q. P and Q lie respectively on OB and AB. if the area of the triangle APQ is 3/8ᵗʰ of the area of the triangle OAB, then AQ/BQ is equal to
a) 2 b) 2/3 c) 1/3 d) 3
4) Lines L₁: x + √3 y = 2, and L₂: ax + by = 1, meet at P and enclose an angle of 45° between them, Line L₃: y= √3 x also passes through P then
a) a²+ b²= 1 b) a²+ b²= 2 c) a²+ b²= 3 d) a²+ b²= 4
5) A triangle is formed by the lines 2x - 3y -6=0, 3x - y +3 =0 and 3x +4y - 12=0. if the points P(α,0) and Q(0,β) always lie on or inside the ∆ ABC, then
a) α∈ [1,2] & β∈ [-2,3]
b) α∈ [-1,3] & β∈ [-2,4]
c) α∈ [-2,4] & β∈ [-3,4]
d) α∈ [-1,3] & β∈ [-2,3]
6) The line x+ 3y -2= 0 bisects the angle between a pair of straight lines of which one has equation x. - 7y +5=0. The equation of the other line is
a) 3x+ 3y - 1= 0 b) x- 3y +2= 0 c) 5x+ 5y -3= 0 d) none
7) S ray of light passing through the point A(1,2) is reflected at a point B on the x-axis and then passes through (5,3). Then the equation of AB is
a) 5x+ 4y - 13= 0 b) 5x- 4y +3= 0 c) 4x+ 5y - 14= 0 d) 4x - 5y + 6= 0
8) Let the algebraic sum of the perpendicular distance from the points (3,0), (0,3) and (2,2) to a variable straight line be zero, then the line passes through a fixed point whose coordinates are
a) (3,2) b) (2,3) c) (3/5,3/5) d) (5/3, 5/3)
9) The image of the pair of line represented by ax² + 2hxy + by²= 0 by the line mirror y= 0 is
a) ax² - 2hxy + by²= 0
b) bx² - 2hxy + ay²= 0
c) bx² + 2hxy + ay²= 0
d) ax² - 2hxy- by²= 0
10) The pair of straight lines x² - 4xy + y²= 0 together with the line x+ y + 4√6=0 form a triangle which is
a) right angled but not isoscr
b) right isosceles
c) scalene d) equilateral
11) Let A(3,2) and B(5,1). ABP is an equilateral triangle is constructed on the side AB remote from the origin then the orthocentre of triangle ABP is
a) (4- √3/2, 3/2 -√3)
b) (4+ √3/2, 3/2 +√3)
c) (4- √3/6, 3/2 √3/3)
d) (4+ √3/6, 3/2 +√3/3)
12) The line PQ whose equation is x - y= 2 cuts the x-axis at P and Q is (4,2). The line PQ is rotated about P through 45° in the anticlockwise direction. The equation of the line PQ in the new position is
a) y= -√2 b) y= 2 c) x = 2 d) x = -2
13) Distance between two lines represented by the line pair, x²- 4xy + 4y²+ x - 2y - 6= 0 is
a) 1/√5 b) √5 c) 2√5 d) none
14) The circumcenter of the triangle formed by the lines, xy+ 2x + 2y +4=0 and x+ y+2=0 is
a) (-1,-1) b) (-2,-2) c) (0,0) d) (-1,-2)
15) Area of the Rhombus bounded by the four lines, ax ± by ± c=0 is
a) c²/2ab b) 2c²/ab c) 4c²/ab d) ab /4c²
16) if the lines ax + y +1=0, x + by +1=0 & x + y + c= 0 where a, b, c are distinct real numbers different from 1 are concurrent, then the value of 1/(1- a) + 1/(1- b) + 1/(1- c)=
a) 4 b) 3 c) 2 d) 1
17) If one vertex of an equilateral triangle of side a lies at the origin and the other lies on the line x - √3 y=0 then the coordinates of the third vertex are
a) (0,a) b) (√3a/2, - a/2) c) (0,-a) d) -√3a/2, a/2)
18) The area enclosed by 2|x|+ 3|y|≤ 6 is
a) 3 sq.unit b) 4 sq.unit c) 12 sq.unit d) 24 sq.unit
19) The point (4,1) undergoes the following three transformation successfully-
i) reflection about the line y= x
ii) translation through a distance 2 units along the positive direction of x-axis
iii) rotation through an angle π/4 about the origin in the counter clockwise direction.
The final position of the point is given by the coordinates:
a) (7/√2,-1/√2) b) (7/√2,1/√2) c) (-1/√2,7/√2) d) none
1b 2b 3d 4b 5d 6c 7a 8d 9a 10d 11d 12c 13b 14a 15b 16d 17abcd 18c 19c
SAP- 4
Match the columns:
Observe the following columns:
Column - I
A) If the distance of any point (x,y) from origin is defined as d(x,y)= 2 |x|+ 3|y|, then perimeter and area of the figure bounded by d(x,y)= 6 are
B) Number of integral values of b for which the origin and the point (1,1) lie on the same side of the straight line a⅖x + aby+1=0 for all a∈ R - {0} is
C) The ends of the hypotenuse of a right angled triangle are (6,0) and (0,6) . The third vertex lie on a circle whose radius is equals to
D) If The slope of one of the lines represented by ax²- 6xy+ y²=0 is square of the other, then a is
Column II
p) 3√2
q) 4√13
r) 12
s) 3
t) 8
Assertion & Reason
These questions contains , statement I(Assassin statement) Statement II(reason).
A) statement I is true, statement II is true; statement II is correct explanation for statement I.
B) Statement I is true, Statement II is true; Statement II is not correct explanation for Statement I.
C) statements I is true, Statement II is false.
D) Statement I is false , statement II is true.
1) Statement I: The points (2,1) and (-3,5) lie on opposite side of the line 3x - 2y +1=0.
Because
Statement II: The algebraic perpendicular distances from the given points to the line have opposite sign.
a) A b) B C) C d) D
2) Statement I: The combined equation of L₁, L₂ is 2x²+ 4xy + y²=0 and that of L₁', L₂' is 3x²+ 8xy + y²= 0. If the angle between L₁, L₂' is θ, then angle between L₂, L₁' is also θ.
Because
Statement II: If the pairs of the line L₁L₂ =0, L₁' L₂ =0 are equally inclined , then angle between L₁, L₂' = angle between L₂, L₁'.
a) A b) B C) C d) D
3) Statement I: The equation 2x²+ 3xy - 2y²+ 5x - 5y + 3= 0 represents a pair of perpendicular straight lines.
Because
Statement II: A pair of lines given by given ax²+ 2hxy + by²+ 2gx + 2fy + c= 0 are perpendicular, if a+ b=0
a) A B) B C) C d) D
4) Statement I: The joint of lines 2y= x +1 and 2y = -(x +1) is 4y²= -(x +1)²,
Because
Statement II: the joint equation of two lines satisfy every point on any one of the line.
a) A b) B c) C d) D
Comprehension based questions:
Comprehension # 1
A locus is the curve traced out by a point which moves under certain geometrical conditions:
To find the locus of a point first we assume the co-ordinates of the moving point as (h,k) and then try to find a relation between h and k with the help of the given conditions of the problem. if there is any variable involved in the process then we eliminate them. At last we replaced h by x and k by y and get the locus of the point which will be an equation in x and y.
On the basis of above information, answer the following questions :
1. locus of centroid of the triangle whose vertices are (a cos t, a sin t) and (1,0) where t is a parameter is
a) (3x -1)²+(3y)²= a²- b²
b) (3x -1)²+(3y)²= a² + b²
c) (3x +1)²+(3y)²= a²+ b²
d) (3x +1)²+(3y)²= a²- b²
2) A variable line cuts x-axis at A, y-axis at B where OA= a, OB= b (O as origin) such that a²+ b²= 1 then the locus of circumcenter of ∆ OAB is
a) x²+ y²= 4 b) x²+ y²= 1/4 c) x²- y²= 4 d) x²- y²= 1/4
3) The locus of the point of intersection of the lines x cosα+ y sinα= a and x sinα - y cosα = b where α is variable is
a) x²+ y²=a²+ b²
b) x²+ y²=a²+- b²
c) x²- y²=a²- b²
d) x²- y²=a²+ b²
Comprehension #2
For points P(x₁, y₁) and Q(x₂, y₂) of the co-ordinate plane, a new distance d(P, Q) is defined by d(P,Q)= |x₁ - x₂|+ |y₁ - y₂|
Let O(0,0), A=(1,2), B=(2,3) and C= (4,3) are four fixed points on x - y plane.
On the basis of above information, answer the following questions:
1) Let R(x,y), such that 0≤ y < 2, then R lies on a line segment whose equation is
a) x + y= 3 b) x + 2y= 3 c) 2x + y= 3 d) 2x + 2y= 3
2) Let S(x,y), such that S is equidistant from points O and B respect to new distance and if x≥ 2 and 0≤ y<3, then locus of S is
a) A line segment of finite length
b) a line of infinite length
c) a ray of finite length
d) a ray of infinite length
3) Let T(x,y), such that T is equidistant from O and C with respect to new distance and if T lies in first quadrant, then T consists of the union of a line segment of finite and an infinite rays whose labelled diagram is.
A- qr, B-s C-p D-t
1a 2a 3d 4d
C1) 1b 2b 3a
C2) 1d 2d 3a
SAP- 5
1) Determine the ratio in which the point P(3,5) divides the join of A(1,3) and B(7,9). Find the harmonic conjugate of P w.r.t. A & B.
2) The area of a triangle is 5. Two of its vertices are (2,1) and (3,-2). The third vertex lies on y= x+3. Find the third vertex.
3) A line is such that its segment between the straight lines 5x - y -4=0 and 3x + 4y -4=0 is bisected at the point (1,5). Obtain the equation.
4) two vertices of a triangle are (4,-3) and (-2,5). If the orthocentre of the triangle is at (1,2), find the co-ordinates of the third vertex.
5) A straight linenL is perpendicular to the line 5x - y = 1. The area of the triangle formed by the line L and the co-ordinate axes is 5. Find the equation of the line.
6) The vertices of a triangle OBC are O(0,0), B(-3,-1), C(-1,-3). Find the equation of the line parallel to BC and intersecting the sides OB and OC, whose perpendicular distance from the point (0,0) is half.
7) The point (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices lie on the line y= 2x + c. Find c and the remaining vertices.
8) if a,b,c are all different and the points {r³/(r -1), (r²-3)/(r -1)} where r= a, n,c are colinear, than prove that 3(a+ b +c)= ab + bc+ ca - abc.
9) two equal sides of an isosceles triangle are given by the equation 7x - y +3=0 and x+ y -3=0 and its third side passes through the point (1,-10). Determine the equation of the third side.
10) Find the direction in which a straight line may be drawn through the point (2,1) so that its point of intersection with the line 4y - 4x +4+ 3√2+ 3√10= 0 is at a distance of 3 units from (2,1).
11) a line through A(-5,-4) meets the line x+ 3y+2=0, 2x+ y+4=0, and x- y-5=0 at the points B, C, D respectively. If (15/AB)²+ (10/AC)²= (6/AD)². Find the equation of the line.
12) in a triangle ABC , D is a point on BC such that BD/D. = AB/AC. The equation of the line AD is 2x+ 3y+ 4=0, and the equation of the line AB 3x+ 2y+1=0. Find the equation of the line AC.
13) A pair of straight lines are drawn through the origin form with the line 2x+ 3y =6 an isosceles triangle right angled at the origin. Find the equation of the pair of straight lines and the area of the triangle correct to two places of decimals.
14) Show that all chords of the curve 3x²+ 3y² - 2x+ 4y=0,which subtend a right angle at the origin are concurrent. also find the point of concurrency.
1) 1:2; (-5,-3)
2) (7/2,13/2) or (-3/2,3/2)
3) 83x - 35y +92=0
4) (33,26)
5) x+ 5y+5√2=0 or x+ 5y -5√2=0
6) 2x+ 2y+ √2=0
7) -4; (2,0); (4,4)
9) x- 3y-31=0, or 3x+ y+7=0,
10) 171,99°
11) 2x+ 3y+22=0
12) 9x+ 46y+83=0
13) x - 5y=0, or 5x+ y=0 area= 2.77 14) (1/3,-2/3)
SAP- 6
1) Find the equation of the straight lines passing through.(-2,-7) and having an intercept of length 3 between the straight lines 4x+ 3y= 12, 4x+ 3y= 3.
2) Determine all values of α for which the point (α,α²) lies inside the Triangle formed by the lines;
2x+ 3y= 1; x+ 2y= 3; 5x -6y= 12, 4x+ 3y= 1.
3) Find the co-ordinate of the orthocentre of the triangle, the equations of whose sides are x+ y= 1, 2x+ 3y= 6, 4x - y= -4, without finding the co-ordinate of its vertices.
4) Find the condition that the diagonals of the parallelogram formed by the lines
ax+ by= - c, ax+ by= - c'; a'x+ b'y= -Cl and a'x+ b'y= - C' are at right angles. Also find the equation to the diagonals of the parallelogram.
5) Find the coordinates of the incentre of the Triangle formed by the line x+ y= -1, x- y= -3 and 7x - y +3=0. also find the centre of the circle to 7x -y+3= 0,
6) A triangle is formed by the lines whose equations are AB: x+ y= 5, BC: x+ 7y= 7 and CA: 7x + y-14=0. Find the bisector of the interior angle at B and the exterior angle at C. Determine the nature of the interior angle at A and find the equation of the bisector.
7) The distance of a point (x₁, y₁) from each of two straight lines which passes through the origin of Coordinates is δ; find the combined equation of these straight lines.
8) Equation of a line is given by y+ 2at= t(x - at²), t being the parameter. Find the locus of the point intersection of the lines which are at right angles.
9) A line 4x + y= 1 through the point A( 2,-7) meets the line BC whose equation is : 3x - 4y +1=0 at a point B. Find the equation of the line AC, so that AB= AC.
10) The vertices of a triangle are A(x₁, x₁ tanθ₁), B(x₂, x₂ tanθ₂) arC(x₃, x₃ tanθ₃). If the circumcenter O of the triangle ABC is at the origin & H(x, y) be its orthocentre, then show that x/y= (cosθ₁ + cosθ₂ + cosθ₃)/(sinθ₁ + sinθ₂ + sinθ₃).
11) The ends A, B of a straight line segment of constant length c slide upon the fixed rectangular axes OX & OY respectively. If triangle OAPB be completed then show that the locus of the foot of the perpendicular drawn from P to AB is ³√x²+ ²√y²= ³√c².
1) 7x + 24y+182=0 or x= -2. 2) (-3/2< α<-1 U 1/2 <α<1
3) (3/7,22/7) 4) a²+ b²= a'²+ b'²; (a+ a')x+ (b + b')y + (c + c')=0 ; (a& a')x+ (b - b')y =0
5) (-1,1);(4,1) 6) 3x + 6y -16=0 ; 8x + 8y -21=0 ; 12x + 6y -39=0 7) (y₁²- δ⅖)x² - 2x₁yxy + (x₁²- δ²)y²=0. 8) y²= a(x - 3a) 9) 52x+ 89 y + 519=0
SAP- 7
1A) If P(1,2), Q(4,6), R(5,7) & S(a,b) are the vertices of a parallelogram PQRS , then
a) a=2, b= 4 b) a=3, b= 4 c) a=2, b= 3 d) a=3, b= 5
B) The diagonals of a parallelograms PQRS are along the lines x+ 3y= 4 and 6x -2y= 7. Then PQRS must be a
a) rectangle b) square c) cyclic quadrilateral d) Rhombus
2) Using coordinate geometry, show that the three altitudes of any triangle are concurrent.
3) Let PQR be a right angled isosceles triangle, right angled at P(2,1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is
a) 3x²- 3y²+ 8xy+ 20x + 10y +25=0
b) 3x²- 3y²+ 8xy- 20x - 10y +25=0
c) 3x²- 3y²+ 8xy+ 10x + 15y +20=0
d) 3x²- 3y²- 8xy-10x - 15y -20=0
4) The incentre of the triangle with vertices (1,√3),(0,0),(2,0) is
a) (1,√3/2) b) (2/3,1/√3) c) (2/3,√3/2) d) (1,1/√3)
5) Let PS be the median of the triangle with vertical P(2,2), Q(6,-1) and R(7,3). The equation of the line passing through (1,-1) and parallel to PS is
a) 2x -9y -7=0 b) 2x - 9y -11=0 c) 2x + 9y -11=0 d) 2x + 9y +7=0
6) For points P(x₁, y₁) and Q(x₂, y₂) of the co-ordinate plane, a new distance d(P,Q) is defined by d(P,Q)=]|x₁ - x₂| + |y₁ - y₂|. Let O(0,0) and A=(3,2). prove that the set of points in the first quadrant which are equidistant (with respect to the New distance) from O and A consist of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.
7) area of the parallelogram formed by the lines y= mx, y= mx+1, y= nx and y= nx +1 equal
a) |m + n|/(m - n)²
b) 2/|m + n|
c) 2/(|m +n|
d) 1. |m - n|
6) a) Let P=(-1,0), Q=(0,0 and R= (3,3√3) be three points. Then the equation of the bisector of the angle PQR is
a) √3x/2 + y= 0 b) x+ √3y= 0 c) √3 x + y= 0 d) x+ √3y/2=0
b) A straight line through the origin O meets the parallel lines 4x + 2y=9 and 2x + y=-5 at points P and Q respectively. Then the point O divide the segment PQ in the ratio
a) 1: 2 b) 3: 4 c) 2: 1 d) 4:3
7) A straight line L through the origin meets the line x + y= 1 and x + y= 3 at P and Q respectively. Through P and Q two straight lines L₁ and L₂ are drawn, parallel to 2x - y= 5 and 3x + y= 5 respectively. Lines L₁ and L₂ intersect at R. Show that the locus of R, as L varies, is a straight line.
8) Area of the triangle formed by the angle bisectors of the pair of line x²- y²+ 2y -1=0 and the line x+ y=3 ( in square unit) is
a) 1 b) 2 c) 3 d) 4
9) The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P(h,k) with the line y= x and x+ y=2 is 4h². Find the locus of the point P.
10) Two rays in the first quadrant x+ y =|a| and ax- y= 4 intersect each other in the interval a∈ (a₀,∞) then find the value of a₀.
11) Let O(0,0), P(3,4), Q(6,0) be the vertices of the triangles OPQ. The point R inside the triangle OPQ is such that triangle OPR, PQR, OQR are of equal area. The coordinates of R are
a) (4/3,3) b) (3,2/3) c) (3,4/3) d) ( 4/3, 2/3)
12) Lines L₁: y - x= 0 and L₂: 2x + y=0 intersect the line L₃: y+2=0 at P and Q respectively. The bisector of the acute angle between L₁ and L₂ intersect L₃ at R.
Statement -I: The ratio PR: RQ equals 2√2: √5
Statement - II: In any triangle, bisector of an angle divides the triangle into two similar Triangles .
A) Statement -I is true, Statement II is it true; Statement II is correct explain for statement I
B) Statement I is true, statement II is true, statement II is NOT a correct explanation for Statement I.
C) statement I is true, statement II is false
D) statement I is false, statement II is true.
13) consider 3 points P= (-sin+β - α), - cosβ), Q= (cos(β -α), sinβ) and R= (cos(β - α +θ), sin(β - θ), where 0< α, β,,θ <π/4. Then
a) P lies on the segment RQ.
b) Q lies on the segment PR.
c) PpR lies on the segment QP.
d) P, Q, R are non collinear.
14) consider the lines given by :
L₁: 3y +x-5= 0 ; L₂: 3x -ky -1=0 ; L₃: 5x +2y-12=0
Match the statement/expression in column I with the statements /Expression in column II
Column I
A) L₁, L₂, L₃ are concurrent, if
B) One L₁, L₂, L₃ is parallel to atleast one of the other two, if
C) L₁, L₂, L₃ form a triangle, if
D) L₁, L₂, L₃ do not form a triangle, if
Column II
p) k=-9
q) k=-6/5
r) k= 5/6
s) k= 5
15) The locus of the orthocentre of the triangle formed by the lines
(1+ p)x - py+ p(1+ p)=0,
(1+ q)x - qy+ q(1+ q)=0 and y=0, where p≠ q, is
a) hyperbola b) a parabola c) an ellipse d) a straight line
1a) c b) d c) acd 3b 4a) d b) d 5d 6a) c b) b c) b 7) x - 3y +5=0 8b 9) y= 2x +1, y= -2x +1 10) 2 11c 12c 13d 14) A-s B) PQ c-r d) pqs 15d
Sap-8
1) Let triangle have vertices A(-4,-3), B(6,-1) and C(2,5). The length of the median from C to AB, is
a) √50 b) √53 c) √89 d) √104
2) The length of a line segment AB is 10 units. If the coordinates of one exterility are (2,-3) and the abscissa of the other extremity is 10 then the sum of all possible values of the ordinate of the other extremity is
a) 3 b) -4 c) 12 d) -6
3) if P(1,2), Q(4,.6), R(5,7) & S(a,b) are the vertices of a parallelogram PQRS , then:
a) a= 2, b= 4 b) a= 3, b= 4 c) a= 2, b= 3 d) a= 3, b= 5
4) The Four Points whose coordinates are (2,1),(1,4),( 5,2) form
a) a rectangle which is not a square
b) a trapezium which is not a parallelogram
c) a square
d) a rhombus which is not square
5) if A and B are the point (-3,4) and (2,1) then the coordinates of the point C on AE produced such that AC= 2BC are
a) (2,4) b) (3,7) c) ( 7,-2) d) (-1/2,5/2)
6) The orthocentre of the Triangle ABC is B and the circumcenter is S (a,b). If A is the origin then the coordinates of C are
a) (2a,2b) b) (a/2,b/2) c) (√(a²+ b²),0) d) none
7) A particle begins at the origin and moves successfully in the following manners as shown ,
1 unit to the right, 1/2 unit up, 1/4 unit to the right. 1/8 unit down, 1/6 unit to the right etc.,
The length of each move is half the length of the previous move and movement continues in the 'zigzag' manner infinitelt. The coordinates of the point which the zigzag Converges is
a) (4/3,2/3) b) (4/3, 2/5) c) (3/2, 2/3) d) (2,2/5)
8) Coordinates of the vertices of a triangle ABC are (12,8), (-2,6) and (6,0)?then the correct statement is
a) triangle is a right but not isosceles
b) triangle is isosceles but not right
c) triangle is obtuse
d) the product of the abscissa of the centroid, orthocentre and circumcentre is 160.
9) The area of the quadrilateral ABCD with vertices A(-2,0), B(0,-4), C(4,-2) D(2,2) is
a) 12 sq unit b) 16 sq unit c) 20 sq unit d) 32 sq unit
10) The median of a triangle meet at (0,-3) and its two vertices are at (-1,4) and (5,2). Then the third vertex is at
a) (4,15) b) (-4,-15) c) (-4,15) d) (4,-15)
11) If the two vertices of a triangle are (7,2) and (1,6) and its centroid is (4,6) then the co-ordinate of third vertex are (a,b). The value of (a+ b) is
a) 13 b) 14 c) 15 d) 16
12) if in triangle ABC , A= (1,10), circumcenter= (-1/3, 2/3) and orthocenter=(11/3,4/3) then the coordinates of midpoint of the side opposite to A is
a) (1,-11/3) b) (1,5) c) (1,-3) d) (1,6)
13) suppose ABC is a triangle with three acute angles A, B, C. The point whose coordinates are (cosB - sinA, sinB - cosA) can be in the
a) first and second quadrant
b) second and third quadrant
c) third and fourth quadrant
d) second quadrant only
14) Let ABC be a fixed triangle and P be variable point in the plane of triangle ABC . suppose a,b,c are lengths of sides BC, CA, AB opposite to the angles A, B, C respectively. If a(PA)²+ b(PB)²+ c(PC)² is minimum than point P with respect to ∆ ABC, is
a) centroid b) circumcenter c) ortnocentre d) incentre
15) consider the points P(2,-4); Q(4,-2) and R(7,1). The points P, Q, R.
a) form an equilateral triangle
b) form a right angled triangle
c) form an isosceles triangle which is not equilateral
d) are collinear
16) Column I
A) the points (2,-2),(8,4),( 5,7) and (-1,1) taken in order are the vertices of a
B) The point (0,1),(2,1), (0,3) and (-2,1) taken in order at the vertices of a
C) the points (3,-5),(-5,-4),( 7,10),(15,9) taken in order are the vertices of a
D) the point (-3,4),(-1,0),( 1,0) and (3,4) taken in order are the vertices of
Column II
P) square
Q) rectangle
C) trapezium
D) parallelogram
E) cyclic quadrilateral
17) a) Find the circumcenter and circumradius of the triangle whose vertices are (1,1), (2,-1) and (3,2).
b) Find the incentre of the triangle whose vertices are (2,3),(-2,-5),(-4,6).
c) if a circle passes through the point (9,3),(7,-1) and (1,-1) then find its
i) centre and ii) radius
18) Find the distances between the following pairs of points
a) (t₁², 2t₁) and (t₂², 2t₂) if t₁ and t₂ are the roots of x² - 2√3x +2=0.
b) (a cosθ, a sinθ) and (a cosφ, a sinφ)
19) The points (2,1),(5, 4) and (1,4) are 3 of the corners of the parallelogram. Find the coordinates of the remaining corner which is opposite to (2,1).
20) The line joining the points (1,-2) and (-3,4) is itisected, find the co-ordinates of the points of the tri-section.
21) Find the lengths of the medians of the triangle whose vertices are (1,2),( 0,3),(-1,-2).
22) The vertices of a triangles are A(1,1), B(4,5), C(6,13). Find cosA.
23) ∆ ABC lies in the plane with A(0,0), B(0,1), C(1,0).
Points M and N are chosen on AB and AC, respectively, such that MN is parallel to BC and MN divides the area of ∆ ABC in half. Find the coordinates of M.
24) Find the harmonic conjugates of the point R(5,1) with respect to the point P(2,10) and Q(6,-2).
25) a variable line passes through P(2,3) and cuts the coordinate axes at A and B. if the parallelogram where OACB(where O is the origin) is completed then find number of ordered pairs (x,y) of integers which lie on the locus of point C.
1a 2d 3c 4c 5c 6a 7b 8d 9c 10b 11b 12a 13d 14d 15d
16) A- QST, B- PQST, C-S, D-RT
17)a) (5/2,1/2); √(5/2) b) (-1,2); c) i) (4,3); ii) 5
18) a) 8 b) 2a sin|(θ-φ)/2| 19) (4,7) 20) (-1/3,0),(-5/3,2). 21) 3√2/2,3,3√10/2 22) 63/65 23) (0,1/√2) 24) (8,-8) 25) 7
Sap-9
Let ABCD is a Square with sides of unit length. Points E and F are taken on sides AB and ADmrespectively so that AE= AF. Let P be a point inside the square ABCD.
1) The maximum possible area of quadrilateral CDFE is
a) 1/8 b) 1/4 c) 5/8 d) 3/8
2) The value of (PA)²-?(PB)²+ (PC)¹- (PD)² is equal
a) 3 b) 2 c) 1 d) 0
3) Let a line passing through point A divides the square ABCD into two parts so that area of one portion is double the other, then the length of portion of line inside the square is
a) √10/3 b) √13/3 c) √11/3 d) 2/√3
4) The line x= c cuts the triangle with corners (0,0),(1,1) and (9,1) into two region. For the area of the two regions to be the same c must be equal to
a) 5/2 b) 3 c) 7/2 d) 3 or 15
5) A triangle has two of its vertices at ( 0,1) and (2,2l in the cartesium plane. Its third vertex lies on the x-axis. if the area of the triangle is 2 square units then the sum of the possible abscissa of the third vertex, is
a) - 4 b) 0 c) 5 d) 6
6) A point P(x,y) moves so that the sum of the distance from P to the co-ordinate Axes is equals to the distance from P to the point (1,1). The equation of the locus of P in the first quadrant is
a) (x +1)(y+1)= 1 b) (x +1)(y+1)= 2 c) (x -1)(y-1)= 1 d) (x -1)(y-1)= 2
7) Let A(2,-3) and B (-2,1) be vertices of a ∆ ABC. if the centroid of ∆ ABC moves on the line 2x + 3y=1, then the locus of the vertex C is
a) 2x + 3y=9 b) 2x - 3y=7 c) 3x + 2y=5 d) 3x - 2y= 3
8) A stick of length 10 units rests against the floor and a wall of a room. if the stick begins to slide on the floor then the focus of point is
a) x⅖ + y²=2.5 b) x¹+ y²=25 c) x¹ + y²=100 d) none
9) AB is the diameter of a semicircle k, C is a arbitrary point on the semi circle (other than A or B) and S is the centre of the circle inscribed into triangle ABC , then measure of -
a) angle ASB changes as C moves on k.
b) angle ASB is the same for all position of C but it cannot be determined without knowing the radius.
c) angle ASB= 135 for all S.
d) angle ASB= 150° for all C.
10) Given the points A(0,4) and B(0,-4), the equation of the locus of the point P such that |AP - BO|= 6 is
a) 9x² -7 y²= -63
b) 9x² -7 y²= 63
c) 7x² - 9y²= -63
d) 7x² -9y²= 63
11) Each member of the family of parabola y= ax²+ 2x +3 has a Maximum or a minimum point depending upon the value of 'a' is.
a) a straight line with slope 1 and y intercept 3.
b) a straight line with slope 2 and y intercept 2.
c) a straight line with slope 1 and x intercept 3.
d) a straight line with slope 2 and y intercept 3.
12) Column I
A) the point (2,-2),(-2,1) and (5,2)
B) the points (1,-2),(-3,0) and (5,6)
C) the points (3,7),(6,5) and (15,-1)
D) The points (2,2),(- 2,-2) and (-2√3, 2√3}
Column II
P) are the vertices of a right angled triangle.
Q) are the vertices of a right angle isosceles triangle
R) are the vertices of an equilateral triangle
S) do not form a triangle
13) Two vertices of a triangle are at the points( 3,-1), (-2,3) and the centroid is at the origin. Find the co-ordinates of the remaining vertex.
14) Prove the following results analytically.
i) diagonals of an isosceles trapezium are equal.
ii) l₁¹+ l₂²+ l₃²= (3/4) (a²+ b²+ c²) where l₁, l₂, l₃ are the lengths of median of ∆ ABC)
iii) medians to the equal sides of an isosceles Triangles are equal and thee converse.
15) If x₁, x₂, x₃ are the roots of the equation x³- 3px²+ 3qx -1=0 then find the centroid of the triangle the coordinates of whose vertices are (x₁, 1/x₁)(x₂, 1/x₂) and (x₃, 1/x₃).
16) If P (t²,2t), Q(1/t², -2/t) and S(1,0) be any three points, find the value of (1/SP + 1/SQ).
17) If a and b are real numbers between 0 and 1 such that the point (a,1),(1,b) and (0,0) form an equilateral triangle, find a and b.
18) The he vertices of a Triangles are (1,a),(2,b) and (c²,-3)
i) prove that its centroid cannot lie on the y-axis .
ii) find the condition that the centroid may lie on the x-axis for any value of a,b,c∈ R
19) If ( 3/2,0),(3/2,6) and (-1,6) are mid points of the sides of a triangle, then find
a) centroid of the triangle
b) incentre of the triangle
20) The vertices of a Triangle ABC are A(1,2); B( 2,3), C(3,1). Find the cosines of the interior angles of the triangle and hence or otherwise find the coordinates of
a) orthocentre of the triangle b) circumCentre of the triangle.
21) Find the relation between x and y when the point (x, y) lies on the straight line joining the points (2,-3) and (1,4).
22) Find the area of the Pentagon whose vertices taken in order are (0,4),(3,0),(6,1),(7,5) and (4,9).
23) Consider the triangle with vertices A(-2,4),( B(10,-2), C(- 2,-8). if G is the centroid of the triangle, find the area of the triangle BGC.
24) if the area of the triangle formed by the point (1,2),(2,3)(x 4) is 40 square units,find x.
25) Find the area of the quadrilateral whose vertices are A(1,1), B(3,4) C(5,-2) and D(4,-7).
26) let ∆₁ denotes the area of the triangle formed by the vertices (am₁², 2am₁),((am₂², 2am₂), (am₃², 2am₃) and ∆₂ denote the area of the triangle formed by the vertices (am₁m, a(m₁ + m₂)), (am₂m₃, a(m₂+ m₃)) and (am₃m, a(m₃+ m₁). Find ∆₁/∆₂.
27) A(0,1) and B(0,-1) are two points. If a variable point P moves such that sum of distance from A and B is 4. Then the locus of P is the equation of the form of x²/a²+ y²/b²=1. Find the value of (a²+ b²).
28) If O be the origin, and if the coordinates of any two points P₁ and P₂ be respectively (x₁, y₁) and (x₂, y₂). prove that OP₁. OP₂ cosP₁OP₂= x₁x₂+ y₁y₂.
29) The ends of the hypotenuse of a right angled triangle are (6,0) and (0,6). Find the locus of the third vertex and interpret the locii geometrically.
30) a) Find the locus of a point which is equidistant from the points (3,4), ( 5,2).
b) if the distance of a point P from the point (2,1) and (1,2) ate in the ratio 2:1, find the locus of the point P.
c) a triangle ABC is formed by 3 lines x+ y +2=0, x-2y +5=0, 7x+ y -10=0. P is a point inside the triangle ABC such that areas of the triangle of the triangle PAB, PBC, PCA are equal. If the coordinates of the point P are (a,b) and the area of the triangle ABC is δ then find (a+ b +δ).
1c 2d 3b 4b 5a 6b 7a 8b 9c 10a 11a 12 A) PQ B) p C) s D) r
13) (-1,-2) 15) (p,q) 16) 1 17) a= b = 2-√3 18) a+ b= 3
19) i) (2/3,4) ii) (1,2)
20)a) (5/3,7/3) b) (13/6,11/6)
21) 7x + y= 11 22) 36.5 23) 24 24) 83,-77
25) 41/2 26) 2 27) 7 29) x²+ y²- 6x - 6y= 0
30) x - 3y= 1, 3x² +3y -4x - 14y +15=0, 15
SAP- 10
1) A line passes through (22) and cuts a triangle of area 9 square units from the first quadrant. The sum of all possible values for the slope of such a line, is
a) - 2.5 b) - 2 c) - 1.5 d) - 1
2) a variable straight line passes through the points of intersection of the lines x+ 2y= 1 and 2x - y= 1 and meets the co-ordinate axes in A and B. The locus of the middle point of AB is :
a) x+ 3y= 10xy b) x - 3y= -10xy c) x+ 3y= -10xy d) none
3) a variable straight line passes through a fixed point (a,b) intersecting the coordinates axes at A and B. If O is the origin then the locus of the centroid of the triangle OAB is
a) bx+ 2ay - 3xy= 0 b) bx+ ay - 2xy= 0 c) ax+ by - 3xy= 0 d) none
4) The equation of L₁ and L₂ are y= mx and y= nx, respectively . Suppose L ₁ makes twice as a large of an angle with horizontal (measured in counterclockwise from the positive x-axis) as L₂ and L₁ has 4 times the slope of L₂. If L₁ is not horizontal, then the value of the product (mn) equals
a) √2/2 b) -√2/2 c) 2 d) - 2
5) The extremities of the base of an isosceles triangle ABC are the points A(2,0) and B(0,1). if the equation of the side AC is x= 2. Then the slope of the side BC is
a) 3/4 b) 4/3 c) 3/2 d) √3
6) The graph of the function, y= cosx cos(x +2) - cos²(x +1) is
a) a straight line passing through (0, - sin²1) with slope 2
b) a straight line passing through (0,0).
c) a parabola with vertex (1, - sin²1)
d) a straight line passing through the point (π/2, - sin²1) and parallel to the x-axis .
7) A and B are any two points on the positive x and y axis respectively satisfying 2(OA)+ 3(OB)= 10. If P is the middle point of AB then the locus of P is
a) 2x +3y=5 b) 2x +3y=10 c) 3x +2y=5 d) 3x +2y=10
8) a line with gradient 1 intersects a line with gradient 6 at the point (40,30). The distance between x intercept of these lines is
a) 6 b) 8 c) 10 d) 12
9) Locus of a point which is equidistant from the point (3,4) and (5,-2) is a straight line whose x-intercept is
a) 1/3 b) 2/3nc) 1 d) -1/3
10) The diagonals of a parallelogram PQRS are along the lines x +3y=4 and 6x - 2y=7. Then PQRS must be a
a) rectangle b) square c) cyclic quadrilateral d) rhombus
11) The sides of a triangle ABC lie on the line 3x +4y=0, 4x +3y=0 and x = 3. Let (h,k) be the centre of the circle inscribed in ∆ ABC. The value of (h + k) equals
a) 0 b) 1/4 c) -1/4 d) 1/2
12) If m and b are real numbers and mb> 0, then the line whose equation is y= mx + b cannot contain the point
a) (0,2009) b) (2009,0) c)(0,-2009) d) (20,-100)
13) If the vertices P and Q of a triangle PQR are given by (2,5) and (4,-11) respectively, and the point R moves along the line N: 9x +7y= -4, then the locus of the centroid of the triangle PQR is a straight line parallel to
a) PQ b) QR c) RP d) N
14) The co-ordinate of the orthocentre of the triangle by the lines, 4x -7y+10 =0, x +y=5 and 7x +4y= 15 is
a) (2,1) b) (-1,2) c) (1,2) d) (1,-2)
15) If the X intercept of the line y= mx +2 is greater than 1/2 then the gradient of the line lies in the interval
a) (-1,0) b) (-1/4,0) c) (-∞, -4) d) (-4,0)
16) Let coordinates of the points A and B be (1,2) and (7,5) respectively . The line AB rotated through 45° in anticlockwise direction about the point of tri-section of AB which is nearer to B. The equation of the line in new position is
a) 2x - y -6= 0 b) 2px - y - 1= 0 c) 3x - y - 11= 0 d) none
17) The greatest slope along the graph represented by the equation 4x²- y⅖+ 2y -1=0 is
a) -3 b) -2 c) 2 d) 3
18) 18 565 780 coordinates of the centre of the triangle is 3134 334 Teri of the triangle is 20 square units the coordinates a vertex are 5030 the vertex lie on the line the coordinates of 5135 5775
24) Three vertices of a triangle are A(4,3), B(1,-1) and C(7,k). Value/s of k for which centroid, orthocentre, incentre and circumcenter of the ∆ ABC lie on the same straight line is/are
a) 7 b) -1 c) -19/8 d) none
25) Column I
A) Which cuts off an intercept 4 on the x-axis and passes through the point (2,-3).
B) Which cuts off equal intercepts on the co-ordinate axes and passes through (2,5)
C) Which makes an angle of 135° with the axis of x and which cuts the axis of y at a distance - 8 from the origin and
D) Through the point (4,1) and making with axes in the first quadrant a triangle whose area is 8.
Column II
P) 2x + y +1 = 0
Q) x -p+ y -7 = 0
R) 3x - 2y - 12= 0
S) x +y - 8= 0
T) x + y +8 = 0
26) Find area of the triangle formed by the straight line whose equations are 2y + x - 5= 0, 2x + y - 7 = 0 and x - y +1 = 0
27) a) Find the equation of the straight line which passes through the point (1,2) and is such that the given point bisects the part intercepted between the axes.
b) Find the equations to the straight lines which join the origin and the points of tri-section of the portion of the line x +3y -m12= 0 intercepted between the axes of coordinates.
c) Find the equations to the straight lines each of which passes through the point (3,2) and intersects the x-axis and y axis in A, B respectively such that OA- OB=2.
28) a) Find the equation of the straight line passing through (3,4) and the intersecting point of the two lines 5x - y -9= 0 and x +6y -8= 0.
b) Find the equation to the straight line which go through the origin and trisect the portion of the straight line 3x +y - 12= 0 which is intercepeted between the axes of coordinates.
c) Find the equation to the straight line which passes through the point (-5,4) and is such that the portion of it between the axes is divided by the point in the ratio 1:2.
29) Find the equation of the sides of the medians of the triangle formed by the joining the points (2,4),(4,6) and (-6,-10).
30) Find the equation to the straight line which passes through the point (5,6) and has intercepts on the axes
a) equal in magnitude and both positive.
b) equal in magnitude but opposite in sign
1a 2a 3a 4c 5a 6d 7a 8c 9c 10d 11a 12b 13d 14c 15d 16c 17c 18d 19a 20c 21d 22d 23bd 24bc 25 AR BQ CT DS
26) 3/2
27) 2x + y -4 = 0 , b) 2x - 3y = 0; x - 6y= 0 c) 2x +3y -12 b = 0 ; x - y = 1
28) 3x - y -5= 0 b) 2x - y = 0 , 3x - 2y= 0 c) -8x +5y -6⁰= 0
29) x - y +2= 0 , 7x - 4y +2= 0 , 8x - 5y -2 = 0 , 2x - y= 0, 3x - 2y= 0 , 5x - 3y = 0
30) x + y - 11= 0 ii) -x + y -1= 0
SAP- 11
1) Number of lines that can be drawn through the point (4,-5) so that its distance from (-23) will be equal to 12 is equal to
a) 0 b) 1 c) 2 d) 3
2) two mutually perpendicular straight lines through the origin from an isosceles triangle with the line 2x + y -5= 0. Then the area of the triangle is
a) 5 b) 3 c) 5/2 d) 1
3) Let the lines (y -2)= m₁(x -5) and (y +4)= m₂(x -3) intersect at right angles at P(where m₁ and m₂ are parameters) if locus of P is x²+ y²+ gx + fy +7=0, then (f- g) equals
a) 1 b) 2 c) 8 d) 10
4) P lies on the line y= x and Q lies on y= 2x. The equation for the locuss of the midpoint of PQ, if |PQ|= 4, is
a) 25x² +36xy+ 13y²= 4
b) 25x² - 36xy+ 13y²= 4
c) 25x² - 36xy- 13y²= 4
d) 25x² +36xy- 13y²= 4
5) The vertex of the right angle of a right angle triangle lies on the straight line 2x - y- 10= 0 and the two other vertices, at points (2,-3) and (4,1) then the area of triangle in sq, unity is
a) √10 b) 3 c) 33/5 d) 11
6) Point P lies on the line l{(x,y)| 2x +5 y- 15=0}. If P also equidistant from the co-ordinate Axes. then P can be located in which of the four quadrants .
a) I only b) II only c) I or II only d) IV only
7) If each of the points (a,6) and (3,b) lies in the line joining the points (3,2) and (5,1) then the point (a,b) lies on the line:
a) 3x +4y +7= 0 b) 2x +3y -7= 0 c) 4x -3y -7= 0 d) 3x -2y -7= 0
8) The line L₁ given by x/5 + y/b= 1 passes through the point M(13,32). The line L₂ is parallel to L₁ and has the equation x/c + y/3= 1. Then the distance between L₁ and L₂ is
a) √17 b) 17/√15 c) 23/√17 d) 23/√17
Let M(2,13.8) is the circumcentre of ∆ PQR whose side PQ and PR are represent by the straight lines 4x -3y = 0 and 4x +y -16 = 0 respectively.
9) The orthocentre of ∆ PQR is
a) (7/3?4/3) b) (4/3,7/3) c) (3,3/4) d) 3/4,3)
10) If A, B, C are the midpoint of the sides PQ , QR, PR of ∆ PQR respectively, then the area of ∆ ABC equals
a) 1 b) 2 c) 3 d) 4
11) If PB be the median of the ∆ PQR, then the equation of the straight line passing through N(-2,3) and perpendicular to PB is
a) 4x +y +5= 0 b) x - 4y + 14= 0 c) 4x - y +11= 0 d) x +4y -10= 0
Paragraph for Question 12-14
In the diagram, a line is drawn through the point A(0,16) and B(8,0). Point P is chosen in the first quadrant on the line through A and B. Points C and D are chosen on the x and y axis respectively, so that PDOC is a rectangle.
12) Perpendicular distance of the line AB from the point (2,2) is
a) √4 b) √10 c) √20 d) √50
13) Sum of the coordinates of point P if PDOC is a square is
a) 32/3 b) 16/3 c) 16 d) 11
14) Number of possible ordered pair/s of all positions of the point P on AB so that area of the rectangle PDOC is 30 square.units is
a) three b) two c) one d) zero
Paragraph for Question 15-17
Consider a ∆ ABC whose sides are BC , CA and AB are represent by the straight lines x - 2y +5=0, x+ y +2=0 and 8x - y -20=0 respectively.
15) The area of ∆ ABC equals
a) 41/2 b) 43/2 c) 45/2 d) 47/2
16) If AD be the median of the ∆ ABC then the equation of the straight line passing through (2,-1) and parallel to AD is
a) 4x - 3y -11=0 b) 13x - 4y -30 =0 c) 4x +13y +5 =0 d) 13x +4y -22=0
17) The orthocentre of the ∆ ABC is
a) (-3,1) b) (-1/3,2/3) c) (-2,4) d) (-2/3,4/3)
18) Consider the line L₁: x/2 + y/4 -1=0, L₂: x/4 + y/3 -1=0, L,₃: x/3 + y/4 -2=0, and L₄: x/4 + y/3 -2=0,
Statement 1: The quadrilaterals formed by these four lines is a rhombus.
Statement 2: if diagonals of a quadrilateral formed by any four lines are unequal and intersect at right angle then it is rhombus.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
19) Statement -1: Centroid of the triangle whose vertices are A(-1,11); B(-9,-8) and C(15,-2) lies on the internal angle bisector of the vertex A.
Statement -2: triangle ABC is isosceles with B and C as base angles.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
20) The equation of the attitude of the ∆ ABC whose vertices are A(-4,2), B(6,5) and C(1,-4) can be
a) 10x +3y +2=0 b) 5x +9y +2=0 c) 6x - 5y =0 d) 5x -6y =0
21) Two vertices of the ∆ ABC are at the point A(-1,-1) and B(4,5) and the third vertex lies on the straight line y= 5(x -3). if the area of the ∆ is 19/2 then the possible coordinates of the vertex C are
a) (5,10) b) (3,0) c) ( 2,-5) d) (5,4)
22) line x/a. + y/b = 1 cuts the co-ordinate axes at A(a,0)!and B(0,b) and the line x/A' + y/B' = -1 at A'(-a',0) and B'(0,-b'). if the points A,B,A',B' are concyclic then the orthocentre of the triangle ABA' is
a) (0,0)(0,b')(0,aa'/b) d) (0,bb'/a)
23) a line passes through the origin and makes an angle of π/4 with the line x - y+1. Then:
a) equation of the line is x= 0
b) the equation of the line is y=0
c) the point of intersection of the line with the given line is (-1,0)
d) the point of intersection of the line with the given line is (0,1).
24) equation of a straight line passing through the point (2,3) and inclined at an angle of arc tan(1/2) with the y+ 2x = 5 is
a) y=3 b) x=2 c) a) 3x +4y -18 =0 d) 4x +3y -17 =0
25) Consider the equation y= y₁= m(x - x₁). If m and x₁ are fixed and different lines are drawn for different values of y₁, then:
a) the lines will pass through a fixed point
b) there will be a set of parallel lines.
c) all the lines intersect the line x= x₁
d) all the lines will be parallel to the line y= x₁.
26) Consider the line Ax+ By + C=0
Column I
A) x intercept is finite and y intercept is infinite
B) x intercept is infinite and y intercept is finite
c) both x and y intercept are zero
d) both x and y intercepts are infinite
Column II
P) A= 0, B, C≠ 0
Q) C= 0, A, B≠ 0
R) A, B= 0, and C≠ 0
S) B= 0, A, C≠ 0
27) column I
A) Four lines x +3y -10 =0 , x +3y - 20=0, 3x -y +5 =0 and 3x -y -5 =0 form a figure which is
B) The point A(1,2), B(2,-3), C(-1,-5) and D(-2,4) in order are the vertices of
C) The lines 7x +3y -33 =0 , 3x -7y +19=0, 3x -7y -10=0 and 7x +3y -4 =0 form a figure which is
D) four lines 4x -3y -7 =0 , -4x +3y +7=0, -3x +4y -21 =0 and 3y - 4x +14 =0 form a figure which is
P) a quadrilateral which is neither a parallelogram nor a trapezium nor a kite
Q) a parallelogram
R) a rectangle of area 10 square units
S) a square
28) find the sum of the abscissas of all points of the line x+ y= 4 that lies a unit distance from the line 4x + 3y -10=0.
29) Find the equation of the sides of a square whose each side is of a length 4 units and Centre is (1,1). Given that one pair of sides is parallel to 3x - 4y= 0.
30) let (xᵣ, yᵣ)r= 1, 2, 3 are the co-ordinates of the vertices of a triangle ABC . If D is the point on BC dividing it in the ratio 1:2 reckoning from the vertex B, prove that the equation of the line AD is
2|x y 1| + | x y 1
x₁ y₁ 1 x₁ y₁ 1=0
x₂ y₂ 1. x₃ y₃ 1
also find the equation of the line AE in the smaller form where E is the harmonic conjugate of X w.r.t the point B and C.
1a 2a 3d 4b 5b 6c 7a 8c 9c 10b 11d 12c 13a 14b 15c 16d 17b 18c 19a 20abd 21ab 22bc 23abcd 24bc 25bc 26) AS CQ DR 27) AQRS BO CQ DQ 28) -4 29) 3x -4y +11 =0 , 3x -4y - 9=0, 4x +3y +3 =0, 4x +3y -17 =0
30) 2|x y 1| + | x y 1
x₁ y₁ 1 x₁ y₁ 1=0
x₂ y₂ 1. x₃ y₃ 1
Sap-11
1) The area of the parallelogram formed by the lines 3x +4y -7a =0; 3x +4y -7b =0; 4x +3y -7c =0and 4x +3y -7d =0 is
a) |(a- b)(c - d)|/7
b) |(a- b)(c - d)|
c) |(a- b)(c - d)|/49
d) 7|(a- b)(c - d)|
2) If x₁, y₁ are the roots of x²+ 8x -20=0, x₂, y₂ are the roots of 4x²+ 32x - 57=0 and x₃, y₃ are the roots of 9x²+ 72x -112=0, then the points (x₁,y₁), (x₂,y₂),(x₃,y₃)
a) are colllinear
b) form an equilateral triangle
c) form a right angled isosceles triangle
d) are concyclic
3) Let (x₁,y₁), (x₁,y₂),(x₃,y₃) are the vertices of a triangle ABC respectively. D is a point on BC such BC= 3BD. The equation of the line through A and D, is
a) | x y 1| + 2|x y 1|
x₁ y₁ 1 x₁ y₁ 1 = 0
x₂ y₂ 1 x₃ y₃ 1
b) 3| x y 1| + |x y 1|
x₁ y₁ 1 x₁ y₁ 1 = 0
x₂ y₂ 1 x₃ y₃ 1
c) | x y 1| + 3|x y 1|
x₁ y₁ 1 x₁ y₁ 1 = 0
x₂ y₂ 1 x₃ y₃ 1
d) 2| x y 1| + 2|x y 1|
x₁ y₁ 1 x₁ y₁ 1 = 0
x₂ y₂ 1 x₃ y₃ 1
4) Equation of a straight line passing through the origin and making with x-axis an angle twice the size of the angle made by the line y= 0.2x with the x-axis , is
a) y= 0.4x b) y= 5x/12 c) 6y - 5x = 0 d) none
5) A triangle ABC is formed by the lines 2x - 3y -6=0; 3x - y +3=0 and 3x +4y -12=0. if the points P(α,0) and Q(0,β) always lie on or inside the ∆ ABC, then
a) α∈ [-1,2] and β∈[-2,3]
b) α∈ [-1,3] and β∈[-2,4]
c) α∈ [-2,4] and β∈[-3,4]
d) α∈ [-1,3] and β∈[-2,3]
6) The co-ordinates of a point P on the line 2x - y +5=0 such that |PA - PB| is maximum where A is (4,-2) and B(2,-4) will be
a) (11,27) b) (-11,-17) c(- 11, 17) d) (0,5)
7) If the lines λx + (sinα)y + cosα=0
x + (cosα)y + sinα=0
x -(sinα)y + cosα=0
Pass through the same point where α ∈ R then λ lies in the interval
a) [1,1] b) [-√2,√2] c) [-2,2] d) (-∞, ∞)
8) Two points A(x₁,y₁) and B(x₂,y₂) are chosen on the graph of f(x)= log x with 0<x₁,<x₂. The points C and D trisect line segment AB with AC< CB. Through C horizontal line is drawn to cut the curve at E(x₃,y₃). If x₁ = 1 and x₂ = 1000 then the value of x₃ equals
a) 10 b) √10 c) ³√10² d) ³√10
9) Area of the quadrilateral formed by the lines |x|+ |y|=2 is:
a) 8 b) 6 c) 4 d) none
10) The number of possible straight lines, passing through (2,3) and forming with coordinate axes, whose area is 12 sq unit is
a) one b) two c) three d) four
11) Let A=(3,2) and B=(5,1) is an equilateral triangle constructed on the side of AB remote from the origin then the orthocentre of triangle ABP is
a) (4-√3/2, 3/2 -√3)
b) (4 + √3/2, 3/2 + √3)
c) (4-√3/6, 3/2 -√3/2)
d) (4 + √3/6, 3/2 + √3/2)
12) Family of lines represent by the equation (cosθ + sinθ)x + (cosθ - sinθ)y - 3(3 cosθ + sinθ)=0 passes through a fixed point M for all real values of θ. The reflection of M in the line x - y=0, is
a) (6,3) b) (3,6) c) (-6,3) d) (3,-6)
13) A is a point on either of two lines y+ √3 |x|=2 at a distance of 4/√3 units from their point of intersection. The coordinate of the foot of the perpendicular from A on the bisector of the angle between them are
a) (-2/√3,2) b) (0,0) c) 2/√3,2) d) (0,4)
14) the line (k +1)²x + ky - 2k² - 2 =0 passes through a point regardless of the value k. Which of the following the line with slope 2 passing through the point ?
a) y= 2x -8 b) y= 2x - 5 c) y= 2x - 4 d) y= 2x + 8
15) Given the family of lines , a(2x+ y + 4)+ b(x - 2y -3)=0. Among the lines of the family, the number of lines situated at a distance of √10 from the point M(2,-3) is
a) 0 b) 1 c) 2 d) ∞
16) The coordinates of the point of reflection of the origin (0, 0l in the line 4x - 2y-5=0 is
a) (1,-2) b) (2,-1) c) (4/5, -2/5) d) ( 2,5)
17) m,n are integer with 0< n < m. A is the point (m,n) on the Cartesian plane. B is reflection of A in the line y= x. C is the reflection of B in the y-axis, D is a reflection of C in the x-axis and E is the reflection of D in the y-axis . The area of the pentagon ABCDE is
a) 2m(m + n) b) m(m + 3n) c) m(2m + 3n) d) 2m(m + 3n)
Comprehensive type 18-19
An equilateral triangle ABC has its centroid at the origin and the base BC lies along the line x + y= 1.
18) Area of the equilateral ∆ ABC
a) 3√3/2 b) 3√3/4 c) 3√2/2 d) 2√3/4
19) gradient of the other two lines are
a) √3,√2 b) √3, 1/√3 c) √2+1,√2-1 d) 2+ √3, 2 - √3
Reasoning Type
20) Given the lines y+ 2x = 3 and y+ 2x = 5 cut the axes at A, B and C, D respectively.
Statement 1: ABCD forms quadrilateral and point (2,3) lies inside the quadrilateral.
Statement 2: point lies on the same side of the lines.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
21) let points A, B, C are represented by
(a cosθᵢ, a sinθᵢ)i= 1,2,3 and cos(θ₁ - θ₂)+ cos(θ₂ - θ₃)+ cos(θ₃ - θ₁)= -3/2.
Statement 1: Orthocentre of ∆ ABC is at origin.
Statement 2: ∆ ABC is an equilateral triangle.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
22) Statement 1: Let u,v,w satisfy the equation uvw= -6, uv+ vw+ wu= -5, a+ v+ w= 2 where u> v > w, then the set of value/s of 'a' for which the points P(u,-w) and Q(v, a²) lies on the same side of the line 4x - y+5=0 are given by (-3,3).
Statement 2: If two points M(x₁, y₁) and N(x₂, y₂) lies on the same side of the line ax+ by + c=0, then (ax₁ + by₁ + c)+ax₂ + by₂ +c)> 0.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
23) A line segment AB is divided internally and externally in the same ratio at P and Q respectively and M is the midpoint of AB.
Statement 1: MP, MB, MQ are in GP
Statement 2: AP, AB and AQ are in HP
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
24) If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x - √3 y= 0 then the coordinates of the third vertex are
a) (0,a) b) (√3a/2, - a/2) c) (0,-a) d) (-√3a/2, a/2)
25) if x/c + y/d = 1 is a line through the intersection of x/a + y/b = 1 and x/b. + y/a = 1 and the lengths of the perpendiculars drawn from the origin to these lines are equal in lengths then
a) 1/a²+ 1/b²= 1/c²+ 1/d²
b) 1/a² - 1/b²= 1/c²- 1/d²
c) 1/a + 1/b = 1/c + 1/d d) none
26) The sides of a triangle are the straight lines x + y= 1; 7y= x, √3 y + x=0. Then which of the following is an interior point of the triangle ?
a) circumcenter b) centroid c) incentre d) orthocentre
27) Column I
A) Rhe lines y=0; x - 6y+4=0 and x + 6y- 9 =0 constitute a figure which is
B) the points A(a,0), B(0,b), C(c,0) and D(0,d) are such that ac= bd and a,b,c,d are all non zero.
C) the figure formed by the four lines the line ax± by ± c= 0 (a≠ b), is
D) The line of pairs x² - 8x +12=0 and y²- 14y +45=0 constitute a figure which is
Column II
P) a cyclic quadrilateral
Q) a rhombus
R) a square
d) a trapezium
Integer Type
28) The equation of a line through the midpoint of the sides AB and AD of rhombus ABCD, whose one diagonal is 3x - 4y +5=0 and one vertex is A(3,1) is ax + by + c=0. Find the absolute value of (a+ b + c) where a, b, c are Integers expressed in lowest form.
29) consider three lines : 5x - y +4 =0 ; 3x - y +5=0 ; x + y + 8 =0
if these lines enclose a triangle ABC and sum of the square of the tangent of the interior angles can be expressed in the form p/q where p and q are relative prime numbers, compute the value of (p+ q).
1d 2a 3d 4b 5d 6b 7b 8a 9a 10c 11d 12b 13b 14a 15b 16b 17b 18a 19d 20d 21a 22a 23a 24abcd 25ac 26bc 27) A- PS B- P C- Q D- PQR
28) 1 29) 465
Sap-12
1) Consider a parallelogram whose sides are represented by the lines 2x +3y =0 ; 2x +3y -5=0 ; 3x - 4y =0 and 3x - 4y= 3. The equation of the diagonal not passing through the origin is
a) 21x - 11y + 15 =0 b) 9x - 11y +15=0 c) 21x -29y -15 =0 d) 21x -11y -15=0
2) In a triangle ABC , if (2,-1) and 7x - 10y +1 =0 ; 3x - 2y +5=0 are equations of an altitude and an angle bisector respectively drawn from B, then equation of BC is
a) x +y + 1 =0 b) 5x +y +17=0 c) 4x +9y +30 =0 d) x -5y - 7=0
3) A variable line L= 0 is drawn through O(0,0) to meet the lines x +2y -3 =0 and x +2y +4=0 at points M and N respectively. A point P is taken on L= 0 such that 1/OP² = 1/OM²+ 1/ON². locus of P is
a) x² +4y² = 144/25 b) (x +2y)² = 144/25 c) 4x² +y² =144/25 d) (x -2y)² =144/25
4) if the straight lines ax + amy + 1 =0 , bx + (m+1)by +1=0 and cx +(m+2)cy +1 =0 , m≠ 0 are concurrent then a,b,c are in
a) AP only for m= 1
b) AP for all m
c) GP for all m
d) HP for all m
Paragraph (5-7)
The base of an isosceles triangle is equal to 4, the base angle is equal to 45°. A straight line cuts the extension of the base at a point M at the angle θ and bisect the lateral side of the triangle which is nearest to M
5) The area of the quadrilateral which the straight line cuts off from given triangles is
a) (3+ tanθ)/(1+ tanθ)
b) (3+ 2tanθ)/(1+ tanθ)
c) (3+ tanθ)/(1- tanθ)
d) (3+ 5tanθ)/(1+ tanθ)
6) The range of values of A for different values of θ, lies in the interval .
a) (5/2,7/2) b) (4,5) c) (4,9/2) d) (3,4)
7) The length of portion of straight line inside the triangle may lie in the range:
a) (2,4) b) (3/2,√3) c) (√2,2) d) (√2,√3)
Paragraph Type+8-10)
Consider two points A= (1,2) and B(3,-1). Let M be a point on the straight line L: x+ y=0.
8) If M be a point on the line L=0 such that AM+ BM is minimum, then the reflection of M in the line x= y is
a) (1,-1) b) (-1,1) c) (2,-2) d) (-2,2)
9) If M be a point on the line L=0 such that |AM - BM| is maximum, then the distance of M from N=(1,1) is
a) 5√2 b) 7 c) 3√5 d) 10
10) If M be a point on the line L= 0 such that |AM - BM| is minimum, then the area of ∆ AMB equals
a) 13/4 b) 13/2 c) 13/6 d) 13/8
Paragraph Type (11-13)
Consider a family of the lines (4a +3)x - (a+1)y - (2a +1)=0 where a belongs to R.
11) The locus of the foot of the perpendicular from the origin on each member of this family is
a) (2x -1)²+ 4(y +1)²= 5
b) (2x -1)²+ (y +1)²= 5
c) (2x +1)²+ 4(y +1)²= 5
d) (2x -1)²+ 4(y -1)²= 5
12) A member of this family with the positive gradient making an angle of π/4 with the line 3x - 4y=2, is
a) 7x - y= 0 b) 4x - 3y +2= 0 c) x +7y= 15 d) 5x - 3y -4= 0
13) Minimum area of the triangle which a member of this family with negative gradient can make with the positive semi axes, is
a) 8 b) 6 c) 4 d) 2
Reasoning Type
14) Consider the following statements
Statement 1: The area of the triangle formed by the points A(20,22); B(21,24); C(22,23) is the same area of the triangle formed by the point P(0,0); Q(1,2) R(2,1).
Statement 2: The area of the triangle is invariant w.r.t the translocation of the coordinates axes.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
15) Statement 1: The quadrilateral formed by the lines x+ √3 y= 5, √3x + y=3; x + √3 y= 3; and √3 x + y=5 is a rhombus.
Statement 2: if the angle between the diagonals of a quadrilateral is 90° then it is a rhombus.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
16) Statement 1: Incentre of the triangle formed by the lines whose sides are 3x + 4y=0, 5x - 12y = 0 and y -15=0 is the point P whose coordinates are (1,8).
Statement 2: Point P is equidistant from the 3 lines forming the triangle.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
17) Point Q is symmetric to point P(4,-3) with respect to bisectors of first and third quadrant then the length of PQ is 7√2.
Statement 2: Bisectors of the first and third quadrant is perpendicular bisector of PQ.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
18) Consider a triangle whose vertices are A(-2,1), B(1,3) C(3x, 2x -3) where x is real number.
Statement 1: The area of the triangle ABC is independent of x.
Statement 2: The vertex C of the triangle ABC always moves on a line parallel to the base AB.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
19) If the diagonals of quadrilateral formed by the lines px + qy+ r=0, p'x + q'y+ r=0, px + qy+ r'=0, p'x + q'y+ r'=0 are at right angles, then p²+ q²= p'²+ q'⅖.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
20) If a²+ 9b²- 4c²= 6ab then the family of lines ax+ by +c=0 are concurrent
a) (1/2,3/2) b) (-1/2, -3/2) c) (-1/2,3/2) d) (1/2,-3/2)
21) The straight lines x+ y= 0, 3x+ y= 4, x+ 3 y= 4 form a triangle which is
a) isosceles b) right angled c) obtuse angled d) equilateral
22) the x-coordinates of the vertices of a square of unit area are the roots of the equation x²- 3|x|+2=0 and the y coordinates of the vertices are the roots of the equation y²- 3y +2=0 then the possible vertices of the square is/are
a) (1,1),(2,1),(2,2),(1,2)
b) (-1,1),(-2,1),(-2,2),(-1,2)
c) (2,1),(1,-1),(1,2),(2,2)
d) (-2,1),(-1,-1),(-1,2),(-2,2)
23) P(x,y) moves such that the area of the triangle formed by P, Q (a,2a) and R(-a,-2a) is equal to the area of the triangle formed by P, S(a,2a) and T(2a,3a). The locus of P is straight line given by-
a) 3x - y= a b) 5x - 3y= - a c) 5x - 5y= -a d) 2y= ax
24) Let u= ax + by + a ³√b= 0, v= bx - ay + b ³√a = 0. a, b belongs to R be two straight lines . The equation of the bisectors of the angle formed by k₁u - k₂v= 0 and k₁u + k₂v = 0 for non zero real k₁ and k₂ are
a) u=0 b) k₂u + k₁v = 0 c) k₂u - k₁v = 0 d) v=0
25) The bisectors of angle between the straight lines y - b = 2m(x - a)/(1- m²) and y - b = 2m'(x - a)/(1- m'²) are
a) (y - b)(m + m')+ (x - a)(1- mm') =0
b) (y - b)(m + m') - (x - a)(1- mm') =0
c) (y - b)(1- mm')+ (x - a)(m + m') =0
d) (y - b)(1- mm')- (x - a)(m + m') =0
26) Set of family of lines are described in column I and their mathematical are given in column II. Match the column (m,n are parameters).
Column I
A) Having gradient 3
B) Having y intercept 3 times the x intercept
C) Having x intercept (-3)
D) concurrent at (2,3)
Column II
P) mx - y + 3 - 2m=0
Q) mx - y + 3m=0
R) èx + y - 2a=0
S) 3x - y + a =0
27) Column I
A) if the line x +2ay + a =0, x + 3by + b =0 and x + 4cy + c =0 are concurrent, then a,b,c are in
B) the lines, ax + by + (aα+ b)=0; bx + cy + (bα+ c)=0; and (aα+ b)x + (bα+ c)y =0
are concurrent if:
C) If the line ax +2y + 1 =0, bx + 3y + 1 =0 and cx + 4y + 1 =0 passes through the same point then a,b,c are in
D) Let a,b,c be distinct non negative numbers. If the lines ax +ay + c =0, x+ 1 =0 and cx + cy + b =0 pass through the same point then a, c, b are in
Column II
P) AP
Q) GP
R) HP
S) (x - α) is a factor of ax²+ 2bx + c=0
Integer Type
28) If the equation of the diagonals of the parallelogram formed by the lines, 2x - 3y +7 = 0,2x - y -5 = 0, 3x +2y -5= 0 and 3x +2y + 4= 0are ax + by - 5 = 0 and px + qy +1= 0, where a, b,p, q are integers. Find the value of a+ b + p+ q.
29) Theparallelogram is bounded by the lines y= ax + c, y= ax + d, y= bx + c and y= bx+ d and has the area equals to 18. The parallelogram bounded by the lines y= ax + c, y= ax - d, y= bx + c and y= bx - d has area 72. Given that a,b,c and d are positive integers , find the smallest possible value of a+ b+ c+ d.
30) A variable line passing through the origin intersects two given straight lines 2x + y= 4 and x +3y= 6 at R and S respectively. A point P is taken on this variable line. Find the equation of the locus of the point P if-
a) OP is the arithmetic of OR and OS
b) OP is the Geometric mean of OR, OS.
1d 2b 3b 4d 5d 6d 7c 8b 9d 10a 11d 12a 13c 14a 15c 16b 17a 18a 19a 20cd 21ac 22ab 23ab 24ad 25ad
26) As Br CQ dp
27) ar bqs cp dq
28) 40 29) 16
30) a) 2x²+ 7xy + 3y²- 8x - 9y=0 b) 2x²+ 7xy + 3y²- 24 =0
Sap-12
1) P Is a point inside the triangle ABC . Lines are drawn through P, parallel to the sides of the triangle. The three resulting Triangles with the vertex at P have areas 4,9 and 49 sq units. Tha area of the triangle ABC is
a) 2√3 b) 12 c) 24 d) 144
2) The position vectors of vertices of ∆ ABC are (1,-2),(-7,6) and (11/5, 2/5) respectively. The measure of the interior angle A of the ∆ ABC is
a) acute and lies in (75°,90°)
b) acute and lies in (60°,75°)
c) acute and lies in (45°,60°)
d) obtuse and lies in (120°,150°)
3) The area of the triangular region in the first quadrant bounded on the left by the y-axis, bounded above by the line 7x + 4y = 168 and bounded below by the line 5x + 3y= 121, is
a) 50/3 b) 52/3 c) 53/3 d) 17
4) Let A(5,12), B(-13cosθ, 13 cosθ) and C(13 sinθ , -13 cosθ) are angular of ABC where θ∈ R. The locus of orthocentre of ∆ ABC is
a) x - y+7=0 b) x - y-7=0 c) x + y-7=0 d) x + y+7=0
5) Number of straight line passing through the point (4,3) whose x intercept is a prime number and whose y intercepts is a positive integer, is equal to
a) 0 b) 1 c) 2 d) more than 2 but finite
6) Let PQR be a right angled isosceles triangle, right angled at P(2,1). If the equation of the line QR is 2x + y= 3, then the equation representing the pair of lines PQ and PR is
a) 3x²- 3y²+ 8xy + 20x + 10y +25=0
b) 3x²- 3y²+ 8xy - 20x - 10y +25=0
c) 3x²- 3y²+ 8xy + 10x + 15y +20=0
d) 3x²- 3y²- 8xy - 10x - 15y -20 =0
7) if the straight line joining the origin and the points of intersection of the curve 5x²- 6y²+ 12xy + 4x -2y +3 =0 and x + ky -1=0.
are equally inclined to the co-ordinate Axes then value of k is
a) is equal to one
b) is equal -1
c) is equal to 2
d) does not exist in the set of real numbers.
8) The angles between the straight line joining the origin to the points common to 7x² + 8y² - 4xy + 2x - 4y -8=0 and 3x - y=2 is
a) tan⁻¹√2 b) π/3 c) π/4 d) π/2
9) A pair of perpendicular straight lines is drawn through the origin forming with the line 2x + 3y=6 and isosceles triangle right angled at the origin. The equation of the line pair is
a) 5x²- 5y² - 24xy =0 b) 5x²- 5y² - 26xy =0 c) 5x²- 5y² + 24xy =0 d) 5x² - 5y² + 26xy =0
10) If the line y= mx bisects the angle between the lines ax² + by² 2hxy =0 then m is a root of the equation
a) hx² + (a - b)x - h =0
b) x² + h(a - b)x - 1 =0
c) (a - b)x²+ hx - (a - b) =0
d) - hx + (a - b)x² - (a - b) =0
11) if the equation ax² + 6xy + 2gx + 2fy + c =0 represents a pair of lines whose slopes are m and m², then sum of all possible values of a is
a) 17 b) -19 c) 19 d) -17
12) Through a point on the x-axis a straight line is drawn parallel to y-axis so as to meet the pair of straight lines ax² + 2hxy + b¹ =0 in B and C. If AB= BC then
a) h² = 4ab b) 8h² = 9ab c) 9h² = 8ab d) 4h² = ab
13) suppose that a ray of light leaves the point (3,4) reflects off the y-axis towards the Axis,
reflects off the x-axis, and finally arrives at the point (8,2). The value of x, is
a) 9/2 b) 13/3 c) 13/3 d) 16/3
14) If A(1, p²); B(0,1) and C(p,0) are the coordinates of 3 points then the value of p for which the area the triangle ABC is a minimum, is
a) 1/√3 b) -1/√3 c) 1/√3 or -1/√3 d) none
15) Let S={(x,y)| x² + y² +2xy -3x - 3y +2 =0}, then S
a) consist of 2 coincident lines.
b) consists of two parallel Lines which are non coincident.
c) consist of two intersecting lines .
d) is a parabola.
16) The complete set of values of the parameter α so that the point P(α, (1+ α²)⁻¹) does not lie outside the triangle formed by the lines L₁ : 15y= x +1, L₂: 78y = 118 - 23x and L₃: y+2=0, is
a) (0,5) b) [2,5] c) [1,5] d) [0,2] e) (2,5]
Reasoning type
17) Statement 1: the equation 2xy + 3x - 4y= 12 does not represent a line pair.
Statement 2: A general equation of degree 2 in which coefficient of x²=0 and coefficient y²= 0 and coefficient of xy≠ 0 can not represent a line pair .
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
18) Statement 1: The lines represented by x²- y²+ 2x +1=0 are equally inclined with the coordinates axes.
Statement 2: in a general equation of degree two ax² + by²+ 2hxy +2gx + 2fy + c =0 representing two lines, if coefficient of xy=0 then the lions are equally inclined with the co-ordinate axes.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
19) Consider the following statements
Statement 1: the equation x² + 2y²+ 4y -2√3x + 5=0 represent two real lines on the cartesian plane
Statement 2: A general equation of degree 2 ax² + by²+ 2hxy +2gx + 2fy + c=0 denotes a line pair of
abc+ 2fgh - af² - bg² - ch²=0
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
20) Given ∆ ABC whose vertices are A(x₁, y₁), B(x₂, y₂), C(x₃, y₃)
Let there exists a point P(a,b) such that 6a= 2x₁ + x₂ + 3x₃, 6b= 2y₁ + y₂ + 3y₃.
Statement 1: Area of triangle PBC must be less than the area of ∆ ABC.
Statement 2: P lies the triangle ABC.
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
21) If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is/are always rational point/s?
a) centroid b) incentre c) circumcenter d) orthocentre
22) Two equal sides of an isosceles triangle are given by the equations 7x - y+3=0 and x + y=3 and its third side passes through the point (1,-10). The equation of the third side can be
a) x +3y+29 =0 b) x - 3y -31 =0 c) 2x -y-13 =0 d) 3x + y+ 7 =0
23) Straight lines 2x +y -5 =0 and x -2y-3 =0 intersect at the point A. Points B and C are chosen on these two line such that AB = AC. Then the equation of line BC passing through the point (2,3) is
a) 3x -y-3 =0 b) x +3y -11 =0 c) 3x +y-9 =0 d) x - 3y+7=0
24) The lines L₁ and L₂ denoted by 3x² + 10xy + 8y² + 14x + 22y +15=0 intersect at the point P and have gradient m₁ and m₂ respectively. The acute angles between them is θ. Which of the following relations hold Good?
a) m₁ + m₂ = 5/4 b) m₁m₂ = 3/8
c) acute angle between L₁ and L₂ is sin⁻¹(2/5√5)
d) Sum of the abscissa and ordinate of the point P is -1
25) Let B(1,-3) and D(0,4) represent two vertices of rhombus ABCD in (x,y) plane, then coordinates of vertex A if angle BAD=60° can be equal to
a) ((1-7√3)/2, (1- √3)/2))
b) ((1+7√3)/2, (1+ √3)/2))
c) ((1- 14√3)/2, (1- 2√3)/2))
d) ((1+ 14√3)/2, (1+ 2√3)/2))
26) Column I
A) The four lines 3x - 4y +11=0, 3x -4y -9 =0, 4x + 3y +3 =0 and 4x +3y -17=0 enclose a figure which is
B) The lines 2x + y- 1 =0 , x + 2y -1 =0 , 2x +y- 3=0 and x + 2y- 3=0 form a figure which is
C) If O is the origin, P is the intersection of the lines 2x½ - 7xy+ 3y² +5x +10y-25 =0, A and B are the points in which these lines are cut by the line x + 2y -5 =0, then the points I,A,P,B (in some order) are the vertices of
Column II
P) a quadrilateral which is neither a parallelogram nor a trapezium nor a kite.
Q) a parallelogram which is neither a rectangle nor a rhombus.
R) a rhombus which is not a square
S) a square
27) Consider the three line equations ax + by + c =0 , bx + cy + a =0 cx + ay + b =0 where a, b, c belongs to R
Column I
A) If a+ b + c = 0 and a²+ b²+ c²≠ ab + bc+ ca then
B) If a+ b + c = 0 and a²+ b²+ c²= ab + bc+ ca then
C) If a+ b + c ≠ 0 and a²+ b²+ c²≠ ab + bc+ ca then
d) If a+ b + c = 0 and a²+ b²+ c²= ab + bc+ ca then
Column II
P) entire xy plane
Q) the lines are concurrent
R) lines are coincident
S) lines are neither coincident nor concurrent
Integer Type
28) The equations 9x³+ 9x²y - 45x²= 4y³+ 4xy²- 20y² represents 3 straight lines, two of which pass through the origin. Find the area of the triangle formed by these lines (in sq units).
29) Find the value of K for which the equation 2x² + 8x + 7y- xy + Ky² -10=0 may represent a pair of lines. For this value of K show that this equation can be transformed into a homogeneous equation of second degree by translating the origin to a properly chosen point. Also find the acute angle between the line pair represented by the given general equation.
30) A variable line passing through the origin intersects the lines 2x + 5y -3=0 and 4x + 7y -3=0 at P and Q respectively. if a point R is taken on the variable line such that OP, OR, OQ are in harmonical progression then find the locus of R.
31) If the straight line joining the origin to the point of intersection 3x² - xy + 3y²+ 2x - 3y +4=0 and 2x + 3y =k are at right angles , then find the value of 5k - 6k².
32) Find the area enclosed by the graph of x²y²- 9x² -25y² +225=0.
33) A ray of light is sent along the line x - 2y - 3=0. Upon reaching to the line 3x - 2y - 5 =0, the ray is reflected from it. Find the equation of the line containing the reflected ray.
34) A square ABCD lying in I quadrant has area 36 sq. units and is such that its side AB is parallel to x-axis . Vertices A, B and C are on the graph of y= logₐx, y= 2 logₐx and y= 3 logₐx respectively then find the value of a⁶.
35) in a triangle ABC if the equation of the median AD and BE are 2x + 3y - 6 =0 and 3x - 2y - 10 =0 respectively and AD= 6, BE= 11, then find the area of the triangle ABC.
1d 2b 3a 4a 5c 6b 7b 8d 9a 10a 11b 12b 13b 14d 15b 16b 17c 18a 19d 20a 21acd 22bd 23ab 24bcd 25ab
26) As Br CQ
27) aq bp cs dr
28) 30
29) k= -1; (-1,4); 2x²- xy - y²=0; θ = tan⁻¹(3)
30) x + 2y - 1=0 31) 32 32) 60 33) 29x - 2y -31=0 34) 3 35) 44
₁₂₃ ²² ∈ ∈₁₂²²₁₂²²₁₂θθ₁₂ θ ₁₂₁₂₁₂₂₁²²²²²²²²²²²²²²²²₁₁₂₂₁₂₁₂
₁₂ ∞∞∞∞∞∞∞∞∞
PARABOLA
Sap-1
1) Find the vertex, Axis, directrix , focus, latus rectum and the tangent at vertex for the parabola 9y² - 16x - 12y -57=0. (-61/16,2/3), y= 2/3, x= -613/144, (-485/144,2/3), 16/9, x= -61/16
2) The length of latus rectum of a parabola, whose focus is (2,3) and directrix is the line x - 4y +3=0 is
a) 7/√17 b) 14/√21 c) 7/√21 d) 14/√17. D
3) Find the equation of the parabola whose focus is (-6,6) and vertex is (-2,2). (2x - y)² + 194x + 148y - 124=0.
4) The extreme points of the latus rectum of a parabola are (7,5) and (7,3). Find the equation of the parabola. (y-4)² = 2(x - 6.5) meets at x-axis (14.5,0) and (y -4)² = 2(x - 6.5) and meets metts the x-axis at (14.5,0) and the equation of second parabola is (y- 4)² = -2(x - 7.5). Metts at x-axis at (-0.5,0)
5) Name the conic represented by the equation √(ax)+ √(by)=1, where a, b, ∈R, a,b> 0. Parabola
6) Find the vertex, Axis, ficus, directorix, latus rectum of the parabola. (-7/2,5.2), y= 5/2, (-17/4, 5/2), x= -11/4, 3
7) Find the equation of the parabola focus is (1,-1) and whose vertex is (2,1). also find its axis and the latus rectum. 4x²+ y²- 4xy + 8x + 46y - 71=0, 2x - y=3, 4√5
8) Find the equation of the parabola whose latus rectum is 4 units, axis is the line 3x + 4y= 4 and the tangent at the vertex is the line 4x - 3y +7=0. (3x + 4y -4)²= 20(4x - 3y +7)
9) Through the vertex O of a parabola y²= 4x chords OP and OQ are drawn at right angles to one another. Show that for all position of P, PQ cuts the axis of the parabola at a fixed point.
10) Find the value of a for which the point (a²-1, a) lies inside the parabola y²= 8x. (-∞-√(8/7) U (√(8/7, ∞)
11) The focal distance of a point on the parabola (x -1)²= 16(y -4) is 8. Find the coordinates. (-7,8),(9,8)
12) Show that the focal chord of the parabola y²= 4ax makes an angle α with x-axis is of length 4a cosec²α.
13) Find the condition that the straight line ax + by + c=0 touches the parabola y²= 4kx. kb²= ac
14) Find the length of the chord of the parabola y²= 8x, whose equation is x + y = 1. 8√3
15) A tangent to the parabola y²= 8x makes an angle of 45° with the straight line y= 3x +5. Find its equation and its point of contact. y= x/2 +4 , (8,8)
16) Find the equation of the tangents to the parabola y²= 9x which go through the point (4,10). y= x/4 + 9, y= 9x/4 +1
17) Find the locus of the point P from which tangents are drawn to the parabola y²= 4ax having slopes m₁ and m₂ such that
i) m₁²+ m₂²= λ (constant)
ii) θ₁ - θ₂ = θ₀ (constant)
Where θ₁ and θ₂ are the inclination of the tangents from positive x-axis. y²- 2ax = λ x², y²- 4ax = (x + a)² tan²θ₀
18) Find the equation of the tangent to the parabola y²= 12x, which passes through the point (2,5). Find also the coordinates of their points of contact. x - y+3=0, (3,6); 3x - 2y+4=0, (4/3,4)
19) Find the equation of the tangents to the parabola y²= 16x, which are parallel and perpendicular respectivaly to the line 2x - y+5=0. Find also the coordinate of their points of contact. 2x - y +2=0, (1,4); x + 2y +16=0, (16,-16)
20) Prove that the locus of the point of intersection of tangents to the parabola y²= 4ax which meets at an angle θ is (x + a)² tan²θ= y²- 4ax.
Sap-2
1) Prove that the normal chord to a parabola y²= 4ax at the point whose ordinates is equal to the abscissa suntends a right angle at the focus.
2) if two normal drones from any point to the parabola y¹= 4ax make an angle α and β with the axis such that tanα. tan β=2, then find the locus of this point. y²= 4ax
3) If 3 distinct and real normals can be drawn to y²= 8x from the point (a,0), then
a) a> 2 b) a ∈ (2,4) c) a> 4 d) none. C
4) Find the number of distinct normal that can be drawn from (-2,1) to the parabola y²- 4x - 2y -3=0. 1
5) If 2x + y + k =0 is a normal to the parabola y² = -16x, then find the value of k. 48
6) three normals are drawn from the point (7,14) to the parabola x²- 8x - 16y=0. Find the co-ordinate of the feet of the normals. (0,0),(-4,3) and (16,8)
7) The angle between the tangents drawn from a point (-a,2a) to y²= 4ax is
a) π/4 b) π/2 c) π/3 d) π/6
8) The circle drawn with variable chord x + ay -5=0 (a being a parameter) of the parabola y²= 20x as diameter will always turns the line
a) x+5=0 b) y +5=0 c) x+ y+ 5=0 d) x - y +5=0. A
9) If the equation λ²x + λy - λ¹+ 7=0 represents a family of lines where λ is parameter, then find the equation of the curve to which these lines will always be tangents . (y+2)²= 28(x -1)
10) Find the angle between the tangents drawn from the origin to the parabola y²= 4a(x - a). π/2
11) Uf the line x - y -1=0 intersect the parabola y²= 8x at P and Q, then find the point of intersection of tangents at P and Q. (-1,4)
12) Find the locus of point whose chord of contact w.r.t. to the parabola y²= 4bx is the tangent of the parabola y²= 4ax. y²= 4b²x/a
13) Find the locus of the middle point of the chord of the parabola y²= 4ax which passes through a given (p,q). y²- 2ax - qy + 2ap=0.
14) Find the locus of the middle point of a chord of a parabola y²= 4ax which subtends a right angle at the vertex. y²= 2a(x - 4a)
15) Find the equation of the chord of contacts of tangents drawn from a point (2,1) to the parabola x²= 2y. 2x = y +1
16) Find the coordinates of the middle point of the chord of the parabola y²= 16x, the equation of which is 2x - 3y +8= 0. (14,12)
17) Find the locus of the midpoint of the chords of the parabola y²= 4ax such that tangent at the extremities of the chords are perpendicular. y²= 2a(x - a)
18) The parabola y²= 4x and x²= 4y divide the square region bounded by the line x= 4, y= 4 and the coordinates axes. If S₁, S₂, S₃ are respectively the areas of these parts numbered from top to bottom then find S₁ : S₂ : S₃. 1:1:1
19) Let P be the point (1,0) and Q a point on the parabola y²= 8x, then find the locus of the midpoint of PQ. y²- 4x +2=0
20) The common tangent of the parabola y²= 8ax and the circle x²+ y²= 2a² is
a) y= x+ a b) x+ y+ a=0 c) x+ y+ 2a=0 d) y= x+ 2a
21) If the tangent to the parabola y²= 4ax meets the axis in T and tangent at the vertex A in Y and the rectangle TAYG is completed, show that the locus of G is y²+ ax=0.
22) If P(-3,2) is one end of the focal chord PQ of the parabola y²+ 4x + 4y=0, then the slope of the normal at Q is
a) -1/2 b) 2 c) 1/2 d) -2
23) Show that the two parabolas y²= 4ax and y²= 4c(x - b) can not have common normal, other than the axis unless b/(a- c) > 2.
24) If r₁, r₂ be the length of the perpendicular chord of the parabola y²= 4ax drawn through the vertex, then show that (r₁r₂)⁴⁾³ = 16a²(r₁²⁾³ + r²⁾³).
25) The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangent at these points.
26) Show that the orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.
ELLIPSE
Sap-1
1) LR of an ellipse is half of its minor axis, then its eccentricity is
a) 3/2 b) 2/3 c) √3/2 d) √2/3
2) Find the equation of the ellipse whose foci are (2,3),(-2,3) and whose semi minor axis is of length √5. x²/9 + (y-3)²/5=1
3) Find the equation of the ellipse having centre at (1,2), one focus at (6,2) and passing through the point (4,6). (x -1)²/45 + (y-2)²/20 =1
4) If LR of an ellipse x²/a² + y²/b² =1, (a< b) is half of its major axis, then find its eccentricity. 1/√2
5) Find the equation of the ellipse whose foci are (4,6) and (16,6) and whose semi minor axis is 4. (x-10)²/52 + (y-6)²/16 =1
6) Find the eccentricity, foci and the length of the latus rectum of the ellipse x²+ 4y²+ 8y - 2x +1=0. √3/2, (1±√3,-1); 1
6) The equation of the ellipse with respect to coordinate axes whose axis is equal to the distance between its foci and whose LR= 10, will be
a) x²+ y² =100 b) x²+ 2y² =100 c) 2x²+ 3y² =80 d) none. AB
7) The foci of an ellipse are (0,±2) and its eccentricity is 1/√2. Find its equation. x²/4 + y²/8 =1
8) Find the centre, the length of the axes, eccentricity and the foci of ellipse 12x²+ 4y² + 24x - 16y +25 =0. (-1,2),√3, 2a=1, √(2/3); (-1, 2± 1/√2)
9) The equation x²/(8- t) + y²/(t -4) =1, will represent an ellipse
a) t∈ (1,5) b) t∈ (2,8) c) t∈ (4,8) - {6} d) t∈ (4,10) - {6} c
10) For what value of λ does the line y= x + λ touches the ellipse 9x²+ 16y² =144. ±5
11) If α, β are eccentricity angles of end points of a focal chord of the ellipse x²/a² + y²/b² =1, then tan(α/2). tan(β/2) is equal to
a) (e -1)/(e+1)
b) (1- e)/(1+ e)
c) (e + 1)/(e-1)
s) (1 + e)/(1- e). Ac
12) Find the position of the point (4,3) relative to the ellipse 2x²+ 9y² =113. On
13) A tangent to the ellipse x²/a² + y²/b² =1, (a> b) having slope -1 intersect the axis of x and y in point A and B respectively. If O is the origin then find the area of triangle OAB. (1/2)(a²+ b²)
14) Find the condition for the line x cosθ + y sinθ = P to be a tangent to the ellipse x²/a² + y²/b² =1, P¹= a² cos²θ + b²sin²θ.
15) Find the equations of the tangents to the ellipse 3x²+ 4y²=12 which are perpendicular to the line y+ 2x=4. x - 2y ±4=0
16) The tangent at a point P on an ellipse intersects the major axis in T and N is the foot of the perpendicular from P to the same axis. Show that the circle drawn on NT as diameter intersects the auxiliary circle orthogonally.
17) Find the equation of the tangents to the ellipse 9x²+ 16y²=144 which are parallel to the line x+ 3y + k=0. 3y+ x ±√97=0
18) Find the equation of the tangent to the ellipse 7x²+ 8y²=100 at the point (2,-3). 7x - 12y=50
19) Find the condition that the line lx + my = n may be a normal to the ellipse x²/a² + y²/b² =1. n²/(a²- b²)². ( a²/l² + b²/m²)
20) If the normal at an end of a latus rectum of an ellipse x²/a² + y²/b²=1 passes through one extremity of the minor axis , show that the eccentricity of the ellipse is given by e= √{(√5 -1)/2}.
21) P and Q are corresponding points on the ellipse x²/a² + y²/b² =1 and the auxiliary circles respectively. The normal at P to the ellipse meets CQ in R, where C is the centre of the ellipse. Prove that CR= a+ b.
Sap-2
1) Find the equation of the normal to the ellipse 9x²+ 16y²=288 at the point (4,3). 4x - 3y=7
2) let P be a variable point on the ellipse x²/a² + y²/b² =1 with foci F₁ and F₂. If A is the area of the triangle PF₁F₂, then find maximum value of A. abe
3) If the normal at the point to the point P(θ) to the ellipse x²/3+ y²/2=1 intersects it again at the point Q(2θ), then find cosθ. -1
4) Show that for all real values of 't' the line 2tx + y √(1- t²)= 1 touches a fixed ellipse. Find the eccentricity of the ellipse. √3/2
5) If the tangents to the parabola y²= 4ax intersect the ellipse x²/a²+ y²/b² = 1 at A and B, then find the locus of the point of intersection of tangents at A and B. y²= b⁴x/a³.
6) Find the equation of the chord of contact to the ellipse x²/16 + y²/9 = 1 at the point (1,3). x/16 + y/3=1
7) if the chord of contact of tangents from 2 points (x₁, y₁) and (x₂, y₂) to the ellipse x²/a² + y²/b² = 1 are at right angles , then find x₁x₂/(y₁, y₂). - a⁴/b⁴
8) If a line 3x - y = 2 intersect ellipse x²/8 + y²/4 = 1 at a point A and B, then find coordinates of point of intersection of tangents at point A and B. (12,-2)
9) A tangent to the ellipse to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x + 2y² = 6 are at right angles.
10) Find the locus of the midpoint of the focal chords of the ellipse x²/a² + y²/b² = 1. - ex/a = x²/a² + y²/b²
11) Find the equation of the chord of the ellipse x²/16 + y²/9 = 1 whose mid point be (-1,1). -9x + 16y = 25
12) A man running round a race course note that sum of the distance of two flag posts from him is always 20 metres and distance between the flag posts is 16 m. Find the area of the path be encloses in square metres. 60π
13) if chord of contact of the tangent drawn from the point (α,β) to the ellipse x²/a². + y²/b² = 1 touches the circle x² + y² = k², then find the locus of the point (α,β). x²/a⁴ + y²/b⁴ = 1/k²
14) A point moves so that the sum of the squares of its distances from two intersecting straight lines is constant. prove that its locus is an ellipse.
15) Find the condition on a and b for which two distinct chords of the ellipse x²/2a² + y²/2b² = 1 passing throughn( a,-b) are bisected by the line x + y= b. a²+ 6ab ≥ 7b²
16) Any tangent to an ellipse is cut by the tangents at the ends of the major axis in T and T'. Prove that circle on TT' as diameter passes through foci.
17) A variable point P on an ellipse of eccentricity e, is joined to its foci S, S'. Show that the locus of the incentre of the triangle PSS' is an ellipse whose eccentricity is √{2e/(1+ e)}.
SAP- 3
HYPERBOLA
SAP-1
₁₂₃ ²² ∈ ∈₁₂²²₁₂²²₁₂θθ₁₂ θ ₁₂₁₂₁₂₂₁²²²²²²²²²²²²²²²²₁₁₂₂₁₂₁₂
₁₂ ∞∞∞∞∞∞∞∞ ∞ λ αβ
LIMIT
Things to Remember
1. Limit of a function f(x) is said to exist as, x -> a when lim ₓ→₀⁻ f(x)= lim ₓ→ₐ⁺ f(x)= finite quantity.
2. FUNDAMENTAL THEOREMS ON LIMIT
Let limₓ→ₐ f(x)= l & limₓ→ₐ g(x)= m. If l & m exists then:
i) limₓ→ₐ f(x) ± g(x)= l ± m
ii) limₓ→ₐ f(x). g(x)= l.m
iii) limₓ→ₐ f(x)/g(x)= l/m, provided m≠ 0
iv) limₓ→ₐ k f(x)= k limₓ→ₐ f(x); where k is a constant.
v) limₓ→ₐ f[g(x)]= f(limₓ→ₐg(x))= f(m); provided f is continuous at g(x)= m.
For example limₓ→ₐ lnf(x)= ln[limₓ→ₐf(x)]= ln l(l > 0).
3) STANDARD LIMITS
a) limₓ→₀ (sinx)/x= 1 = limₓ→₀ (tanx)/x=limₓ→₀ (tan⁻¹x)/x = limₓ→₀ (sin⁻¹x)/x
(Where x is measured in radians)
b) limₓ→₀ (1+ x)¹⁾ˣ = e= limₓ→∞ (1 + 1/x)ˣ note however
There are limₕ→₀ and limₙ→∞ (1- h)ⁿ= 0
and limₓ→₀ and limₙ→∞ (1+ h)--> ∞
c) limₓ→ₐf(x)= 1 and limₓ→ₐ φ(x)->∞, then
limₓ→ₐ[f(x)]ᵠ⁽ˣ⁾ = ₑlimₓ→ₐ φ(x)[(x) -1]
d) limₓ→ₐ f(x)= A> 0 & limₓ→ₐ φ(x)= B (a finite quantity) then ;
limₓ→ₐ [f(x)]ᵠ⁽ˣ⁾ = e² where z =limₓ→ₐ φ(x). ln[f(x)] = eBˡⁿᴬ = Aᴮ
e) limₓ→₀ (aˣ -1)/x = ln a(a> 0). In particular limₓ→₀ (eˣ -1)/x = 1
f) limₓ→ₐ (xⁿ - aⁿ)/(x - a)= n a ⁿ⁻¹.
4) SQUEEZE PLAY THEOREM
If f(x)≤ g(x)≤ h(x) ∀ x & limₓ→ₐ f(x)= l = limₓ→ₐ g(x)= l.
5) INDETERMINANT FORMS:
0/0, ∞/∞, 0 x ∞, 0⁰, ∞⁰, ∞ - ∞ and 1^∞
Note:
i) We can not help ∞ on the paper, Infinity (∞) is a symbol and not a number. It doesn't obey the law of elementary algebra.
ii) ∞ + ∞ = ∞
iii) ∞ x ∞ = ∞
iv) (a/∞)= 0 if a is finite
v) a/0 is not defined, if a≠ 0
vi) ab= 0, if and only if a= 0 or b= 0 and a & b are finite.
6) The following strategies should be born in mind for evaluating the limits:
a) Factorisation
b) Rationalization or double rationalization
c) Use trigometric transformation ;
appropriate substitution and using standard formula
d) Expansion of function like Binomial expansion, exponential and logarithmic expansion, expansion of sinx, cosx, tanx should be remembered by heart and are given below:
i) aˣ = 1+ x ln a/1! + x² ln² a/2! + x³ ln³ a/3! + ....., a> 0
ii) eˣ = 1+ x/1! + x²/2! + x³/3! + ....., x ∈R
iii) ln (1+ x) = x - x²/2 + x³/3 - x⁴/4 + ....., for -1< x≤1.
iv) sinx = x - x³/3! + x⁵/5! + x⁷/7! + ....., x ∈(-π/2,π/2)
v) cosx = 1 - x²/2! + x⁴/4! + x⁶/6! + ....., x ∈(-π/2,π/2)
vi) tanx = x + x³/3 + 2x⁵/15 + ....., x ∈(-π/2,π/2)
vii) tan⁻¹x = x - x³/3 + x⁵/5 - x⁷/7 + .....,
SAP -1
1) lim ₓ→₁ (x²- x log x + log x-1)/(x -1). 2
2) lim ₓ→₁ {[¹⁰⁰ₖ₌₁∑ xᵏ] -100}/(x -1). 5050
3) limbx->π/4 (1- tanx)/(1- √2 sinx). 2
4) lim ₓ→₁ {p/(1- xᵖ) - q/(1- xᑫ), p, q ∈N. (p- q)/2
5) Find the sum of an infinite geometric series whose first term is the limit of the function f(x)= (tanx - sinx)/sin³x as x--> 0 and whose common ratio is the limit of the function g(x)= (1- √x)/(cos⁻¹x)² as x--> 1. a=1/2; r=1/4; S=2/3
6) lim-->∞ (x + ln cosh x) where cosh= (eᵗ + ⁻ᵗ)/2. ln 2
7) a) Limₓ→₁/√₂ cos⁻¹{2x √(1- x²)}/(x - 1/√2). Does not exist
b) Limₓ→π/4 √{1- √sin2x)}/(π - 4x). Does not exist
c) Limₓ→ ₋₇ {[x²] + 15[x]+ 56}/{sin(x +7) sin(x +8)}. Where [ ] denotes the greatest integer function. 0
8) Limₓ→3π/4 (1+ ³√tanx)/(1- 2 cos²x). -1/3
9) Limₓ→₀ (8/x⁸) [1- cos(x²/2) - cos(x²/4) + cos(x²/2) cos(x²/4)]. 1/32
10) Lim θ->π/4 (√2- cosθ - sinθ)/(4θ -π)². 1/16√2
11) limₕ->₀ [sin(π/3+ 4h) - 4 sin(π/3+ 3h) + 6 sin(π/3+ 2h) - 4 sin(π/3+ h) + sin(π.3)]/h⁴. √3/2
12) Lim x-> ∞ x²[√{(x +2)/x} - ³√{(x +3)/x}]. 1/2
13) Lim x-> ∞ [3x⁴+ 2x²) sin(1/x) + |x³| +5]/[|x²+ |x²|+ |x| +1]. -2
14) If l=Lim x-> ∞ ⁿᵣ₌₂∑ {(r +1) sin{π/(r +1)} - r sin(π/3)} then find {l}. Where {} denotes the fractional part function). π-3
15) i) Find a, b if :Lim x-> ∞[(x ² +1)/(x +1) - ax - b]= 0. a=1, b=-1
ii) Lim x-> ∞ [√(x² - x +1) - ax - b]= 0. a=-1, b= 1/2
16) lim ₓ→₀ [ln (1+ sin²x). cot(ln² (1+ x))]. 1
17) lim ₓ→₁ [+ln (1+ x) - ln2) (3. 4ˣ⁻¹ - 3x)}/[{(7 + x)¹/³ - (1+ 3x)¹/²} sin(x -1)]. -9/4 ln(4/e)
18) lim ₓ→₀ (27ˣ - 9ˣ - 3ˣ +1)/{√2 - √(1+ cosx)}. 8√2(ln 3)²
19) Let
f(x)= x/sinx, x> 0 & g(x)= x+3, x< 1
= 2- x, x≤ 0 = x²-2x -2, 1≤x<1
= x -5, x≥ 2
Find LHL and RHL of g(f(x) at x= 0 and hence find lim ₓ→₀ g(f(x)). -3,-3,-3
20) a) lim ₓ→₀ tan⁻¹(a/x²), where a ∈ R. π/2 if a> 0; 0 if a= 0 and -π/2 if a< 0
b) Plot the graph of the function f(x)= limₜ→₀ (2x/π tan⁻¹(x/t²)). f(x)=|x|
21) Let Pₙ = ₐPₙ₋₁ -1 , ∀ n= 2,3,..... and Let P₁ = aˣ -1 where a∈ R⁺ then evaluate lim ₓ→₀ (Pₙ/x). (ln a)ⁿ
22) If the lim ₓ→₀ (1/x³){1/√(1+ x) - (1+ ax)/(1+ bx)} exists and has the value equal to l, then find the value of 1/a - 2/l + 3/b. 72
23) Lim ᵧ→₀ [limₓ→∞ {exp(x ln(1+ ay/x)) - exp(x ln (1+ by/x))}/y]. a- b
24) Let {a}, {bₙ} , {cₙ} be sequences such that
i) aₙ + bₙ + cₙ = 2n +1
ii) aₙbₙ + bₙcₙ + cₙaₙ = 2n -1
iii) aₙbₙcₙ = -1
iv) aₙ< bₙ< cₙ.
Then find the value of lim ₙ₋∞ (naₙ). -1/2
25) Let f(x)= ax³+ bx²+ cz + d and g(x)= x²+ x -2.
If lim ₓ→₁ f(x)/g(x)= 1 and lim ₓ→₋₂ f(x)/g(x)= 4, then find the value of (c² + d²)/(a² + b²). 16
26) lim ₓ→∞ [2x² +3)/(2x² +5)]^(8x²+3). e⁻⁸
27) lim ₓ→∞ {(x + c)/(x - c)}ˣ= 4 then find c. ln 2
28) (tan(πx/4))^tan(πx/2). e⁻¹
29) lim ₓ→₀{(x - 1 + cosx)/x}¹⁾ˣ. e⁻¹/²
30) If n ∈N and aₙ = 2² + 4² + 6² + .....(2n)² and bₙ= 1²+ 3² + 5² + .....+ (2n -1)².
Find the value of lim ₙ→∞ (√aₙ - √bₙ)/√n. √3/2
31) The graph of f and g are given. Use thm to evaluate each limit.
Column I
A) limₓ→₁ f(g(x))
B) lim x->₂ √(3f(x) -2)
C) lim ₓ→₀ f(x)/g(x) + f(x) g(x)
D) lim->ₓ_ ₁⁺ (3 f(x) - g(x))/(f(x) + g(x))
Column II
P) 1
Q) does not exist
R) 0
S) 2. AQ BS CR DO
SAP-2
1) lim ₓ→∞ x² sin(ln √cos(π/x)). -π²/4
2) lim ₓ→∞ [cos{cos{2π{x/(1+ x)}}ᵃ}]^(x²) a ∈ Q. ₑ-2π²a²
3) Let f(x)= [sin⁻¹(1- {x}) cos⁻¹(1- {x})/[√(2{x}). (1- x)] , then find lim ₓ→₀⁺ then find lim ₓ→₀⁻ f(x), where {x} denotes the fractional part function. π/2, π/2√2
4) limₙ-->∞ [{√(n² + n) -1}/n]^2√(n²+ n -1). e⁻¹
5) lim ₓ→∞ {a₁¹⁾ˣ + a₂¹⁾ˣ + a₃¹⁾ˣ + .....+ aₙ¹⁾ˣ}ⁿˣ/n where a₁, a₂, a₃, .....aₙ > 0. (a₁, a₂, a₃, .....aₙ)
6) lim ₓ→₀ [(1+ x)¹⁾ˣ/e]¹⁾ˣ. e⁻¹⁾²
7) lim ₓ→∞ {cosh(π/x)/cos(π/x)}^x² where cosh (eᵗ + e⁻ᵗ)/2. ₑπ²
8) lim ₓ→ₐ 1/(a² - x²)²[(a²+ x²)/ax - 2 sin(aπ/2) sin(πx/2)] where a is an odd integer. (π²a²+4)/16a⁴
9) If L= limₓ→₁ {(1- x)(1- x²)(1- x³).....(1- x²ⁿ)}/{(1- x)(1- x²)(1- x³)....(1- xⁿ)}² then show that L can be equal to
a) ⁿᵣ₌₁ Π (n + r)/r
b) (1/n!) ⁿᵣ₌₁ Π (4r -2)
c) The sum of the coefficients of two middle terms in the expansion of (1+ x)²ⁿ⁻¹.
d) The coefficient of xⁿ in the expansion of (1+ x)²ⁿ.
10) If lim ₓ→∞ {a(2x³ - x²)+ b(x³ + 5x² -1) - c(3x³ + x²)}/{a(5x⁴ - x) - bx⁴ + c(4x⁴ +1) + 2x² + 5x}= 1, then the value of (a+ b+ c) can be expressed in the lowest form as p/q. Find the value of (p + q). 167
11) Let x₀ = 2 cos(π/6) and xₙ =√(2+ xₙ₋₁), n= 1,2,3,..... Find limₙ-->∞ 2ⁿ⁺¹. √(2- xₙ). π/3
12) lim ₓ→₀ [{ln(1+ x)¹⁺ˣ}/x² - 1/x]. 1/2
13) Let ⁿₙ₌₃Π (1- 4/n²) ;
M= ⁿₙ₌₂ Π (n³ -1)/(n³ +1) and
M= ⁿₙ₌₂ Π (n³ -1)/(n³ +1) and
N= ⁿₙ₌₁ Π (1+ n⁻¹)²/(1+ 2n⁻¹), then find the value of L⁻¹ + M⁻¹ + N⁻¹. 8
14) A circular arc of radius 1 subtends an angle of x radians, 0< x < π/2 as shown in the figure.
The point C is the intersection of the two tangent lines at A and B. Let T(x) be the area of triangle ABC and let S(x) be the area of the shaded region. Compute:
a) T(x)
b) S(x)
c) the limit of T(x)/S(x) as x_ 0. T(x)= (1/2) tan²(x/2) sinx or tan(x/2) - (sinx)/2, S(x)= x/2 - (sinx)/2, limit= 3/2
15) Let f(x)= limₙ-->∞ ⁿₙ₌₁∑ 3ⁿ⁻¹ sin³(x/3ⁿ) and g(x)= x - 4 f(x). Evaluate lim ₓ→₀ (1+ g(x))ᶜᵒᵗˣ . g(x)= sinx and l= e
16) If f(n, θ)= ⁿᵣ₌₁ Π(1- tan¹(θ/2ʳ)), then compute limₙ-->∞ f(n, θ). θ/tanθ
17) If lim ₓ→₀ ln(x ln a) ln{(ln ax)/ln(x/a)}= 6, then find the value of a. e³
18) Evaluate lim ₓ→∞ {x/e - x{x/(x +1)}ˣ}. -1/2e
19) f(x) is the function such that lim ₓ→₀ f(x)/x = 1. If lim ₓ→₀ {x(1+ a cosx) - b sinx}/(f(x))³ = 1, then find the value of a and b. -5/2,-3/2
20) Through a point A on a circle, a chord AP is drawn and on the tangent at A a point T is taken such that AT= AP. If TP produced meet the diameter through A at Q. Prove that the limiting value of AQ when P moves upto A is double the diameter of the circle.
21) At the end points A, B of the fixed segment of length L, lines are drawn meeting in C and making angles θ and 2θ respectively with the given segment. Let D be the foot of the altitude CD and let x represents the length of AD. Find the value of x and θ tends to zero i.e., lim θ→₀ x. 2L/3
22) At the end points and the midpoint of a circular arc AB, tangents lines are drawn, and the points A and B are joined with a chord. Prove that the ratio of the areas of the two triangles this formed tends to 4 as the arc AB decreases indefinitely. 4
23) If L= lim ₓ→₀{1/ln(1+ x) - 1/ln(x +√(1+ x²))} then find the value of (L + 153)/L. 307
24) Let f(x)= limₙ-->∞ (2x²ⁿ sin(1/x) + x)/(1+ x²ⁿ) then find
a) lim ₓ→∞x f(x). 2
b) lim ₓ→₁f(x). D. N. E
c) lim ₓ→₀f(x). 0
d) lim ₓ→∞ f(x). 0
25) Using Sandwich theorem, evaluate
a) limₙ-->∞{1/√n² + 1/√+n² +1) + 1/√(n² +2) + .....+ 1/√(n² + 2n)}. 2
b) limₙ-->∞ 1/(1+ n²) + 2/+2+ n²) + ....n/(n + n²). 1/2
Comprehension Type Questions
Let C₁, C₂ be two circles of unit radius with centres A and B and L be their common tangent as shown in adjacent figure. Consider a variable point R on L from which tangents RP and RQ are drawn to C₁ and C₂ respectively. Let area of the shadeed region be S(θ) and that of triangle PQR be T(θ). Let length of the arc CP be l.
On the basis of above information, answer the following questions:
26) lim θ→₀ PQ/l² is
a) 0 b) 1 c) 1/2 d) 4
27) lim θ→₀ S(θ)/k³ is
a) 1/12 b) 1/24 c) 3/4 d) 0
28) lim θ→₀ T(θ)/l is
a) 1 b) 1/2 c) 2 d) 0
26b 27a 28d
Sap-3
1) ) lim ₓ→₀ sin(π cos²x)/x² equal
a) -π b) π c) π/2 d) 1
2) Evaluate: lim ₓ→₀ (aᵗᵃⁿˣ - aˢᶦⁿˣ)/(tanx - sinx), a> 0
3) The integer n for which lim ₓ→₀ {(cosx -1)(cosx - eˣ)}/xⁿ is a finite nonzero number is
a) 1 b) 2 c) 3 d) 4
4) If lim ₓ→₀ {sin(nx) [(a - n)nx - tanx]/x²= 0 (n> 0) then the value of a is equal to
a) 1/n b) n² +1 c) (n²+1)/n d) none
5) Find the value of limₙ-->∞ [2(n +1)/π cos⁻¹(1/n) - n].
6) Let L= lim ₓ→₀ {a - √(a²- x²) - x²/4}/x⁴, a> 0. If L is finite, then
a) a= 2 b) a= 1 c) L = 1/64 d) L= 1/32
1b 2) ln a 3c 4c 5) 1- 2/π 6 ac
CONTINUITY
Things to Remember
1) A function f(x) is said to be continuous at x= c, if lim ₓ→꜀ = f(c). Symbolically f is continuous at x= c if lim ₕ→₀ f(c - h)= lim ₕ→₀ f(c + h)= f(c).
i.e., LHL at x= c = RHL at x= c equals value of 'f' at x = c.
It should be noted that continuity of a function at x= a is meaningful only if the function is defined in the immediate neighborhood of x= a, not necessarily at x= a.
2) REASON OF DISCONTINUITY
i.e lim ₓ→꜀⁻ f(x)≠ lim ₓ→꜀⁺ f(x)
ii) f(x) is not defined at x= c
iii) lim ₓ→꜀ f(x) ≠ f(c)
Geometrically, the graph of the function will exhibit a break at x= c. The graph as shown is discontinuous at x= 1,2 and 3.
3) TYPES OF DISCONTINUTIES:
Type 1: (Removable type of discontinuity)
In case lim ₓ→꜀ f(x) exists but not equal to f(c) then the function is said to have a removable discontiniuty or discontiniuty of the first kind. In this case we can redefine the function such that limit ₓ→ₐ f(x)= f(c) and make it continuous at x= c. Removable type of discontinuity can be further classified as:
a) Missing Point Discontiniuty: Where lim ₓ→ₐ f(x) exists finitely but f(a) is defined.
e.g., f(x) = (1- x)(9- x²)/(1- x) has a missing point discontiniuty at x= 1, and f(x) = (sinx)/x has a missing point of discontinuity at x= 0.
b) Isolated Point Discontiniuty: Where lim ₓ→ₐ f(x) exists and f(a) also exists but; lim ₓ→ₐ ≠ f(a). e.g.f(x)= (x²-16)/(x -4), x≠ 4 and f(4)= 9 has an isolated point discontiniuty at x= 4.
Similarly
f(x) =[x] + [- x]= [ 0 if x ∈ I
-1 if x ∉ I
has an isolated point discontiniuty at x ∈ I.
Type-2: (Non-Removable type of discontinuous)
In case lim ₓ→꜀ f(x) does not exist then it is not possible to make the function continuous by redefining it. Such discontinuous are known as Non-Removable discontiniuty or discontiniuty of the 2nd kind. Non-Removable type of discontiniuty can be further classified as:
a) Finite discontiniuty e.g., f(x)= x - [x] at all integral x; f(x)= tan⁻¹(1/x) at x= 0 and f(x)= 1/(1+ 2¹⁾ˣ) at x= 0 (note that f(0⁺)= 0; f(0⁻)= 1)
b) Infinite discontiniuty e.g., f(x)= 1/(x -4) or g(x)= 1/(x -4)² at x= 4; f(x)= 2ᵗᵃⁿˣ at x=π/2 and f(x)= (cosx)/x at x= 0.
c) Oscillatory discontiniuty e.g., f(x)= sin(1/x) at x= 0.
In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but limit ₓ→ₐ does not exist.
Note: From the adjacent graph note that
--> f is continuous at x= -1
--> f has Isolated discontiniuty at x= -1
--> f has missing point discontiniuty at x= 2
--> f has non removable (finite type)
discontiniuty at the origin.
4) In case of discontinuity of the second kind the non-negative difference between the value of the RHL at x= c and LHL at x= c is called The Jump Of Discontiniuty. A function having a finite number of jumps in a given interval I is called a Piece Wise Continuous Or Sectionally Continuous function in this interval.
5) All Polynomials, Trigometrical functions, exponential and logarithmic function are continuous in their domains.
6) If f and g are two functions that are continuous at x= c then the functions defined by:
F₁(x)= f(x) ± g(x); F₂(x)= Kf(x), K any real number; F₃(x)= f(x). g(x) are also continuous at x= c
Further, if g(c) is not zero, then F₄(x)= f(x)/g(x) is also continuous at x= c
7) THE INTERMEDIATE VALUE THEOREM:
Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if y₀ is a number between f(a) and f(b), their exits a number c between a and b such that f(c)= y₀.
The function f, being Continuous in [a,b) takes on every value between f(a) and f(b)
Note Very Carefully
a) If f(x) is continuous and g(x) is discontinuous at x= a then the product function φ (x)= f(x).g(x) is not necessarily be discontinous at x= a
e.g., f(x)= x and
g(x)= sin(π/x) , x≠ 0
0, x= 0
b) If f(x) and g(x) both are discontinuous at x= a then the product function φ(x)= f(x). g(x) is not necessarily be discontinous at x= a.
e.g., f(x) = - g(x) = 1, x≥ 0
-1, x< 0
c) A continuous function whose domain is closed must have a range also in closed interval.
d) If f is continuous at x= c and g is continuous at x=f(c) then the composite g[f(x)] is continuous at x= c. e.g., f(x) = (x sinx)/(x² +2) and g(x)= |x| are continuous at x= 0, hence the composite (g o f)(x)= |x sinx/(x²+2)| will also be continuous at x= 0.
7) CONTINUITY IN AN INTERVAL:
a) A function f is said to be continuous in (a, b) if f is continuous at each and every point ∈ (a, b).
b) A function f is said to be continuous in a closed interval [ a, b] if:
i) f is continuous in the open interval (am b)
ii) f is right continuous at 'a' i.e., lim ₓ→ₐ⁺ f(x)= f(a) = a finite quantity.
iii) f is left continuous at 'b' i.e., lim ₓ→b⁻ f(x) = f(b) = a finite quantity.
Note that a function f which is continuous in [a,b] possess the following properties:
i) If f(a) and f(b) possess opposite signs, then there exists atleast one solution of the equation f(x)= 0 in the open interval (a,b).
ii) If K is any real number between f(a) and f(b), then there exists atleast one solution of the equation f(x)= K in the open interval (a,b).
8) SINGLE POINT CONTINUITY
Function which are continuous only at one point are said to exhibit single point continuity e.g.,
f(x)= x if x ∈Q
-x if x ∉ Q and
g(x)= x if x ∈Q
0 if x ∉ Q
are both continuous only at x= 0
SAP-1
1) If the function f(x)= (3x² + ax + a +3)/(x² + x -2) is continuous at x= -2. Find f(-2). -1
2) Find all possible values of a and b so that f(x) is continuous for all x∈R if
|ax +3|, if x≤ -1
|3x + a|, if -1< x ≤ 0
f(x= (b sun2x)/x - 1b, if 0< x< π
cos²x -3 , if x≥π. 0,1
3) (6/5)ᵗᵃⁿ⁶ˣ/ᵗᵃⁿ⁵ˣ, if 0< x<π/2
f(x)= b+2, if x=π/2
(1+ |cosx|}ᵃ|ᵗᵃⁿˣ|/ᵇ if π/2< x < π
Determine the value of a and b, if f is continuous at x=π/2. 0,-1
4) Suppose that f(x)= x³ - 3x² - 4x +12 &
h(x)= f(x)/(x -3), x≠ 3
K, x= 3
a) find all zeros if f(x). -2,2,3
b) find the value of K that makes h continuous at x= 3, 5
c) using the value of K found in (b(, determine whether h is an even function. Even
5) Let yₙ(x)= x² + x²/(1+ x²) + x²/(1+ x²)² + .....+ x²/(1+ x²)ⁿ⁻¹ and y(x)= limₙ₋∞ yₙ(x)
Discuss the continuity of yₙ(x) (n∈N) and y(x) at x= 0. yₙCon at x= 0, y(x)dis at x= 0
6) (1- sinπx)/(1+ cos2πx), x< 1/2
f(x)= p, x= 1/2
√(2x -1)/(√(4+ √(2x-1) - 2)), x> 1/2
Determine the value of p, if possible, so that the function continuous at x= 1/2. P not possible
7) Given the function f(x)= √(6- 2x) and h(x)= 2x² - 3x + a. Then
a) evaluate h(g(2)). 4- 3√2+ a
b) if f(x)= g(x), x≤ 1
h(x) x> 1
Find a solution that f is continuous. 3
8) Let f(x)= 1+ x, 0≤ x≤2
3- x, 2< x≤3
Determine the form of g(x)= f[f(x)] and hence find the point of discontinuity of g, if any. g(x)= 2+ x for 0≤x ≤1, 2- x for 1< x≤2, 4- x for 2< x ≤3, g is discontinuous at x =1 and x= 2
9) Let [x] denote the greatest integer function and f(x) be defined in a neighborhood of 2 by
f(x)= exp[{(x +2) ln 4}^[ˣ⁺¹]/⁴ - 16]/4ˣ - 16), x< 2
A{1- cos(x -2)}/{(x -2) tan(x -2)}, x> 2
Find the values of A and f(2) in order that f(x) may be continuous at x= a. 1, 1/2
10 f(x)= ln cosx/{⁴√(1+ x²) -1} if x> 0
(eˢᶦⁿˣ -1)/ln(1+ tan2x) if x< 0. -2,2 hence not possible to define
11) Determine a and b so that f is continuous at x=π/1 where
f(x)= (1- sin³x)/3cos²x if x< π/2
a if x=π/2
b(1- sinx)/(π-2x)² if x>π/2. 1/2,4
12) Determine the values of a, b, c for which the function
f(x)= (sin(a+1)+ sinx)/x, x< 0
{(x + bx²)¹⁾² - x¹⁾²}/bx³⁾², x> 0
Continuous at x= 0. a= -3/2, b≠ 0, c= 1/2
13) If f(x)= (sin3x+ A sin2x + B sinx)/x⁵ (x≠ 0) is continuous at x= 0. Find A and B. Also find f'(0). -4,5,1
14) [(π/2 - sin⁻¹(1- {x}²) sin⁻¹(1- {x}]/[√2 ({x} - {x³}] for x≠ 0
f(x)= π/2 for x= 0
Where {x} is the fraction pal part of x.
Consider another function g(x); such that
g(x) = f(x) for x≥ 0
= 2√2 f(x) for x< 0
Discuss the continuity of the functions f(x) and g(x) at x= 0. π/2,π/4√2 so f is disn at x= 0; g is continuous
15) Discuss the continuity of t in [0,2] where
f(x)= |4x -5[x] for x> 1
[Cosx] for x≤ 1
Where [x] is the greatest integer not greater than x. Also draw the graph. The function f is continuous everywhere in [0,2] except for x= 0,1/2,1,2
16) If y x + {-X} + [x] where [x] is the integral part of {x} is the fractional part of x. Discuss the continuity of f in [-2,2]. Dis
17) Find the locus of (a,b) for which the function
f(x)= ax- b for x≤ 1
3x for 1< x< 2
bx² - a for x≥ 2 is continuous at x= 1 but discontinuous at x= 2. Locus (a,b)--> x,y is y= x -3 excluding the points where y= 3 intersects it.
18) A function f: R--> R is defined as
f(x)= lim ₙ→∞ (ax²+ bx + c + eⁿˣ)/(1+ c. eⁿˣ) where f is continuous on R. Find the values of a, b, c. C= 1, a, b belong to R
19) Let g(x)= lim ₙ→∞ (xⁿf(x) + h(x)+ 1)/(2xⁿ + 3x +3), x≠ 1 and g(1)= limₓ→₁ (sin²(π.2ˣ))/ln(sec(π.2ˣ)) be continuous function at x= 1, find the value of 4g(1)+ 2f(1) - h(1). Assume that f(x) and h(x) are continuous at x= 1. 5
20) Let f(x)= limₙ→∞ (x²ⁿ⁻¹ + ax³ + bx²)/(x²ⁿ +1). If f(x) is continuous for all x ∈ R, find the bisector of angle between the lines 2x + y - 6=0 and 2x - 4y +7=0 which contains the point (a,b). 6x - 2y -5=0
21) f(x)= (aˢᶦⁿˣ - a ᵗᵃⁿˣ)/(tanx - sinx) for x> 0 =
(ln (1+ x + x²)+ ln(1- x + x²)/(secx - cosx) for x< 0, if f is continuous at x= 0, find a, now if g(x)= ln(2- x/a) cot(x - a) for x≠ a, a≠ 0, a> 0. If g is continuous at x= a then show that f(e⁻¹)= - e.
22) Let f(x)= (sinx+ cosx)ᶜᵒˢᵉᶜˣ; -π/2<x< 0
a, x= 0
(e¹⁾ˣ + e²⁾ˣ + e³/|ˣ|)/ae²⁾ˣ + be³/|ˣ|); 0< x<π/2
If f(x) is continuous at x= 0, find the value of (a² + b²). e²+ e⁻²
23) Given f(x)= ⁿᵣ₌₁∑ tan(x/2ʳ) sec(x/2ʳ⁻¹); r, n∈ N
g(x)= limₙ→∞{ln(f(x)+ tan(x/2ⁿ) - (f(x)+ tan(x/2ⁿ)ⁿ [sin(tan(x/2)]}/{1+ (f(x)+ tan(x/2ⁿ)ⁿ
= K for x=π/4 and domain of g(x) is (0,π/2).
Where [ ] denotes the greatest integer function.
Find the value of k, if possible, so that g(x) is continuous at x=π/4. Also state the points of discontinuity of g(x) in (0,π/4), if any. Con. Everywhere
24) Let f(x)= x³ - x² - 3x -1 and h(x)= f(x)/g(x) where h is a rational function such that
a) it is continuous every where except when x= -1,
b) limₓ→∞ h(x)= ∞
c) lim ₓ→₋₁ h(x)= 1/2.
Find ₓ→₀ (3h(x)+ f(x) - 2g(x)). g(x)= 4(x +1) and limit= -39/4
25) a) Let f be a real valud continuous function on R and satisfying f(-X) - f(x)= 0 ∀ ∈ R. If f(-5)= 5, f(-2)= 4, f(3)= -2 and f(0)= 0 then find the minimum number of zeros of the equation f(x)= 0. 5
b) Find the number of points of discontinuity of the function f(x)= [5x] + {3x} in [0,5] where [y] and {y} denote largest integer less than or equal to y and fractional part of y respectively. 30
26) a) If g: [a,b] --> [a,b] is continuous and onro function, then show that there is some c
∈ [a,b] such that g(c)= c.
b) Let f be continuous on the interval [0,1] to R such that f(0)= f(1). Show that there exists a point c in [0,1/2] such that f(c)= f(c + 1/2)
27) Consider the function
g(x)= (1- aˣ + xaˣ ln a)/aˣx² for x< 0
(2ˣaˣ - x ln 1 - x ln a -1)/x² for x> 0 where a> 0
Find the value of a and f(0) so that the function g(x) is continuous at x= 0. a= 1/√2, g(0)= (ln 2)²/8
Match the column
Column I
A) f(u)= 1/(u² + u -2), where u= 1/(x -1). The value of x at which f is discontinuous
B) f(x)= u², where u= x -1, x≥ 0
x +1, x< 0
The number of values of x at which f is discontinuous
C) If the function f(x)= (1- (cos(sinx)))/x⅖ is continuous at x= ó, then f(0) is
D) f(x)= x, x ∈ Q
1- x, x ∉ Q, then the values of x at which f(x) is continuous
Column II
P) 1/2
Q) 0
R) 2
S) 1
Aprs Bq Cp Dp
SAP-2
1) State whether true or false
i) f(x)= lim ₙ→∞ 1/(1+ n sin²πx) is continuous at x= 1.
ii) The function f(x)= ₂-2¹/⁽¹⁻ˣ⁾ if x≠ 1 and f(1)= 1 is not continuous at x= 1.
iii) There exists a continuous function f: [0,1] on to [0,10], but there exists no continuous function g: [0,1] onto (0,10).
iv) If f(x) is continuous in [0,1] and f(x)= 1 for all rational numbers in [0,1] then f'(1/√2) equal to 1.
v) If f(x)= (cosπx + sin(πx/2))/{(x -1)(3x² - 2x -1)} if x≠ 1
k, if x= 1
is continuous at x= 1, then the value of k is 3π²/32.
Only one correct
2) f is continuous function on the real line. Given that
x² + (f(x)- 2)x - √3. f(x)+ 2√3 -3=0. Then the value of f(√3)
a) cannot be determined
b) 2(1-√3) c) 0 d) 2(√3-2)/√3
3) The function f(x)= [x]² - [x²] (where [y] is the greatest integer less than or equal to y), is discontinuous at
a) all integers
b) all integers except 0 and 1
c) all integers except 0
d) all integers except 1
4) Let g(x)= tan⁻¹|x| - cot⁻¹|x|, f(x)= [x]/[x +1], h(x)= |gf(x))| where=x} denotes fractional part and [x] denotes the integral part then which of the following holds good ?
a) h is continuous at x= 0
b) h is discontinuous at x= 0
c) h(0-)=π/2 d) h(0+)=-π/2
5) Consider f(x)= limₙ→∞ +xⁿ - sin xⁿ)/(xⁿ + sin xⁿ) for x> 0x≠ 1,
f(1)= 0 then
a) f is continuous at x= 1
b) f has an infinite or oscillatory discontinuous at x= 1
c) f is a finite discontinuity at x= 1
d) f has a removable type of discontinuity at x= 1
6) f(x)=[{|x|}]ₑx² {[x + {x}]}/(ₑ1/x² -1) sgn(sinx) for x ≠ 0
= 0 for x= 0
Where {x} is the fractional part function; [x] is the step up function and sgn(x) is the signum function of x then, f(x)
a) is continuous at x= 0
b) is discontinuous at x= 0
c) has a removable discontinuity at x= 0
d) has an irremovable discontinuity at x= 0
7) f(x)= x[x]² log₁₊ₓ2 for -1< x<0
ln(ₑx² + 2 √{x})/tan√x for 0< x<1
Where [*] & {*} are the greatest integer function and fractional part function respectively, then
a) f(0)= ln 2 => f is continuous at x= 0
b) f(0)= 2 is continuous at x= 0
c) f(0)= e²=> f is continuous at x= 0
d) f has an irremovable discontinuity at x= 0
8) f(x)= (√(1+ x) - √(1- x))/{x}, x≠ 0
f(x)= cos2x, -π/4< x <0,
h(x)= (1/√2) f(g(x)) for x< 0
-1 for x = 0
f(x) for x> 0
Then, which of the following holds good
Where {x} denotes fractional part function.
a) h is continuous at x= 0
b) h is discontinuous at x= 0
c) f(g(x)) is an even function
d) f(x) is an even function
9) The function f(x)= [x] cos{(2x -1)/2}π, where [•] denotes the greatest integer function, is discontinuous at
a) all x b) all integer point c) no x d) x which is not an integer
10) Consider the function defined on [0,1] --> R, f(x)= (sinx - x cosx)/x² if x≠ 0 and f(0)= 0 then the function f(x)
a) has a removable discontinuity at x= 0
b) has a non removable finite discontinuity at x= 0
c) has a non removable infinite discontinuity at x= 0
d) is continuous at x= 0
11) f(x)= sin{(a - x)/2) tan[πx.2a]} for x> a
[Cos(πx/2a)]/(a - x) for x< a
Where [•] is the greatest integer function of x, and a> 0, then
a) f(a-)< 0
b) f has removable discontinuity at x= a
d) f(a+)< 0
12) Consider the function
f(x)= limₙ→∞(sin πx - x²ⁿ sin(x -1))/(1+ x²ⁿ⁺¹ - x²ⁿ), where n ∈N
Statement 1: f(x) is discontinuous at x= 1
Because
Statement 2: f(1)= 0
A) Statement 1 is true, statement 2 is true and statement 2 is correct explanation of statement 1
B) statement 1 is true, statement 2 is true and statement 2 is not correct explanation for statement 1
C) statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true
13) Consider the function f(x)= sgn(x -1) and g(x)= cot⁻¹[x -1] where [•] denotes the greatest integer function.
Statement 1: The function F(x)= f(x). g(x) is discontinuous at x= 1. Because
Statement 2: if f(x) is discontinuous at x= a and g(x) is also discontinuous at x= a then the product function f(x) g(x) is discontinuous at x= a.
A) Statement 1 is true, statement 2 is true and statement 2 is correct explanation of statement 1
B) statement 1 is true, statement 2 is true and statement 2 is not correct explanation for statement 1
C) statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true
More than 1 are correct
14) If f is defined on an interval [a,b]. Which of the following statement/s is/are incorrect?
a) if f(a) and f(b), have opposite sign, then there must be a point c∈ (a,b) such that f(c)= 0.
b) if f is ron [a,b], f(a)< 0 and f(b)> 0 then there must be a point c ∈(a,b) such that f(c)= 0
c) if f is continuous on [a,b] and there is a point c in (a,b) such that f(c)= 0, then f(a) and f(b) have opposite sign.
d) If f has no zeros on [a,b], then f(a) and f(b) have the same sign.
15) Which of the following functions f has/have a removable discontinuity at the indicated point?
a) f(x)=(x² -2x -8)/(x +2) at x= -2
b) f(x)= (x -7)/|x -7| at x= 7
c) f(x)= (x³ +64)/(x + 4) at x= -4
d) f(x)= (3- √x)/(9- x) at x= 9
16) In which of the following cases the given equations has atleast one root in the indicated interval?
a) x - cosx= 0 in (0,π/2)
b) x + sinx = 1 in +0,π/6)
c) a/(x -1) + b/(x -3) = 0, a, b> 0 in (1,3)
d) f(x) - g(x)= 0 in (a,b) where f and g are continuous on [a, b] and f(a)> g(a) and f(b) < g(b).
17) Indicate all correct alternatives if, f(x) = x/2 -1, then on the interval [0,π]
a) tan( f(x)) and 1/f(x) are both continuous
b) tan(f(x)) and 1/f(x) are both discontinuous
c) tan(f(x)) and f⁻¹(x) are both continuous
d) tan(f(x) is continuous but > 1/f(x) is not
Paragraph questions 18-20
Consider a function f(x)= (2/3) ln³⁾² (sinx + cosx) for x ∈ (0,π/2)
18) The value of ₓ→π/2 f(x)/(π/2 - x)³⁾² is
a) 2/3 b) 1 c) 2/3(2)³⁾⁴ d) non-existent
19) The function g(x) is defined as
g(x)= {(3/2)f(x)}²⁾³ , x ∈(0,π/2)
ₑ{(3/2)f(x - π/2)}²⁾³/√2, x(π/2,π)
Then g(x)
a) is continuous at x=π/2 for g(π/2)= 0
b) has a removable discontinuity at x=π/2
c) g(x) is discontinuous at x=π/2 and jump of discontinuity is equal to √2
d) has a non removable discontinuity at x=π/2
20) The range of g(x) is (g(x) same as Q.19)
a) (0,1] b) (0, (1/2)( ln 2] c) (1/2,1] d) (0,1/2 ln 2] U (1/√2, 1]
1) i f ii t iii t iv t c t
2b 3d 4a 5c 6a 7d 8a 9c 10d 11b 11b 13c 14acd 15acd 16abcd 17cd 18a 19d 20d
SAP-3
Discuss the continuity of the function
f(x)= +ₑ1/(x -1) -2)/(ₑ1/+x -1) +2), x≠ 1 at x= 1
DIFFERENTIABILITY
Things to remember:
1) Right hand & Left hand Derivatives;
By definition: f'(a)= lim ₕ→₀ (f(a+ h) - f(a))/h if it exists
f'(a⁺) lim ₕ→₀⁺ {f(a+ h) - f(a)}/h, provided the limit exists and is finite.
ii) The left hand Derivative of f at x= a denoted by f'(a⁻) is defined by:
f'(a⁻)= lim ₕ→₀⁺{f(a+ h) - f(a)}/h, provided the limit exists and is finite.
We also write f'(a⁺) = f'(a⁺) and f'(a⁻)= f'(a⁻).
* This geometrically means that a unique tangent with finite slope can be drawn at x= a as shown in the figure.
iii) Derivability & Continuity:
a) If f'(a) exists then f(x) is derivable at x= a => f(x) is continuous at x= a.
b) If a function f is derivable at x then f is continuous at x.
For: f'(x)= lim ₕ→₀{f(a+ h) - f(a)}/h exists
Also f(x+ h) - f(x)= {f(x+ h) - f(x)}h/h [h≠ 0]
Therefore:
lim ₕ→₀{f(x+ h) - f(x)}= lim ₕ→₀{f(x+ h) - f(x)}. h/h = f'(x). 0 = 0
Therefore lim ₕ→₀{f(x+ h) - f(x)}= 0 => lim ₕ→₀ f(x+ h) = f(x)=> f is continuous at x.
Note:
if f(x) is derivable for every point of its domain of definition, then it is continuous in that domain. The converse of the above result is not true:
"IF f IS CONTINUOUS AT x, THEN f IS DERIVABLE AT x" IS NOT TRUE.
e.g., the function f(x)= |x| & g(x) = x sin(1/x); x≠ 0 and g(0)= 0 are continuous at x= 0 but not derivable at x= 0.
NOTE CAREFULLY
a) Let f'(a⁺)= p and f'(a⁻)= q where p and q are finite then:
i) p= q => f is derivable at x= a => f is continuous at x= a.
ii) p≠ q => f is not derivable at x= a.
It is very important to note that f may be still continuous at x= a.
In short, for a function f:
Differentiability => Continuity ; Continuity ≠> Derivability
Non Derivability ≠> Discontinuous ; But discontinuity=> non Derivability
b) If a function f is not differentiable but is continuous at x= a it geometrically implies a sharp corner at x= a.
3) DERIVABILITY OVER AN INTERVAL
f(x) is said to be derivable over an interval if it is derivable at each and every point of the interval f(x) is said to be derivable over the closed interval [a,b] if :
i) for the points a and b, f' (a⁺) & f'(b⁻) exists and
ii) for any point c such that a< c < b, f' (x⁺) & f'(c⁻) exists and are equal.
NOTE
1) If f(x) and g(x) are derivable at x= a then the functions f(x)+ g(x), f(x) - g(x), f(x). g(x), will also be derivable at x= a and if g(a) ≠ 0 then the function f(x)/g(x) will also be derivable at x= a.
2) If f(x) is differentiable at x= a and g(x) is not differentiable at x= a, then the product function F(x)= f(x). g(x) can still be differentiable at x= a e.g., f(x)= x and g(x) =|x|.
3) If f(x) and g(x) both are not differentiable at x= a then the product function:
F(x)= f(x)g(x) can still be differentiable at x= a e.g., f(x) = |x| and g(x) =|x|
4) If f(x) and g(x) both are non derivable at x= a then the sum function F(x)= f(x)+ g(x) may be differentiable function. e.g. f(x) = |x| and g(x) = -|x|.
5) If f(x) is derivable at x= a ≠> f'(x) is continuous at x= a.
e.g f(x) = x² sin(1/x) if x≠ 0
0 if x= 0
6) A Surprising result: Suppose that the function f(x) and g(x) defined in the interval (x₁, x₂) containing the point x₀, and if f is differentiable at x= x₀ with f(x₀)= 0 together with g is continuous at x= x₀ then the function F(x)= f(x) g(x) is differentiable at x= x₀
eg., F(x)= sinx. x²⁾³ is differentiable at x= 0.
SAP-1
1) Discuss the continuity and differentiability of the function f(x)= sinx + sin|x|, x∈ R. Draw a rough sketch of the graph of f(x).
2) Examine the continuity and differentiability of f(x)= |x|+ |x -1|+ |x -2| x∈ R. Also draw the graph of f(x).
3) If the function f(x) is defined as
f(x)= -x²/2 for x≤ 0
xⁿ sin(1/x) for x> 0
is continuous but not derivable at x= 0 then find the range of n.
4) A function f is defined as follows:
f(x)= 1 for -∞< x < 0
1+ |sinx| for 0≤ x<π/2
2+ (x - π/2)² for π/2≤ x< + ∞
Discuss the continuity and differentiability at x= 0 and x=π/2.
5) Examine the origin for continuity and differentiability in the case of the function f defined by f(x)= x tan⁻¹(1/x) ≠ 0 and f(0)= 0.
6) Let f(0)= 0 and f'(0)= 1. For a positive integer k, show that
lim ₓ→₀ (1/x) (f(x) + f(x/2)+.....+ f(x/k))= 1+ 1/2+1/3+......+1/k
7) Let f(x)= xₑ-(¹/|ˣ|⁺¹/ˣ) ; x≠ 0, f(0)= 0, test the continuity and differentiability at x= 0
8) If f(x)= |x -1|. ([x] - [-X]), then find f'(1⁺) and f'(1⁻) [x] denotes greatest integer function.
9) If f(x)= ax² - b if |x|< 1
-1/|x| if |x|≥ 1 is derivable at x= 1. Find the values of a and b.
10) Let f(x) be defined in the interval [-2,2] such that
f(x)= -1, -2≤ x ≤ 0
x -1, 0< x ≤ 2 & g(x)= f(|x|)+ |f(x)|. Test the differentiability of g(x) in (-2,2).
11) Let g(x)= a √(x +2), 0< x<2
bx+2, 2 ≤x <5
If g(x) is derivable on (0,5), then find (2a + b).
12) Examine for continuity and differentiability at the points x=1 and x = 2, the function f defined by
f(x)= x[x] , 0≤ x < 2
(x -1)[x], 2≤ x ≤3 where [x]= greatest integer less than or equal to x.
13) ˣˣ²¹⁾ˣ⁻¹⁾ˣ¹⁾ˣ⁻¹⁾ˣᵤ²²lim ₓ→₀α⁺⁻²ₙ→∞ᵏ⁺¹⁰⁰ₙ₌₁₁₂⁴⁴₁⁻₂⁻₁₂²²²²²²²²²²²ₐˣₐ₁·ₓ²²²²²²⁻¹²²ⁿ∑ᵣ₌₁₁₂₂₃₄ₑˣ₁₂₄₃₁²²¹⁾ˣ⁻¹₃₄²²³ᵏᵏ⁻¹ᵏᵏ⁻¹
² ∈ ∞ ∑
DIFFERENTIATION
SAP-1
1) f(x)= lnx. 1/x
2) 1/x. -1/x²
3) If y= eˣ tanx + x logₑx, find dy/dx. eˣ(tanx + sec²x)+ (1+ logₑx)
4) If y= (logx)/x + eˣ sin2x + log₅x, find dy/dx. eˣ (sin2x + 2 cos2x) + 1/(xlog₅x).
5) If x= exp{tan⁻¹{(y - x²)/x²}}, then dy/dx.
a) x(1+ tan(logx))+ sec²x
b) 2x(1+ tan(logx))+ sec²x
c) 2x(1+ tan(logx))+ secx
d) 2x + x(1+ tan(logx))². D
6) logₑ tan⁻¹√(1+ x²). x/[tan⁻¹√{(1+ x²)(2+ x²)√(1+ x²)}}
7) (x +1)(x+2)(x +3). 3x²+ 12x +11
8) e⁵ˣtan(x²+2). e⁵ˣtan(x²+2) + 2xe⁵ˣsec²(x²+2)
9) (sinx)ˡⁿ ˣ. (sinx)ˡⁿ ˣ[ln(sinx)/x + cotx ln x]
10) {x¹⁾² (1- 2x)²⁾³}/{(2- 3x)³⁾⁴ (3- 4x)⁴⁾⁵}. y{1/2x - 4.3(1- 2x) + 9/4(2- 3x) + 16/5(3- 4x)
11) xˣ. xˣ(ln x +1)
12) ₑx ₑx²ₑx³ₑx⁴. y(1+ 2x + 3x¹+ 4x³)
13) xʸ + yˣ= 2. {yˣ ln y + xʸ(y/x)}/(xʸ ln x + yˣ(x/y)}
14) If y= sinx/(1+ cosx)/1+ sinx/(1+ cosx) ..... Show that {(1+ y) cosx + y sinx}/{1+ 2y + cosx - sinx}.
15) find dy/dx, if x+ y = sin(x +y). {cos(x - y) -1}/{cos(x - y)+ 1}
16) If x²+ xʸ + y=0, find y', also find the value of y' at the point (0,0). (2x + eʸ)/(xeʸ +1): -1
17) If y= cos t and x= a(t - sin t) find the value of dy/dx at t=π/2. -1
18) Prove that the function represented parametrically by the equations x= (1+ t)/t²; y= 3/2t²+ 2/t satisfies the relationship: x(y')= 1+ y
19) Find dy/dx at t=π/4 if y= cos⁴t and x= sin⁴t. -1
20) Find the slope of the tangent at a point P(t) on the curve x= at², y= 2at. 1/t
21) logₑ(tanx) with respect to sin⁻¹{eˣ). e⁻ˣ √(1- e²ˣ)/(sinx cosx).
SAP-2
1) If g is inverse of f and f'(x)= 1/(1+ xⁿ), then g'(x) is
a) 1+ xⁿ b) 1+ {f(x)}ⁿ c) 1+ {g(x)}ⁿ d) none
2) Differentiate xˡⁿ ˣ with respect to ln x. 2(xˡⁿ ˣ)(ln x)
3) If f is inverse of f and f(x)= 2x + sinx; then g'(x) is
a) 3/x² + 1/√(1- x²)
b) 2+ sin⁻¹x
c) 2+ cos g(x)
d) 1/(2+ cos(g(x)). D
4) If f(x)= x³+ x² f(1)+ x f''(2)+ f"'(3) for all x belongs to R. Then find f(x) independent of f'(1), f"(2) and f"'(3). -5,2,6
5) If x= a(t+ sin t) and y= a(1- cos t), find d²y/dx². (1/4a) sec⁴(t/2)
6) y= f(x) and x= g(y) are inverse functions of each other then express g'(y) and g"(y) in terms of derivative of f(x). Remember d²x/dy²= - (d²y/dx²)/(dy/dx)³.
7) If y= xₑx¹ then find y". 4y + 2xy'
8) Find y" at x=π/4, if y= tanx. π+4
9) Prove that the function y= eˣ sinx satisfies the relationship y" - 2y'+ 2y= 0.
10) If determinant x x² x³
1 2x 3x²
0 2 6x find f'(x).
11) If determinant eˣ x²
ln x sinx then find f'(1).
12) If determinant
2x x² x³
x²+2x 1 3x +1
2x 1- 3x² 5x then find f'(1)
13) d/dx[sin²{cot⁻¹ √{(1+ x)/(1- x)}}]. 1/2
14) If f(x)= sin⁻¹{2x/(1+ x²)}. Then find
i) f'(2). N-2/4
ii) f'(1/2). 8/5
iii) f'(1). doesn't exist
15) y= cos⁻¹(4x³ - 3x). Then find
i) f'(- √3/2). -6
ii) f'(0). 3
iii) f'(√3/2). -6
16) If √(1- x²) + √(1- y²)= a(x - y), then show that dy/dx= √{(1- y²)/(1- x²)}
17) Find second order derivatives of y= sinx with respect to z= eˣ. -(sinx + cosx)/e²ˣ.
18) If y= (tan⁻¹x)² then show that (1+ x²)² d²y/dx² + 2x(1+ x²) dy/dx = 2.
19) Obtain differential coefficient of
Tan⁻¹✓{√(1+ x²) -1}/x] with respect to cos⁻¹√[{1+ √(1+ x²)}/{2 √(1+ x²)}. 1
Sap-3
1) If y= (secx - tanx)/(secx + tanx) then dy/dx is
a) 2 secx(secx - tanx)
b) - 2 secx(secx - tanx)²
c) 2 secx(secx - tanx)²
d) - 2 secx(secx - tanx)²
2) if y= (1+ x² + x⁴)/(1+ x + x²) and dy/dx = ax+ b, then values of a and b are
a) 2,1 b) -1,1 c) 2,-1 d) -2,-1
3) Which of the following could be the sketch graph of y= d/dx (x ln x)?
4) Let f(x)= x +3 ln(x -2) and g(x)= x + 5 ln(x -1), then the set of x satisfying the inequality f'(x) < g'(x) is
a) (2,7/2) b) (1,2U(7/2,∞) c) [2,∞) d) (7/2, ∞).
5) Differential coefficient of (x⁽ˡ⁺ᵐ⁾/⁽ᵐ⁻ⁿ⁾)¹/⁽ⁿ⁻ˡ⁾ (x⁽ᵐ⁺ⁿ⁾/⁽ⁿ⁻ˡ⁾)¹/⁽ˡ⁻ᵐ⁾) (x⁽ⁿ⁺ˡ⁾/⁽ˡ⁻ᵐ⁾)¹/⁽ᵐ⁻ⁿ⁾ w.r.t.x is
a) 1 b) 0 c) -1 d) xˡᵐⁿ
6) If y= 1/(1+ xⁿ⁻ᵐ + xᵖ⁻ᵐ) + 1/(1+ xᵐ⁻ⁿ + xᵖ⁻ⁿ) + 1/(1+ xᵐ⁻ᵖ + xⁿ⁻ᵖ) then dy/dx at x= ₑᵐ^n^p is equal to
a) eᵐⁿᵖ b) eᵐⁿ/ᵖ c) eⁿᵖ/ᵐ d) none
7) If cos⁻¹{(x² - y²)/(x² + y²)= log a then dy/dx is
a) - x/y b) - y/x c) y/x d) x/y
8) If f(x)= ¹⁰⁰ₙ₌₁Π+x - n)ⁿ⁽¹⁰¹⁻ⁿ⁾ ; then f(101)/f'(101)=
a) 5050 b) 1/5050 c) 10010 d) 1/10010
9) If f(x)= (|x|)|ˢᶦⁿˣ|, then f'(-π/4) is
a) (π/4)¹⁾√²(√2/2 log(4/π) - 2√2/π)
b) (π/4)¹⁾√²(√2/2 log(4/π) - 2√2/π)
c) (π/4)¹⁾√²(√2/2 log(π/4) - 2√2/π)
d) (π/4)¹⁾√²(√2/2 log(π/4) + 2√2/π)
10) If y=
11) If y= ₓx² then dy/dx
a) 2 ln x. ₓx² b) (2 ln x+ 1).ₓx² c) (ln x + 1)ₓx² d) ₓx² ln(ex²)
12) If xᵐ.yⁿ = (x + y)ᵐ⁺ⁿ, then dy/dx is
a) xy b) x/y c) y/x d) (x + y)/xy
13) If x √(1+ y) + y √(1+ x)= 0, then dy/dx is
a) 1/(1+ x)² b) -1/(1+ x)² c) -1/(1+ x) + 1/(1+ x)² d) none
14) If x² eʸ + 2xyeˣ +13=0, then dy/dx is
a) -(2xeʸ⁻ˣ + 2y(x +1))/x(xeʸ⁻ˣ +2)
b) (2xeʸ⁻ˣ + 2y(x +1))/x(xeʸ⁻ˣ +2)
c) -(2xeˣ⁻ʸ + 2y(x +1))/x(xeʸ⁻ˣ +2)
d) none
15) If x= ₑy+ e^ y+....θ, x> 0 then dy/dx is
a) x/+1+ x) b) (1+ x)/x c) (1- x)/x d) 1/x
16) If x= θ - 1/θ and y= θ + 1/θ, then dy/dx
a) x/y b) y/x c) -x/y d) - y/x
17) The derivatives of sin⁻¹{x/√(1+ x²)} w.r.t cos⁻¹{(1- x²)/(1+ x²), (x> 0) is
a) 1 b) 2 c) 1/2 d) -1/2
18) Leg g is the inverse function of f and f'(x)= x¹⁰/(1+ x²). If g(2)= a then g'(2) is equal to
a) 5/2¹⁰ b) (1+ a²)/a¹⁰ c) a¹⁰/(1+ a²) d) (1+ a¹⁰)/a²
19) Let f(x)= sinx ; g(x)= x² and h(x)= logₑx and F(x)= h(g(f(x))) then d²F/dx² is equal to
a) 2 cosec³x b) 2 cot(x²) - 4 x² cosec²(x²) c) 2x cot x² d) -2 cosec²x
20) If f(x)= √(x² +1), g(x)= (x +1)/(x² +1) and y(x)= 2x -3, then f'(h'(g'(x))))=
a) 0 b) 1/√(x² +1) c) 2/√5 d) x/√(x² +1)
21) If f and g are the functions whose graphs are shown, let u(x)= f(g(x)); w(x)= g(g(x)), then the value of u'(1)+ w'(1) is
a) -1/2 b) -3/2 c) -5/4 d) does not exist
22) f'(x)= g(x) and g'(x)= - f(x) for all real x and f(5)= 2= f'(5) then f²(10)+ g²(10 is
a) 1 b) 4 c) 8 d) none
23) If f(x)= xⁿ, then the value of f(1) - f'(1)/1! + f"(1)/2! + f"'(1)/3! + .......(-1)ⁿ fⁿⁿⁿⁿ....ⁿ ᵗᶦᵐᵉˢ(1)/n!
a) 2ⁿ⁻¹ b) 0 c) 1 d) 2ⁿ
24) A function y= f(x) has second order derivatives f"(x)= 6(x -1). If its graph passes through the point (2,1) and at the point the tangent to the graph is y= 3x -5, then the function is
a) (x +1)³ b) (x +1)² c) (x -1)² d) (x -1)³
25) if f(x)= x + x²/1! + x³/2! + ..... xⁿ/(n -1)!, then f(0)+ f'(0)+ f"(0) + .....+ f"'"" .....,ⁿ ᵗᶦᵐᵉˢ(0) is equal to
a) n(n +1)/2 b) (n² +1)/2 c) {n(n +1)²/2 d) n(n +1)2n +1)/6
26) let f(x)= | cosx. x 1
2 sinx x² 2x
tanx x 1. Then lim ₓ→₀ f'(x)/x=
a) 2 b) -2 c) -1 d) 1
27) If f differentiable in (0,6) and f'(4)= 5 then lim₋₂ +f(4) - f(x²))/(2- x)=
a) 5 b) 5/4 c) 10 d) 20
28) If f(4)= g(4)= 2; f'(4)= 9; g'(4)= 6 then lim→₄ {√f(x) - √g(x)}/(√x -2) is
a) 3√2 b) 3/√2 c) 0 d) none
29) The slope (s) of common tangent/s to the curves y= e⁻ˣ and y= e⁻ˣ sinx can be
a) -ₑ -π/2 b) ₑ -π c) π/2 d) 1
30) If y+ ln(1+ x)=0, which of the following is true?
a) eʸ = xy' +1 b) y'= -1/(x +1) c) y'+ eʸ = 0 d) y'= eʸ
31) If ₂3ˣ, then y' equals
a) 3ˣ ln 3 ln 2
b) y(log₂y) ln 3 ln 2 c) ₂3ˣ. 3ˣ ln6 d) ₂3ˣ. 3ˣ ln 3 ln 2
32) if y= 3t² and x= 2t then d²y/dx² equal
a) 3t b) 3 c) 3/2 d) none
33) If g is inverse of f and f(x)= x² + 3x - 3 (x > 0) then g'(1) equals
a) 1/2g(1)+3) b) -1 c) 1/5 d) -f(1)/(f(1))²
1b 2c 3c 4d 5b 6d 7c 8b 9a 10d 11d 12c 13b 14a 15c 16a 17c 18b 19d 20c 21b 22c 23b 24d 25a 26b 27d 28a 29ab 30abc 31bd 32c 33ac
Sap-4
1) If y= fofof(x) and f(0)=0, f'(0)= 2, then find y'(0)
a) 6 b) 7 c) 8 d) 9
2) If y² = px is a polynomial of degree 3, then 2 d/dx (y³ d²y/dx²) is equal to
a) p"'(x). p'x(x) b) p"(x). p"'(x) c) p(x). p"'(x) d) none
3) If y is a function of x then d²y/dx² + y dy/dx = 0. If x is a function of y then the equation becomes
a) d²x/dy² + x dz/dy= 0
b) d²x/dy² + y(dx/dy)³ = 0
c) d²x/dy² - y (dx/dy)²= 0
d) d²x/dy² - x(dx/dy)²= 0
4) If y= tanx tan2x tan3x then dy/dx is
a) 3 sec²3x tanx tan2x + sec²x tan2x tan3x+ 2 sec²2x tan3x tanx
b) 2y(cosec2x + 2 cosec4x+ 3 cosec6x)
c) 3 sec²3x - 2 sec²2x - sec²x
d) sec²x + 2 sec²2x + 3 sec²3x.
5) If y= e√ˣ + e⁻√ˣ then dy/dx equals
a) (e√ˣ + e⁻√ˣ)/2√x
b) (e√ˣ - e⁻√ˣ)/2√x
c) (1/2√x) √(y² -4)
d) (1/2√x) √(y² +4)
6) Let y= √[x + √{x + √x +........∞ then dy/dx
a) 1/(2y -1) b) x/(x - 2y) c) 1/√(1+ 4x) d) y/(2x + y)
7) If 2ˣ + 2ʸ = 2ˣ⁺ʸ then dy/dx has the value equal to
a) -2ʸ/2ˣ b) 1/(1- 2ˣ) c) 1- 2ʸ d) 2ˣ(1- 2ʸ)/2ʸ(2ˣ -1)
8) The function u= eˣ sinx ; v= eˣ cosx satisfy the equation
a) v du/dx - u dv/dx= u²+ v²
b) d²u/dx² = 2v
c) d²v/dx² = -2u d) none
9) Two function f and g have first and second derivatives at x= 0 and satisfy the relations.
f(0)= 2/g(0), f(0)= 2g'(0)= 4g(0), g"(0)= 5 f"(0)= 6 f(0)= 3 then
a) If h(x)= f(x)/g(x) then h'(0)= 15/4
b) If k(x)= f(x). g(x) then sinx k'(0)= 2
c) lim ₓ→₀ g'(x)/f'(x)= 1/2 d) none
10) If y² + b² = 2xy, then dy/dx is
a) 1/(xy - b²) b) y/(y - x) c) (xy - b²)/(y - x)² d) (xy - b²)/y
11) If √(y + x) + √(y - x)= c, then dy/dx is
a) 2x/c² b) x/{y+ √(y² - x²)} c) {y - √(y² - x²)}/x d) c²/2y
12) lim ₓ→₀₊{(ₓxˣ) - (zˣ} is
a) equal to 0 b) equal to 1 c) equal to -1 d) non existent
13) Select the correct statements
a) The function f defined by
f(x)= [ 2x² +3 for x≤ 1
3x +2 for x> 1]
is neither differentiable nor continuous at x= 1.
b) The function f(x)= x²|x| is twice differentiable at x= 0.
c) If f is continuous at x= 5 and f(5)= 2 then lim ₓ→₂ f(4x²-11) exiss.
d) If limₓ→ₐ (f(x) + g(x))= 2 and limₓ→ₐ (f(x) - g(x))= 1 then limₓ→ₐ f(x). g(x) may not exist.
14) Let L= limₓ→₀ xᵐ(ln x)ⁿ where m,n belongs to N then
a) l is independent of m and n.
b) l is independent of m and depends on m
c) l is independent of n and depends on m
d) l is dependent on both m and n
15) limₓ→₀ (logₛᵢₙ²ₓ cosx)/+logₛᵢₙ¹ₓ/₂ cos(x/2) has the value equal to
a) 1 b) 2 c) 4 d) none
1c 2c 3c 4abc 5ac 6acd 7abcd 8abc 9abc 10bc 11abc 12c 13bc 14a 15c
Sap-5
True/False
1) Let u(x) and v(x) are differentiable functions such that (u/v) (x)= 7 if u'(x)/v'(x)= p and {u(x)/v(x)}'= q then (p+q)/(p - q)= 1.
2) If f(x)= |x -2|, then f'f(x)(= 1 for x> 20
3) If f(0)= a, f'(0)= b, g(0)=0 and (fog)'(0)= c, then g'(0)= c/b.
4) The differential coefficient of f(logx)) w.r.t.logx where f(x)= logx is 1/logx.
5) f'(sinx)=(f(sinx))'
6) if x= t² + 3t -8, y= 2t² - 2t -5, then dy/dx at (2,-1) is 6/7
Match the column
1) Following questions contains statements given in two columns, which have to be matched. The statements in column I are levelled A, B, C, D while the statements in column II are labelled as o,q,r,s. Any given statement in column I have correct matching with ONE statement in column II
2) Column I
A) If f(x)= x³ + x +1, then f'(x² +1) at x= 0 is
B) If f(x)= logₓ² (logx), then f'(eᵉ) is equal to
C) For the function y= ln tan(π/4 + x/2)
If dy/dx = secx + p, then p is equal to
D) if f(x)= |x³ - x² + x -1| sinx, then 4f'(28 f(f(π))) is equal to
Column II
p) 1
q) 0
r) 28
s) 4
ASSERTION AND REASON
These questions contains statements I(assertion) and statement II(reason)
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
1) Statement 1: Let f(x) is a continuous function defined from R to Q and f(5)= 3 then differential coefficient of f(x) w.r.t.x will be 0.
Because
Statement II: Differentiation of constant function is always zero.
a) A B) B C) C d) D
2) Statement 1: Derivative of sin⁻¹{2x/(1+ x²) with respect to cos⁻¹{(1- x²)/(1+ x²) is 1 for 0< x < 1.
Because
Statement II: sin⁻¹{2x/(1+ x²)= cos⁻¹{(1- x²)/(1+ x²) for -1≤ x ≤ 1.
a) A b) B c) C d) D
3) Consider f(x)= x/(x² -1) and g(x)= f"(x).
Statement I: Graph of g(x) is concave up for x > 1.
Because
Statement II: dⁿ/dxⁿ (f(x))= (-1)ⁿn!/2 {1/(x +1)ⁿ⁺¹ + 1/(x -1)ⁿ⁺¹}, n ∈ N
a) A B) B C) C d) D
Comprehension based questions
Let f(x+ y)/2 = (f(y) -1)/2 + xy., xy ∈ R. f(x) is differentiable and f'(0)= 1. Let g(x)= be a deriavable function at x= 0 and follows the functional rule g{(x + y)/k} = (g(x) + g(y))/k (k ∈ R. k≠ 0,2)
Let g'(0)= λ ≠ 0.
1) Domain of ln(f(x)) is
a) R⁺ b) R - {0} c) R d) R⁻
2) Range of y= log₃/₄ (f(x))
a) (-∞,1) b) [3/4, ∞) c) (-∞, ∞) d) R
3) If the graph of y= f(x) and y= g(x) intersect in coincident points the λ can take values
a) 3 b) 1 c) -1 d) 4
Comprehension
Limits that lead to the indeterminate form 1^ ∞, 0⁰, ∞⁰ can sometimes be solved taking logarithm first and then using L' Hospital's rule
Let lim ₓ→ₐ (f(x))ᵍ⁽ˣ⁾ is in the form of ∞⁰, it can be written as ₑlim ₓ→ₐ g(x) ln f(x)= e ᴸ
Where L= lim ₓ→ₐ {lnf(x)/1/g(x)} is ∞/∞ form and can be solved using L' Hospital's rule.
1) lim ₓ→₁₊ ₓ{1/(1- x)
a) -1 b) e⁻¹ c) -2 d) e⁻²
2) lim ₓ→∞ [(ln x(¹⁾²ˣ + ₓ¹/xⁿ] ∀ n ∈ N.
a) 2 b) 0 c) e¹⁾² d) e
3) lim ₓ→₀. (Sinx)²ˢᶦⁿˣ
a) 1 b) 0 c) 2 d) does not exist
Comprehension
Left hand Derivative and right hand Derivative of a function f(x) at a point x= a are defined as
f'(a)= lim ₕ→₀₊(f(a) - f(a - h))/h= lim ₕ→₀⁻ (f(a+ h) - f(a))/h and
f'(a⁺)= lim h→₀₊ (f(a + h) - f(a))/h = lim ₕ→₀⁻ (f(a) - f(a - h))/h = limₓ →₀₊ (f(a) - f(x))/(a - x) respectively
Let f be a twice differentiable function. We also know that Derivative of an even function is odd function and Derivative of an odd function is even function.
1) If f is odd, which of the following is left hand Derivative of f at x = - a
a) lim ₕ→₀⁻ (f(a - h) - f(a))/-h
b) lim ₕ→₀⁻ (f(a - h) - f(a))/h
c) lim ₕ→₀₊(f(a) + f(a - h))/-h
d) lim ₕ→₀⁻ (f(- a) - f(-a - h))/-h
2) If f is even, which of the following is right hand Derivative of f' at x = a
a) lim ₕ→₀⁻ (f'(a) - f'(-a+ h))/h
b) lim ₕ→₀ ₊ (f'(a) + f'(-a- h))/h
c) lim ₕ→₀⁻ (- f'(-a) + f'(-a - h))/-h
d) lim ₕ→₀₊(f'(- a) + f'(-a +h))/-h
3) The statement lim ₕ→₀(f(-x) - f(-x - h))/h= lim ₕ→₀ (f(x) - f(x - h))/- h implies that for all x ∈R
a) f is odd
b) f is even
c) f is neither odd nor even
d) nothing can be concluded
Comprehension
An operator ∆ is defined to operate on differentiable functions defined as follows:
If f(x) is a differentiable then ∆(f(x))= lim ₕ→₀(f³(x + h) - f³(x))/h
g(x) is a differentiable function such that the slope of the tangent to the curve y= g(x) at any point (a, g(a)) is equal to 2e³(a +1) also g(0)=0.
1) ∆(g(x) at x= ln 2 is
a) 24 ln2 {2ln 2 +2)
b) ln2 (4e²)ln²2
c) 96ln(4e²) ln²2
d) 192 ln(4e)ln² 2
2) ∆(∆(x +2)ₓ₌₀
a) 2⁵3⁹ b) 2⁹3⁵ c) 2⁴3⁵ d) 2⁶3⁴
3) lim ₓ→₀ ∆g(x)/ln (cos2x)
a) -12 b) 12 c) 24 d) -24
T/F 1t 2t 3t 4t 5f 6t
Match Ag, Bs, Cp Dr
As Bq Cq Ds
A/R- 1a 2c 3a
Com
1c 2a 3ac
1b 2a 3a
1a 2a 3b
1c 2d 3a
SAP- 6
1) If y= (a + bx)³⁾²/x⁵⁾⁴ and dy/dx vanishes at x=5 then find a/b.
2) If y= (x⁴ + 4)/(x² - 2x +2) then find dy/dx|ₓ₌₁/₂
3) If f(x) = √(2x² -1) and y= f(x²) then find dy/dx at x= 1.
4) If x= (1+ ln t)/t² and y= (3+ 2 ln t)/t. Show that y dy/dx = 2x(dy/dx)² +1.
5) If fₙ(x)= ₑfₙ₋₁(x) for all
n ∈ N and f₀(x)= x then show that d/dx {fₙ(x)= f₁(x). f₂(x).......fₙ(x).
6) If y= x²/2 + (x/2) √(x² +1)+ ln √(x + √x² +1)) show that 2y= xy' + ln y', where y' denotes the Derivative of w.r.t.x.
7) Let f(x) = x + 1/(2x + 1/(2x + 1/2x +.....∞
Compute the value of f(100).f'(100).
8) If y= tan⁻¹{u/√(1- u²)} & x = sec⁻¹{1/(2u²-1)}, u ∈ (0, 1/√2) U(1/√2,1); show that 2 dy/dx +1=0.
9) If y= tan⁻¹{x/{1+ √(1- x²)}} + sin[2 tan⁻¹√{(1- x)/(1+ x)}], then find dy/dx for x ∈ (1,1).
10) If x = 2 cost - cos2t & y= 2 sint - sin2t, find the value of (d²y/dx²) when t= (π/2).
11) If y= ax²/{(x - a)(x - b)(x - c)} + bx/{(x - b)(x - c)} + c/(x - c) + 1, show that y'/y = (1/x) a/(a - x) + b/(b - x) + c/(c - x).
12) If √(x² + y²) = ₑᵃʳᶜ ˢᶦⁿ{y/√(x²+ y²). Prove that d²y/dx² = 2(x² + y²)/(x - y)³, x > 0.
13) Let f(x)= x² - 4x - 3, x> 2 and let g be the inverse of f. Find the value of g' at f(x)= 9.
14) If y= x ln [(ax)
⁻¹ + a⁻¹], show that x(x + 1) d²y/dx² + x dy/dx = y - 1.
15) If x= secθ - cosθ; y= secⁿθ - cosⁿθ, then show that (x² +4)(dy/dx)² - n²(y² + 4)= 0.
16) a) Differentiate y= cos⁻¹{(1- x²)/(1+ x²)} w.r.t. tan⁻¹x, satisfying clearly where function is not differentiable.
b) If y= sin⁻¹(3x - 4x³) find dy/dx stating clearly where the function is not deriavable in (-1,1).
17) Suppose f and g are two functions such that f, g: R --> R,
f(x)= ln(1+ √(1+ x²)) and g(x)= ln(x + √(1+ x²))
Then find the value of x.ₑᵍ⁽ˣ⁾(f(1/x))' + g'(x) at x= 1.
18) Determine the values of a, b and c so that
lim ₓ→₀ {(a + b cosx)x - c sinx)}/x⁵ = 1
Solve using L' Hospital's rule of series expansion (18-21)
19) lim ₓ→₀ {x cosx - ln(1+ x))/x².
20) lim ₓ→₀ [1/x² - 1/sin²x].
21) If lim ₓ→ₐ (aˣ - xᵃ)/(xˣ - aᵃ)= -1 find a.
22) lim ₓ→₀ logₜₐₙ²ₓ (tan²2x).
23) If f(x)=|(x - a)⁴ (x - a)³ 1
(x -b)⁴ (x - b)³ 1
(x - c)⁴ (x - c)³ 1| then
f'(x)= λ|(x - a)⁴ (x - a)² 1
(x - b)⁴ (x - b)² 1
(x - c)⁴ (x - c)² 1|
Find the value of λ.
1) √5. 2) 3. 3) 2 7) 100 9) (1-2x)/2√(1- x²) 10) -3/2 13) 1/8 16) a) not differentiable at x=0 b) not deriavable at x= ± 1/2 17) zero
18) 120,60,180 19) 1/2 20) -1/3 21) 1 22) 1 23) 3
Sap-7
1) If x= 1/z and y= f(x), show that: d²f/dx² = 2z³ dy/dx + z⁴ d²y/dx².
2) Show that if |a₁ sinx + a₂ sin2x + ....+ aₙ sin nx| ≤ |sinx| for x ∈ R, then |a₁+ 2a₂ + 3a₃ + ....+ naₙ|≤ 1.
3) Show that the substitution z= ln{tan(x/2)} changes the equation d²y/dx² + cotx dy/dx + 4y cosec²x = 0 to (d²y/dx²)+ 4y= 0.
4) Find a polynomial function f(x) such that f(2x)= f'(x)= f"(x).
5) If Y= sX and Z= tX, where all the letters denotes the function of x and suffixes denotes the differentiation w.r.t.x then show that
|X Y Z
X₁ Y₁ Z₁ = X³|s₁ t₁
X₂ Y₂ Z₂| s₂ t₂|
6) If √(1- x⁶) + √(1- y⁶)= a³(x³- y³), show that dy/dx = (x²/y²) √{(1- y⁶)/(1- x⁶)}.
7) If α be a repeated root of a quadratic equation f(x)= 0 and A(x), B(x), C(x) be the polynomials of degree 3,4,5 respectively, then show that
|A(x) B(x) C(x(
A(α) B(α) C(α)
A'(α) B'(α) C'(α)
is divisible by f(x), where dash denotes the Derivative.
8) If tan⁻¹{1/(x²+ x +1)} + tan⁻¹{1/(x²+ 3x +3)}+ tan⁻¹{1/(x²+ 5x +7)} + tan⁻¹{1/(x²+ 7x +13)} + .....upto n terms. Find dy/dx, expressing your answer in 2 terms.
9) Let g(x) be a polynomial, of degree one and f(x) be defined by
f(x)= [ g(x) , x≤ 0
{(1+x)/(2+ x)}¹⁾ˣ, x> 0
Find the continuous function f(x) satisfying f'(1)= f(-1).
10) Let f(x + y)/2= (f(y) - a)/2 + xy for all real x and y. If f(x) is differentiable and f'(0) exists for all real permissible values of 'a' and is equal to √(5a - 1- a²). Show that f(x) is positive for all real x.
11) Find the value f(0) so that the function f(x)= 1/x - 2/(e²ˣ -1), x ≠ 0 is continuous at x= 0 and examine the differentiability of f(x) at x = 0.
12) If lim ₓ→₀ (a sinx - bx + cx² + x³)/(2x². ln (1+ x) - 2x³+ x⁴) exists and is infinity, find the value of a, b, c and the limit.
4) 4x³/9 8) 1/(1+ (x + n)²)) - 1/(1+ x²).
9) f(x)=[-(2/3)(1/6 + ln(3/2))x if x ≤ 0
{(1+ x)/+2+ x)}¹⁾ˣ, if x> 0
11) f(0)=1, differentiable at x= 0, f'(0⁺)= -(1/3); f'(0⁻)= -(1/3)
12) 6,6,0,3/40
Sap-8
1) a) If x² + y² = 1, then
a) yy" - 2(y')² +1=0
b) yy" + (y')² +1=0
c) yy" - 2p(y')² -1=0
d) yy" + 2(y')² +1=0.
b) Suppose p(x)= a₀+ a₁x + a₂x² + ....+ aₙxⁿ. If |p(x)|≤ |eˣ⁻¹ -1| for all x≥ 0 show that |a₁ + 2a₂ + ....naₙ|≤ 1
2)a) If y= y(x) and it follows the relation x cos y + y cosx =π then y"(0)
a) 1 b) -1 c) π d) -π
b) If P(x) is a polynomial of degree less than or to 2 and S is the set of all such polynomial so that P(1)= 1, P(0)= 0 and P'(x)> 0 ∀ x∈ [0,1], then
a) S= φ
b) S= {(1- a)x²+ ax, 0< a < 2}
c) S= {(1- a)x² + ax, a ∈ (0,∞)}
d) S={(1- a)x² + ax, 0< a<1}
3) For x> 0, lim ₓ→₀ ((sinx)¹⁾ˣ + (1/x)ˢᶦⁿˣ) is
a) 0 b) -1 c) 1 d) 2
4) If f'(x)= - f(x) and g(x)=f'(x) and F(x)=(f(x/2))² + (g(x/2))² and given F(5)= 5, then F(10) is equal to
a) 5 b) 10 c) 0 d) 15
5) d²x/dy² equals
a) (d²y/dx²)
b) -(d²y/dx²)⁻¹(dy/dx)⁻³
c) (d²y/dx²)⁻²
d) -(d²y/dx²)(dy/dx)⁻³
6) Let g(x)= log f(x) where f(x) is a twice differentiable positive function on (0,∞) such that f(x +1)= x. f(x). Then for N= 1,2,3,.....
g"(N + 1/2) - g"(1/2)=
a) -4{1+ 1/9 + 1/25+ .....+ 1/(2N -1)²}
b) 4{1+ 1/9 + 1/25+ .....+ 1/(2N -1)²}
c) -4{1+ 1/9 + 1/25+ .....+ 1/(2N +1)²}
d) 4{1+ 1/9 + 1/25+ .....+ 1/(2N +1)²}
7) If the function f(x)= x³ + eˣ⁾² and g(x)= f⁻¹(x), then the value of g'(1) is...
1a) b 2a) c b) b 3c 4a 5d 6a 7) 2
⁽ ⁾
L' HOSPITAL RULE
1) lim ₓ→₀ |x|ˢᶦⁿˣ. 1
2) lim ₓ→₀ logₛᵢₙₓ sin2x. 1
3) limₙ→∞ (eⁿ/π)¹⁾ⁿ. e
4) lim ₓ→₀ (tanx - x)/x³. 1/3
5) lim ₓ→₀ (eˣ - x -1)/x². 1/2
6) lim ₓ→₀. (Sinx - tanx)/x³. 1/2
7) lim ₓ→₀ ln(1+ x)/x. 1
Sap-9
Differentiate:
1) tanx. sec²x
2) eˢᶦⁿˣ. eˢᶦⁿˣ cosx
3) ln x. 1/x
4) 1/x. -1/x²
5) eˣ tanx + x logₑx. eˣ(tanx + sec²x)+ (logx +1)
6) (logx)/x + eˣ sin2x + logₑx. (1- logx)/x² + eˣ (sin2x + 2 cos2x)+ 1/(x logₑ5).
7) If x= exp{tan⁻¹{(y - x²)/x²}) then dy/dx is
a) x[1+ tan(logx)+ sec²x]
b) 2x[1+ tan(logx)]+ sec²x
c) 2x[1+ tan(logx)] + secx
d) 2x + x[1+ tan(logx)]². D
8) logₑx[tan⁻¹{√(1+ x²)}]. x/[tan⁻¹{√(1+x²)}(2+x²)√(1+ x²)]
9) (x +1)(x +2)(x +3). 3x²+ 12x +11
10) e⁵ˣ tan(x²+2). 5e⁵ˣ tan(x²+2)+ 2xe⁵ˣ sec²(x²+2)
11) (sinx)ˡⁿ ˣ. (sinx)ˡⁿ ˣ[ln(sinx)/x + cotx ln x]
12) {x¹⁾²(1- 2x)²⁾³}/{(2- 3x)³⁾⁴(3- 4x)⁴⁾⁵}.
13) xˣ. xˣ(ln x +1)
14) ₑx.ₑx².ₑx³.ₑx⁴. y(1+ 2x + 3x²+ 4x³)
15) If y= a cos t and x= a(1- sin t) find the value of dy/dx at t=π/2. -1
16) Prove that the function represented parametrically by the equations x= (1+ t)/t²; y= 3/2t²+ 2/t satisfies the relationship x(y')³= 1+ y'
17) Differentiate logₑ(tanx) w.r.t sin⁻¹(ₑx). e⁻ˣ√(1- e²ˣ)/(sinx cosx)
18) Find dy/dx at t=π/4 if y= cos⁴t and x= sin⁴t. -1
19) Find the slope of the tangent at a point P(t) on the curve x= at², y= 2at. 1/t
20) xˡⁿ ˣ w.r.t. ln x. 2(xˡⁿ ˣ)(ln x)
21) If xʸ + yˣ = 2. {yˣ ln y +xʸ (y/x)}/{xʸ ln x + yˣ(x/y)}
22) y= sinx/(1+ cosx/(1+ sinx/1+ cosx..... {(1+ y)cosx+ y sinx}/(1+ 2y + cosx - sinx)
23) x+ y= sin(x + y). {cos(x - y) -1}/{cos(x - y)+1}
24) If x²+ xeʸ+ y= 0, find dy/dx, also find the value of y' at point (0,0). -{(2x + eʸ)/(xeʸ +1)}, -1
SAP- 10
1) If g is inverse of f and f'(x)= 1/(1+ xⁿ), then g'(x) equals
a) 1+ xⁿ b) 1+ (f(x))ⁿ c) 1+ (g(x))ⁿ d) none. C
2) If g is inverse of f and f(x)= 2x + sinx, then g'(x) equals
a) -3/x² + 1/√(1- x²)
b) 2+ sin⁻¹x
c) 2+ cos g(x)
d) 1/(2+ cos(g(x)). D
3) If f(x)= x³+ x² f'(1)+ x f'(2)+ f"(3 for all x ∈ R. Then find f(x) independent of f'(1), f"(2) and f"(3).
4) If x= a(t + sin t) and y= a(1- cos t), find d²y/dx². (1/4a). sec⁴(1/2)
5) y= f(x) and x= g(y) are inverse functions of each other then express g'(y) and g"(y) in terms of derivative of f(x).
6) If y= (tan⁻¹x)² then show that (1+ x²)² d²y/dx² + 2x(1+ x²) dy/dx=2.
7) If y= xₑx² then find y". 4y + 2xy'
8) Find y" at x=π/4, if y= x tanx. π+4
9) Show that the function y= eˣ sinx satisfies the relationship y" - 2y' + 2y=0.
10) If f(x)=|x x² x³
1 2x 3x²
0 2 6x| find f'(x)
11) If f(x)= |eˣ x²
ln x sinx| then find f'(1). e(sun 1 + cos1)- 1
12) If f(x)= |2x x² x³
x²+2x 1 3x +1
2x 1- 3x² 5x| then find f'(1). 9
13) lim ₓ→₀ |x|ˢᶦⁿˣ. 1
14) lim ₓ→₀ logₛᵢₙₓ(sin2x). 1
15) limₙ→∞ (eⁿ/π)¹⁾ⁿ. e
16) lim ₓ→₀ (tanx - x)/x³. 1/3
17) lim ₓ→₀ (eˣ- x -1)/x². 1/2
18) lim ₓ→₀ (sinx - tanx)/x². -1/2
19) lim ₓ→₀ {ln(1+ x)}/x. 1
20) d/dx[sin²{cot⁻¹√{(1+ x)/(1- x)}}
a) -1/2 b) 0 c) 1/2 d) -1
21) If f(x)= sin⁻¹{2x/(1+ x²)} then find
i) f'(2).
ii) f'(1/2)
iii) f'(1).
22) Find the interval for which f(x)= x²+ x +1 is
i) concave upwards. (0,∞)
ii) concave downwards. (-∞,0)
23) If y= cos⁻¹(4x³- 3x), then find
i) f'(-√3/2). -6
ii) f'(0). 3
iii) f'(√3/2). -6
24) If √(1- x²) + √(1- y²)= a(x. - y), then show that dy/dx= √{(1- y²)/(1- x²)}
25) Find second order derivatives of y= sinx with respect to z= eˣ. -(sinx+ cosx)/e²ˣ
26) Obtain differential coefficient of tan⁻¹{(√(1+ x²) -1))/x} w.r.t. cos⁻¹√[{1+ √(1+ x²)}/2√(1+ x²)]. 1
Sap-11
1) Let f, f and g are differentiable functions. If f(0)= 1; g(0)= 2, h(0)=3 and the deriavatives of their pair wise products at x= 0 are (f g)'(0)= 6; (g h)'(0)= 4 and (h f)'(0)= 5 then compute the value of (fgh)'(0). 16
2) a) If y= (cosx)ˡⁿˣ + (ln x)ˣ. (cosx)ˡⁿˣ[ln(cosx)/x - tanx ln x]+ (ln x)ˣ[1/(ln x) + ln(ln x)]
b) y= ₑxᵉ^x+ ₑxᵉ^x+ ₓeˣ^x. ₑxᵉ^xxᵉ^x[eˣ/x+ eˣ ln x] + ₑxᵉ^x+ xᵉ⁻¹ ₓxᵉ[1+ e ln x]+ ₓeˣ^xₑxᵉ[1/x + ₑx ln x]
3) If y= x²/2 + (x/2) √(x² +1) + ln√{x + √(x²+1)}
4)
5) If y= 1+ x₁/(x - x₁) + (x₂x)/{(x - x₁)(x - x₂)} + (x₁x²)/{(x - x₁)(x - x₂)(x - x₃)} + .... upto (n +1) terms then show that dy/dx = (y/x)[x₁/(x₁ - x). + x₂/(x₂ - x) + x₃/(x₃ - x)+.....+ xₙ/(xₙ - x)]
6) If x= cosec θ - sinθ ; y= cosecⁿθ - sinⁿθ, then show that (x²+ 4)(dy/dx)² - n²(y² + 4)= 0.
7) If a curve is represented parametrically by the equations
x= sin(t + 7π/12) + sin(t - π/12)+ sin(t + 3π/12),
y= cos(t + 7π/12) + cos(t - π/12)+ cos(t + 3π/12)
Then find the value of d/dt (x/y - y/x) at t=π/8. 8
8) If x= (1+ ln t)/t² and y= (3+ 2 ln t)/t. Show that y dy/dx = 2x(dy/dx)²+ 1.
9) Differentiate [{√(1+ x²)+ √(1- x²)}/{√(1+ x²) - √(1- x²)}] w.r.t √(1- x⁴). (1+ √(1- x⁴))/x⁸
10) Let g(x) be a polynomial, of degree one and f(x) be defined by
f(x)= [g(x), x<0
{(1+ x)/(2+x)}¹⁾ˣ, x> 0
Find the continuous function f(x) satisfying f"(1)= f'(-1).
11) If √(1- x⁶)+ √(1- y⁶) = a³(x³ - y³), show that dy/dx = (x²/y²) √{(1- y⁶)/(1- x⁶)}.
12) If y= x+ 1/(x + 1/(x +..... show that dy/dx = 1/(2- x/(x + 1/(x + .....
13)
14) Find the deriavative with respect to x of the function:
(Log꜀ₒₛₓ sinx)(Logₛᵢₙₓ cosx)⁻¹ + arcsine{2x/(1+ x²) at x=π/4. 32/(16+π²). - 8/ln 2
15) Suppose f(x)= tan(sin⁻¹(2x))
a) Find the domain and range of f. (-1/2,1/2),(-∞, ∞)
b) express f(x) as an algebraic function of x. f(x)= 2x/√(1- 4x²)
c) find f'(1/4). 16√3/9
16) a) let f(x)= x² - 4x -3, x> 2 and let g be the inverse of f. Find the value of g' where f(x)= 2. 1/6
b) Let f: R---> R be defined as f(x)= x³ + 3x² + 6x - 5+ 4e²ˣ and g(x)=f⁻¹(x), then find g'(-1). 1/14
c) Suppose f⁻¹ is the inverse function of a differentiable function and f and G(x)= 1/f⁻¹(x).
If f(3)= 2 and f'(3)= 1/9, find G'(2). -1
17) If y= tan⁻¹{1/(x² + x +1)} + tan⁻¹{1/(x² + 3x +3)} + tan⁻¹{1/(x² + 5x +7)} + tan⁻¹{1/x² + 7x +13)} + .....to n terms. Find dy/dx, expressing your answer in 2 terms. 1/{1+ (x + n)²} - 1/(1+ x²)
18) If y= tan⁻¹{u/√(1- u²)} and x= sec⁻¹{1/(2u² -1)}, u ∈(0,1/√2) ∪(1/√2,1) show that 2 dy/dx + 1=0.
19) If y= cot⁻¹[{√(1+ sinx)+ √(1- sinx)}/{√(1+ sinx) - √(1- sinx)}], find dy/dx if x∈ (0,π/2) U (π/2,π). 1/2 or -1/2
20) If y= tan⁻¹{x/{1+ √(1- x²)} + sin[2 tan⁻¹√{(1- x)/(1+ x)}] then find dy/dx for x ∈ (-1,1). (1- 2x)/2√(1- x²)
21) a) If y= f(x) and it follows the relation eˣʸ + y cosx= 2, then find i) y'(0) ii) y"(0). y'(0)= -1, y"(0)=2,
b) A twice differentiable function f(x) is defined for all real numbers and satisfies the following conditions
f(0)= 2; f'(0)= 5 and f"(0)= 3
The function g(x) is defined by g(x)= eᵃˣ + f(x)∀ x ∈ R, where 'a' is any constant
If g'(0) + g"(0)= 0. Find the value/s of a. 1,-2
22) If x= 2 cos t - cos2t and y= 2 sin t - sin2t, find the value of (d²y/dx²) when t=π/2. -3/2
23) Find the value of the expression y³ d²y/dx² on the ellipse 3x² + 4y² = 12. -9/4
24) If f: R---> R is a function such that f(x)= x³ + x²f'(1) + x f"(2)+ f"'(3) for all x ∈ R, then show that f(2)= f(1) - f(0).
25) Let P(x) be a polynomial of degree 4 such that P(1)= P(3)= P(5)= P'(7)= 0. If the real number x≠ 1,3,5 is such that P(x)= 0 can be expressed as x= p/q where p and q are relatively prime, then p+ q equals. 100
10) [-(2/3)(1/6 + ln(3/3))x, if x<0
{(1+ x)/(2+x)}¹⁾ˣ, x> 0
Sap-12
1) If √(x² + y²)= e^{arcsine{y/√(x²+ y²)}, show that d²y/dx² = 2(x²+ y²)/(x - y)³, x> 0.
2) Find a polynomial function f(x) such that f(2x) = f'(x) f"(x). 4x³/9
3) If 2x = y¹⁾⁵ + y⁻¹⁾⁵ then (x² -1) d²y/dx² + x dy/dx = ky, the find the value of k. 25
4) Let y= x sin kx. Find the possible value of k for which the differential equation d²y/dx² + y = 2k cos Kx holds true for all x ∈ R. 1,-1or 0
5) Show that if |a₁sinx + a₂ sin2x + .....+ aₙ sin nx|≤ |sinx| for x ∈ R, then |a₁ + 2a₂ + 3a₃ + .....naₙ|≤ 1
6) The function f: R---> R satisfies f(x²).f"(x) = f'(x) f'(x²) for all real x. Given that f(1) = 1 and f"'(1) =8, compute the value of f'(1) + f"(1). 6
7) a) Show that the substitution z= ln(tan(x/2)) changes the equation d²y/dx² + cot x dy/dx + 4y cosec²x= 0 to (d²y/dz²)+ 4y=0
b) If the dependent variable y is changed to z by the substitution y= tan z then the differential equation d²y/dx² = 1+ 2(1+ y)/(1+ y²) (dy/dx)² is changed to d²z/dx² = cos²z + k(dz/dx)², then find the value of k. 2
8) Show that R= {1+ (dy/dx)²)³⁾²/d²y/dx² can be reduced to the from R²⁾³ = 1/(d²y/dx²)²⁾³ + 1/(d²x/dy²)²⁾³.
9) Show that f(x)= (sinx)/x if x≠ 0 and f(0)= 1. Define the function f'(x) for all x and find f"(x) if it exist.
10) Suppose f and g are two functions such that f, g : R---> R,
f(x)= ln {1+ √(1+ x²)} and fpg(x)= ln{x + √(1+ x²)} then find the value of x eᵍ⁽ˣ⁾(f(1/x))' + g'(x) at x = 1. Zero
11) Let f(x)= [xeˣ x≤ 0
x+ x² - x³ x> 0
Then show that
i) f is continuous and differentiable for all x.
ii) f' is continuous and differentiable for all x.
12) f: [0,1] --> R is defined as
f(x)= [ x³(1- x) sin(1/x)) if 0< x≤1
0. If x= 0
Then show that
i) f is differentiable in [0,1]
ii) f is bounded in [0,1]
iii) f' is bounded in [0,1]
13) Let f(x) be a deriavable function at x= 0 and f((x+y)/k)= (f(x) + f(y))/k (k ∈R, k≠ 0,2). Show that f(x( is either a zero or an odd linear function.
14) Let (f(x + y) - f(x))/2 = (f(y) - a)/2 + xy for all real x and y. If f(x) is differentiable and f'(0) exists for all real permissible values of 'a' and is equal to √(5a - 1 - a²). Show that f(x) is positive for all real x.
15) Column- I
A) f(x)= [ln(1+ x³) sin(1/x), if x> 0
0, if x≤ 0
B) g(x) = [ln²(1+ x³) sin(1/x), if x> 0
0, if x≤ 0
C) u(x)=[ ln(1+ (sinx)/2, if x> 0
0, if x≤ 0
D) v(x)= lim ₓ→₀ (2x/π) tan⁻¹(2/t²)
Column II
P) continuous every where but not differentiable at x= 0
Q) differentiable at x=0 but deriavative is discontinuous at x= 0
R) differentiable and has continuous deriavative
S) continuous and differentiable at x= 0.
16) If f(x)=|(x - a)⁴ (x - a)³ 1
(x - b)⁴ (x - b)³ 1
(x - c)⁴ (x - c)³ 1 then
f'(x)= λ|(x - a)⁴ (x - a)² 1
(x - b)⁴ (x - b)² 1
(x - c)⁴ (x - c)² 1
Find the value of λ. 3
17) If f(x)= |cos(x + x²) sin(x + x²) -cos(x + x²)
sin(x - x²) cos(x - x²) sin(x - x²)
sin2x 0 sin2x²
Then find f'(x). 2(1+ 2x) cos2x(x + x²)
18) If α be a repeated root of a quadratic equation f(x)= 0 and A(x), B(x), C(x) be the polynomials of degree 3,4,5 respectively, then show that that
|A(x) B(x) C(x)
A(α) B(α) C(α)
A'(α) B'(α) C'(α) is divisible by f(x), where dash denotes the deriavative.
19) Let f(x)= a+ x b+ x c+ x
l+x m+x n+x
p+x q+x r+x.
Show that f"(x)= 0 and that f(x)= f(0) + kx where k denotes the sum of all the cofators of the elements in f(0).
20) If Y= sX and Z= tX, where all the letters denotes the functions of x and suffixes denotes the differentiation w.r.t.x then show that
|X Y Z
X₁ Y₁ Z₁ = X³|s₁ t₁
X₂ Y₂ Z₂ s₂ t₂
ARS, BRS, CP DRS
SAP- 13
Evaluate the following limits using L' Hospital's rule or otherwise:
1) lim ₓ→₀ [1/(x sin⁻¹x) - (1- x²)/x²]. 5/6
2) lim ₓ→₀ [x + ln(√(x² +1) + x)/x³]. 1/6
3) lim ₓ→₀ [1/x² - 1/sin²x]. -1/3
4)a) lim ₓ→₀ ₓ(xⁿ -1). 1
b) lim ₓ→₀ (cot x - ln²x). does not exist
5) lim ₓ→₀ (1+ sinx - cosx + ln(1- x))/(x tan²x). -1/2
6) lim ₓ→π/2 (sinx - (sinx)ˢᶦⁿˣ)/(1- sinx + ln(sinx)). 2
7) Find the value of f(0) so that the function f(x)= 1/x - 2/(e²ˣ -1), x≠ 0 is continuous at x= 0 and examine the differentiability of f(x) at x= 0. f(0)=1; differentiable at x= 0, f'(0)= -(1/3); f'+0)= -1/3
8) Let f(x)=| |xˣ, if x≠ 0
1, if x=0
Check the continuity and derivability of f(x) at x= 0. Continuous but not deriavable at x=0
9) If lim ₓ→₀ (a sinx - bx + cx² + x³)/(2x² ln(1+ x) - 2x³ + x⁴)
exists and is finite, find the values of a, b, c and the limit. 6,6,0, 3/40
10) lim ₓ→₀ (x⁶⁰⁰⁰ - (sinx)⁶⁰⁰⁰)/(x²(sinx)⁶⁰⁰⁰). 1000
11)a) lim ₓ→₀ (1- cosx cos2x cos3x .....cos x)/x² has the value equal to 253, find the value of n(where n ∈ N). 11
b) lim ₓ→₀ (1- cos⁵x cos³2x cos³3x)/x². 22
c) lim ₓ→₀ (1- cos3x cos9x cos27x.....cos3ⁿx)/(1- cos(x/3) cos(x/9) cos(x/27).....cos(x/3ⁿ). 4
12) Given a real valued function f(x) as follows:
f(x)= (x²+ 2 cosx -2)/x⁴ for x< 0; f(0)= 1/12 and f(x)= (sinx - ln(eˣ cosx))/6x² for x> 0. Test the continuity and differentiability of f(x) at x=0. f is continuous, but not deriavable at x=0
13) Let a₁ > a₂ > a₃......aₙ> 1; p₁> p₂ > p₃.....> pₙ> 0 ; such that p₁ + p₂ + p₃ + .....pₙ = 1. Also F(x)= (p₁a₁ˣ + p₂a₂ˣ+.....pₙaₙˣ)¹⁾ˣ. Compute
i) lim ₓ→₀ F(x). a₁ᵖ¹ a₂ᵖ²....aₙpⁿ
ii) lim ₓ→∞ F(x). a₁
iii) lim ₓ→-∞ F(x). aₙ
14) If x₁, x₁, x₂, x₃,.......xₙ₋₁ be n zero's of the polynomial P(x)= xⁿ +αx +β, show that the value of Q(x)= (x₁ - x₂)(x₁ - x₃)(x₁ - x₄)......(x₁ - xₙ₋₁), is equal to ⁿC₂x₁ⁿ⁻².
15) Column I contains function defined on R and Column II contains their properties. Match them
Column I
A) lim ₓ→ ∞ [(1+ tan(π/2n))/(1+ sun(π/3n))]ⁿ equal
B) lim ₓ→₀ 1/(1+ cosecx)¹/ˡⁿ ˢᶦⁿˣ equals
C) lim ₓ→₀ ((2/π) cos⁻¹x)¹⁾ˣ equals
Column II
P) e
Q) e²
R) 2^(-2/π)
S) e^(π/6)
AS BP CR
SAP-14
1)a) if ln(x + y)= 2xy, then y'(0) is
a) 1 b) -1 c) 2 d) 0. a
b) { b sin⁻¹{(x + c)/2}, -1/2< x<0
f(x)= 1/2, at x= 0
(eᵃˣ -1)/x, 0< x<1/2.
If f(x) is differentiable at x=0 and |c|< 1/2 then find the value of a and prove that 64b² = 4 - c². 1
2) a) If y= y(x) and it follows the relation x cos y + y cosx =π, then y"(0) is
a) 1 b) -1 c) π d) -π. C
b) If P(x) is a polynomial of degree less than or equal to 2 and S is the set of all such polynomials so that P(1)= 1, P(0)=0 and P'(x)> 0 ∀ x ∈ [0,1], then
a) S= φ b) S=(1- a)x² + ax, 0< a<2 c) (1- a)x² + ax, a ∈(0,∞) d) S= (1- a)x²+ ax, 0< a<1. B
c) If f(x) is a continuous and differentiable function and f(1/n)= 0, ∀ n≥ 1 and n ∈I, then
a) f(x)= 0, x ∈(0, 1]
b) f(0)=0, f'(0)= 0
c) f'(x)= 0= f"(x), x ∈ (0,1]
d) f(0)=0 and f'(0) neeed not to be zero b
d) If f(x - y)= f(x).g(y) - f(y). f(x) and g(x - y)= g(x). g(y)+ f(x). f(y) for all x, y ∈R. If right hand Derivative at x=0 exists for f(x). Find deriavative of g(x) at x= 0. g'(0)= 0
3) For x> 0, lim ₓ→₀ ((sinx)¹⁾ˣ+ (1/x)ˢᶦⁿˣ) is
a) 0 b) -1 c) 1 d) 2. C
4) d²x/dy² equals
a) (d²y/dx²)⁻¹
b) -(d²y/dx²)⁻¹ (dy/dx)⁻³
c) (d²y/dx²)(dy/dx)⁻²
d) -(d²y/dx²)(dy/dx)⁻³. D
5) a) let g(x)= ln f(x) where f'(x) is twice differentiable positive function on (0, ∞) such that f(x+1)=x f(x). Then for N= 1,2,3
g"(N + 1/2) - g"(1/2) =
a) -4{1+ 1/9 + 1/25 + ......1/(2N -1)²
b) 4{1+ 1/9 + 1/25 + ......1/(2N -1)²
c) - 4{1+ 1/9 + 1/25 + ......1/(2N +1)²
d) 4{1+ 1/9 + 1/25 + ......1/(2N + 1)². A
b) Let f and g be real valued functions defined on interval (-1,1) such that g"(x) is continuous, g(0)≠ 0, g'(0)= 0, g"(0)≠ 0, and f(x)= g(x) sinx.
Statement 1: lim ₓ→₀ [g(x) cotx - g(0) cosecx]= f"(0)
And
Statement 2: f'(0)= g(0)
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true. A
6) If the function f(x)= x³+ eˣ⁾² and g(x)= f⁻¹(x), then the value of g'(1) is. 2
7) Let f(θ)= sin{tan⁻¹{sinθ/√cos2θ))}, where -π/4< θ <π/4. Then the value of d/d(tanθ) (f(θ)) is 1
TANGENT, NORMAL,
1) Find the equation of the tangent to the curve y= (x³-1)(x -2) at the points where the curve cuts the x-axis . y+ 3x -3=0 and 7x - y -14=0
2) The equation of the tangent to the curve x= a cos³t, y= a sin³t at t point is
a) x sec t - y cosec t= a
b) x sec t + y cosec t= a
c) x cosec t - y sec t= a
d) x cosec t + y sec t= a. B
3) The equation of the normal to the curve y= x+ sinx at x=π/2 is
a) x=2 b) x=π c) x + π= 0 d) 2x= π. D
4) The equation of normal to the curve x+ y = xʸ, where it cuts x-axis is
a) y= x+1 b) y= - x+1 c) y= x - 1 d) y= - x - 1 c
5) Find the distance between the point (1,1) and the tangent to the curve y= e²ˣ+ x² drawn from the point where the curve cuts y-axis . 2/√5
6) Find the equation of a line passing through (-2, 3) and parallel to tangent at origin for the circle x²+ y²+ x - y=0. x - y +5=0
7) The angle of intersection between the curve x² - 32y=0 and y² - 4x =0 at the point (16, 8) is
a) 60° b) 90° c) tan⁻¹(3/5) d) tan⁻¹(4/3) c
8) Check the orthogonally of the curves y²= x & x²= y. No
9) If curve y= 1- ax² and y= x² intersect orthogonally then the value of a is
a) 1/2 b) 1/3 c) 2 d) 3. B
10) if two curves y= aˣ and y= bˣ intersect at an angle α, then find the value of tanα. |(ln a - ln b)/(1+ ln a ln b)|
11) Find the angle of intersection of curves y= 4 - x² and y= x². tan⁻¹(4√2/7)
12) The length of the normal to the curvenx= a(θ+ sinθ), y= a(1- cosθ) at θ=π/2.
a) 2a b) a/2 c) √2 a d) a/√2. C
13) The length of the tangent to the curve x= a(cos t + log tan(t/2), y= a sin t is
a) ax b) ay c) a d) xy. C
14) prove that at any point of a curve (length of sub tangent)( length of sub normal) is equals to square of the ordinates of point of contact.
15) Find the length of sub tangent to the curve x²+ y²+ xy= 7 at the point (1, -3). 15
APPROXIMATION
1) Find the approximate value of √25.2. 5.02
2) find the approximate value of ³√0.009. 0.208
RATE MEASUREMENT
1) The volume of a cube is increasing at a rate of a 9 cm³/s. How fast is the surface area increasing when the length of an edge is 10 cm. 3.6cm²/s
2) x and y are the sides of 2 squares such that y= x - x². Find the rate of change of the area of the second square with respect to the first square. 2x²- 3x+1
3) What is the rate of change of the area of a circle with respect to radius r at r= 6cm. 12π
4) A stone is dropped into a quite lake and waves move in circles at the speed of 5cm/s. At the instant when the radius of the circular wave is 8cm, how fast is the enclosed area increasing ? 80π cm²/s
5)
MONOTONICITY, MAX & MIN
APPLICATION OF DERIAVATIVES
SAP- 1
1) If the relation between subnormal SN and subtangent ST at any point S on the curve by²= (x + a)³ is p(SN)= q(ST)², then find the value of p/q in terms of b and a.8b/27
2) Find the possible values of 'a' such that the inequality 3 - x²> |x - a| has atleast one negative solution. a∈(-13/4,3)
3) Find the angle of the intersection of the curve, y= [|sinx|+ |cosx|] and x²+ y²= 5 where [.] denotes greatest integral function. tan⁻¹(2)
SAP-2
1) If a variable tangent to the curve x²y= c³ makes intercepts a, b on x and y axis w, then the value of a²b is
a) 27c³ b) 4c³/27 c) 27c³/4 d) 4c³/9
2) The number of values of c such that the straight line 3x + 4y= c touches the curve x⁴/2= x + y is
a) 0 b) 1 c) 2 d) 4
3) Let f(x)= x³+ ax + b with a≠ b and suppose the tangent lines to the graph of f at x= a and x= b have the same gradient. Then the value of f(1) is equal to
a) 0 b) 1 c) -1/3 d) 2/3
4) The tangent to the curve 3xy²- 2x²y = 1 at (1,1) meets the curve again at the point
a) (16/5, 1/20) b) (-16/5, 1/20) c) (1/20, 16/5) d) (-1/20, 16/5)
5) The curve y- eˣʸ + x=0 has a vertical tangent at
a) (1,1) b) (0,1) c) (1,0) d) no point
6) Suppose f and g both are linear function with f(x)= -2x +1 and f(g(x))= 6x -7, then slope of line y= g(x) is
a) 3 b) -3 c) 6 d) -2
7) A curve with equation of the form y= ax⁴+ bx³+ cx + d has zero gradient at the point (0,1) and also touches the x-axis at the point (-1,0) then the values of x for which the curve has a negative gradient are
a) x>-1 b) x< 1 c) x < -1 d) -1≤ x ≤ 1
8) The line which is parallel to x-axis and crosses the curve y=√x at an angle of π/4 is
a) y=-1/2 b) x = 1/2 c) y= 1/4 d) y= 1/2
9) The lines tangent to the curve y³- x²y + 5y - 2x =0 and x⁴ - x³y²+ 5x + 2y=0 at the origin intersect at an angle θ equal to
a) π/6 b) π/4 c) π/3 d) π/2
10) The angle of intersection of the curve 2y = x³ and y²= 32x at origin is
a) π/6 b) π/3. c) π/2 d) π/4
11) The angle of intersection of the curve √y = x and 7= x + 6y at (1,1) is
a) π/5 b) π/4. c) π/3 d) π/2
12) At any point of a curve √{subnormal/sub tangent} is equal to
a) the abscissa of that point
b) the ordinate of that point
c) slope of the tangent at that point
d) slope of the normal at that point
13) The length of the tangent to the curve x= a(θ+ sinθ), y= a(1- cosθ) at θ points is
a) 2 sin(θ/2) b) a sinθ c) 2a sinθ d) a cosθ
14) The length of the subnormal of the curve y²= 8ax (a> 0) is
a) 2a b) 4a c) 6a d) 8a
15) A 13 ft ladder is leaning against a wall when its base starts to slide. At the instant when the base is 12 ft.
a) 12/13 radius/sec b) - 1 radius/sec c) -13/1² radius/sec d) -10/13 radius/sec
16) Water is poured into an inverted conical vessel of which the radius of the base is 2m and height 4m at the rate of 77 litre/minute. The rate at which the water level is rising at the instant when the depth is 70 cm is
a) 10 cm/min b) 20 cm/min c) 40 cm/min d) none
17) A point is moving along the curve y³= 27x. The interval in which the abscissa changes at slower rate than ordinate is
a) (-3,3) b) (-∞,∞) c) (-1,1) d) (-∞,-3) U (3,∞)
18) A particle moves along the curve y= x³⁾² in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units per second. The value of dx/dt when x= 3 is
a) 4 b) 9/2 c) 3√3/2 d) none
19) Which of the following pair/s of curves is/are orthogonal.
a) y²= 4ax; y= e⁻ˣ⁾²ᵅ
b) y²= 4ax; x²= 4ay
c) xy= a², x²- y²= b²
d) y= ax, x² + y²= c²
20) If x/a + y/b = 1 is a tangent to the curve x= Kt, y= K/t, K> 0 then
a) a> 0, b> 0 b) a> 0, b< 0 c) a< 0, b> 0 d) a< 0, b< 0
21) The coordinates of the point/s on the graph of the function, f(x)= x³/3 - 5x²/2+ 7x - 4 where the tangent drawn cut off intercepts from the coordinates axes which are equal in magnitude but opposite in signs, is
a) (2,8/3) b) (3,7/2) c) (1,5/6) d) none
22) For the curve represented parametrically by equations , x= 2 ln cot t +1 and y= tan t + cot t
a) tangent at t=π/4 is parallel to x-axis
b) normal at t=π/4 is parallel to y-axis
c) tangent at t=π/4 is parallel to the line y= x
d) normal at t=π/4 is parallel to the line y= x
23) Consider the curve f(x)= x¹⁾³, then
a) the equation of tangent at (0,0) is x= 0
b) the equation of normal at (0,0) is y = 0
c) normal to the curve does not exist at (0,0).
d) f(x) and its inverse meet at exactly 3 points.
24) Equation of common tangent/s of x²= 12 and xy= 8 is/are
a) y= 3x + 4√6 b) y=- 3x + 4√6 c) 3y= x + 4√6 d) y= - 3x - 4√6
1c 2b 3b 4b 5c 6b 7c 8d 9d 10c 11d 12c 13a 14b 15b 16b 17c 18a 19acd 20ad 21ab 22ab 23abd 24bd
Sap-3
The angle at which the curve y= Keᴷˣ intersects the y-axis is
a) tan⁻¹k² b) cot⁻¹(k²) c) sin⁻¹√{1/√(1+ k⁴)} d) sec⁻¹√+1+ k⁴)
2) The coordinates of point/s at each of which the tangents to the curve y= x³- 3x²- 7x +6 cut off on the positive semi axis OX a line segment half that on the negative semi axis OY is/are given by
a) (-1,9) b) (3,-15) c) (1,-3) d) none
3) The abscissa of the point on the curve √(xy)= a+ x, the tangent at which cuts off equal intercepts from the co-ordinate axes is (a> 0)
a) a/√2 b) -a/√2 c) a√2 d) -a√2
4) A cubic polynomial f(x)= ax³+ bx²+ cx + d has a graph which is tangent to the x-axis at 2, has another x-intercept at -1 and has y-intercept at -2 as shown.
The value of a+ b + c+ d equals
a) -2 b) -1 c) 0 d) 1
5) Equation of a tangent to the curve y cotx = y³ tanx at the point where the abscissa is π/4 is
a) 4x + 2y =π +2
b) 4x - 2y =π +2 c) x=0 d) y=0
6) Consider the curve represented parametrically by the equation x= t³- 4t²- 3t and y= 2t²+ 3t -5 where t ∈ R
if H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then
a) H=2, V=1 b) H=1, V=2 c) H=2, V=2 d) H=1, V=1
7) If y= f(x)= be the equation of a parabola which is touched by the line y= x at the point where x=1. Then
a) f'(1)= 1 b) f'(0)=f'(1) c) f(ó)= 1 - f'(0) d) f(0)+ f'(0) + f''(0)= 1
8) At the point P(a, A') on the graph of y= xⁿ (n ∈ N) in the first quadrant a normal is drawn. The normal intersects the y-axis at the point (0,b), if lim ₓ→₀ b= 1/2, then n equals
a) 1 b) 3 c) 2 d) 4
9) A horse runs along a circle with the speed of 20 km/he. A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse move along the fence at the moment when it covers 1/8 of the circle in km/he is
a) 20 b) 60 c) 30 d) 40
10) Equation of the line through the point (1/2,2) and tangent to the parabola y= -x²/2 +2 and secant to the curve y= √(4- x²) is
a) 2x + 2y -5=0 b) 2x + 2y - 9 =0 c) y - 2=0 d) none
11) if the tangent at P of the curve y²= x³ intersect the curve again at Q and the straight lines OP, OQ make an angles α, β with the x-axis where O is origin then tanα/tanβ has the value equals to
a) -1 b) -2 c) 2 d) √2
12) Let f(x) be a non zero function whose all successive derivatives exist and are nonzero. If f(x), f'(x) and f"(x) are in GP and f(0)= 1, f'(0)= 1, then
a) f'(x)< 0 ∀x ∈ R
b) f''(x)< 1 ∀x ∈ R
c) f''(x) ≠ f"'(0)
d) f''(x)> 0 ∀x ∈ R
13) If the line ax + by + c=0 is a normal to the curve xy= 1, then
a) a> 0, b >0 b) a> 0, b < 0 c) a< 0, b >0 d) a< 0, b <0
1bc 2b 3ab 4b 5abd 6b 7ac 8c 9d 10a 11b 12d 13bc
Sap-4
1)
Column I
A) The angle of intersection of y²= 4x and x²= 4y is 90° and tan⁻¹(m/n) then |m+ n| is equal to (m and n are co prime)
B) The area of Triangle formed by normal at the point (1,0) on the curve x= eˢᶦⁿʸ with axes is
C) If the angle between the curves x²y=1 and y= e²⁽¹⁻ˣ⁾ at the point (1,1) is θ then tanθ is equal to
D) The length of subtangent at any point on the curve y= beˣ⁾³ is equal
Column II
p) 0
q) 1/2
r) 7
s) 3
2) Column I
A) The slope of the curve 2y²= ax²+ b at (1,-1) is -1, then
B) If (a,b) be the point on the curve a normal to the curse next equal interested to the axis then if the tangent at a point 9y²= x³ where normal to the curve makes equal intercepts with the axes, then
C) If the tangent at a point (1,2) on the curve y= ax²+ bx + 7/2 be parallel to the normal at (-2,2) on the curve y= x²+ 6x +10, then
D) if the tangent to the curve xy+ ax+ by=0 at (1,1) is inclined an angle tan⁻¹2 with x-axis , then
Column II
p) a- b= 2
q) a- b = 7/2
r) a- b = 4/3
s) a- b = 3
Assertion and Reason
A) Statement -1 is true, statement -2, is true; statement -2 is a correct explanation for statement 1.
B) statement 1 is true, statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
C) Statement 1 is true, statement 2 is false
D) statement 1 is false, statement 2 is true.
1) statement 1: The ratio of length of tangent to length of normal is proportional to the ordinate of the point of tangency at the curve y²= 4ax.
Because
Statement 2: length of normal and tangent to a curve y= f(x)= |y√(1+ m²| and|y√(1+ m²)/m|, where m= dy/dx.
a) A B) B) c) C d) D
2) Statement 1: The product of the ordinates to the point of tangency to the curve (1+ x²)y = 2- x, where the tangent makes equal intercept with cordinate axes is equals to 1.
Because
Statement 2: slope of straight line making equal intercept with coordinate axis is equal to 1.
a) A B) B C) C d) D
3) Statement 1: any tangent to the curve y= x⁷+ 8x³+ 2x +1 makes an acute angle with the positive x-axis.
Because
Statement 2: any tangent to the curve y= a₀x²ⁿ⁺¹ + a₁x²ⁿ⁻¹ + a₂x²ⁿ⁻³ + ....+ aₙx +1 makes an acute angle with the positive x-axis where a₁, .....aₙ₋₁ ≥ 0; a₀, aₙ > 0 and n ∈ N.
a) A B) B C) C d) D
Comprehensive type questions
consider the function f(x)= x²f(1) - x f'(2)+ f''(3) such that f(0)= 2
1) The value of f(1)
a) 0 b) 1 c) 2 d) -1
2) Equation of tangent to y= f(x) at x= 3 is
a) y= x -7 b) y= x/4 -7 c) y= 4x -7 d) none
3) the angle of intersection of y= f(x) and y= 2e²ˣ is
a) tan⁻¹(3/4) b) tan⁻¹(4/3) c) 0 d) tan⁻¹(6/7)
Comprehensive type questions
Let y = f(x) be a differentiable function which satisfies f'(x) = f²(x) and f(0)= -1/2. The graph of the differentiable function y= g(x) contains the point (0,2) and has the property that for each number P, the line tangent to y= g(x) at (P, g(P)) intersects x-axis at P+2.
1) if the tangent is drawn to the curve y= f(x) at a point where it crosses the y-axis then its equation is
a) 2 = x - 4y b) 2 = x + 4y c) 0 = x + 4y -2 d) none
2) The function y= g(x) is given by
a) e⁻ˣ⁾²/2 b) e⁻ˣ⁾² c) 2 e⁻ˣ⁾² d) e⁻ˣ⁾² +2
3) The number of point of intersection of y= f(x) and y= g(x)
a) 4 b) 0 c) 2 d) 1
1) Ar Bq Cp Ds
2) Ap Br Cq Ds
1A 2C 3A
1) 1a 2c 3d
2) 1a 2c 3d
SAP- 5
1) Find the equation of the tangents drawn to the curve y²- 2x³- 4y +8=0 from the point (1,2).
2) Find the equation of normal to the curve x²= 4y passing through the point (1,2).
3) Prove that the length intercepted by the co-ordinate axes on any tangent to the curve, x²⁾³ + y²⁾³ = c²⁾³ is constant.
4) if tangent to the curve y= x²- 5x +6 passes through the point M(a,6), find the set of values of a.
5) Find all the tangents to the curve y= cos(x + y), -2π≤ x ≤ 2π, that are parallel to the line x+ 2y=0.
6) Find the equation of the normal to the curve y= (1+ x)ʸ + sin⁻¹(sin²x) at x=0.
7) Prove that the segment of the normal to the curve x= 2a sin t+ a sin t cos²t; y= - a cos³t contained between the coordinate axes is equals to 2a.
8) A function is defined parameteically by the equations
x=[2t + t² sin(1/t) if t≠ 0
0 if t= 0
And
y=[(1/t) sin t² if t≠ 0
0. If t=0
Find the equation of the tangent and normal at the point for t=0 if they exist.
9) Find the point of intersection of the tangents drawn to the curve x²y = 1- y at the points where it is intersected by the curve xy= 1- y.
10) a) Find the angle of intersection between the curves y²= 2x/π and y= sinx at x=π/2.
b) Find the angle of intersection between the curves y²= 4x and x²+ y²= 5.
11) Show that the angle between the tangent at any point A of the curve ln(x²+ y²)= C tan⁻¹(y/x) and the line joining A to the origin is independent of the position of A on the curve.
12) a) Find the condition that the curve x²/a + y²/b = 1 and x²/a' + y²/b' = 1 may cut originally .
b) Show that the curves x²/(a²+ K₁) + y²/(b²+ K₁) = 1 andmx²/(a²+ K₂) + y²/b²+ K₂) = 1 intersect orthogonally (K₁ ≠ K₂).
13) a) Find the value of n so that the sub normal at any point on the curve xyⁿ = aⁿ⁺¹ may be constant.
b) Show the curve y= a ln(x²- a²), sum of the length of tangent subtangent varies as the product of the coordinates of the point of contact.
14) Water is flowing out at the rate of 6 m³/min from a reservoir shaped like a hemispherical bowl of radius R= 13m. The volume of water in the hemisperical bowl is given by V= πy²(3R - y)/3 when the water is y metre deep. Find
a) At what rate is the radius of the surface changing when the water is 8m deep.
b) At what rate is the radius of the water surface changing when the water is 8m deep .
15) Sand is pouring from a pipe at the rate of 12cc/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always 1/6th of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm.
16) if in a triangle ABC , the side c and angle C remain constant, while the remaining elements are changed slightly, show that da/cosA + db /cosB =0.
17) a) use the differentiation to a approximate the value of
a) √36.6
b) ³√26.
b) if the radius of a sphere is measured 9 cm with an error of 0.03cm, then find the approximate error in calculating its volume.
18) A man 1.5m tall walks away from a lamp-post 4.5m high at the rate of 4 kmph.
a) How fast is the farther end of the shadow moving on the pavement ?
b) How fast is his shadow lengthening ?
1) 2√3 x - y =2(√3-1) or 2√3 x + y =2(√3+1)
4) a belongs to (-∞,0] U [5, ∞)
5) x + 2y = π/2 or x + 2y = -3π/2
6) x + y = 1
8) tan: 2y - x = 0 , nor: 2x + y = 0 9) (0,1)
10)a) cot⁻¹π b) tan⁻¹3
11) tan⁻¹(2/C)
12) a- b = A' - b'
13) -2
14) a) -1/24π m/min b) -5/288π m/min
15) 1/48π cm/s 17a)i) 6.05 ii) 80/27 b) 9.72π cm²
18) a) 6 kmph b) 2 kmph
SAP- 6
1) The curve y= ax³+ bx²+ cx+ 5, touches the x-axis at P (-2,0) in cuts the yaxis at a point Q, where its gradient is 3. Find a, b, c.
2) at time t> 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t= 0, the radius of the sphere is 1 unit and at t= 15 the radius is 2 units.
a) Find the radius of the sphere as a function of time t.
b) at what time t will the volume of the sphere be 27 times its volume at t= 0.
3) If p₁ and o₂ be the length of the perpendicular from the origin on the tangent and normal respectively at any point (x,y) on a curve, then show that
p₁= |x sinψ - y cosψ|
p₂= |x cosψ + y sinψ|
Where tanψ = dy/dx. If in the above case, the curve be x²⁾³ + y²⁾³ = a²⁾³ then show that: 4p₁²+ p₂²= a².
4) A and B are points of the parabola y= x². The tangents at A and B meet at C. The median of the triangle ABC from C has length m units. Find the area of the triangle in terms of m.
5) tangent at a point P₁ (other than (0,0)) on the curve y= x³ meets the curve again at P₂, The tangent at P₂ meets the curve at P₃ and so on. Show that the abscissa of P₁, P₂, P₃, ...... Pₙ, form a GP. Also find the ratio {area (∆ P₁P₂P₃)}/{Area(∆P₂P₃P₄)}.
6) The chord of the parabola y= - a²x² + 5ax - 4 touches the curve y= 1/(1- x) at the point x= 2 and is bisected by the point. Find a.
7) Prove that the segment of the tangent to the curve y= (a/2) ln {a+ √(a² - x²)}/{a - √(a² - x²)} - √(a² - x²) contained between the y axis and the point of tangency has a constant length.
8) if the tangent at the points +x₁, y₁) to the curve x³ + y³ = a³ meets the curve again in (x₂, y₂( then show that x₂/x₁ + y₂/y₁ = -1.
9) A variable in the xy plane has its orthocentre at vertex B, a fixed vertex A at the origin and the third vertex C restricted to lie on the parabola y= 1+ 7x²/36. The point B starts of the point (0,1) at time t=0 and moves upward along the y-axis at a constant velocity of 2 cm/sec. How fast is the area of the triangle increasing when t= 7/2 sec.
10) What normal to the curve y= x² forms the shortest chord?
1) -1/2,-3/4,3
2) a) r= (1+ t)³⁾⁴ b) 80
4) m√m/√2 5) 1:64 6) 1 9) 66/7 10) x + √2 y =√2 or x - √2 y = - √2
SAP- 7
1) If the normal to the curve , y= gpf(x) at the point (3,4) makes an angle 3π/4 with the positive x-axis. thennf'(3)
a) -1 b) - 3/4 c) 4/3 d) 1
2) The point/s on the curve y³+ 3x²= 12y where the tangent is vertical, is/are
a) (±4/√3,-2) b) (± √11/3), 1) c) (0,0) d) (±4/√3,2)
3) tangent to the curve y= x²+ 6 atma point P(1,7) touches the circle x²+ y²+ 16x + 12y + c=0 at a point Q. Then coordinates of Q are
a) (-6,-11) b) (-9,-13) c) ( -10, -15) d) (-6,-7)
4) If |f(x₁) - f(x₂)|< (x₁ - x₂)½, for all x₁, x₂ ∈ R. Find the equation of tangent to the curve y= f(x) at the point (1,2).
5) The tangent to the curve y= eˣ drawn at the point (c, eᶜ) intersect the line joining the points (c -1, eˣ⁻¹) and (c +1, eˣ⁺¹)
a) on the left of x= c
b) on the right of x= c
c) at no points
d) at all points
1d 2d 3d 4) y -2=0 5a
MONOTONICITY
1) Let f(x)= x³- 3x +2. Examin the nature function at points x= 0, 1 and 2. D, neither nor, increasing
2) If function f(x)= x³+ Kx² -Kx +1 is increasing at x= 0 & decreasing at x= 1, then find the greatest integral value of K. -1
3) Prove that the function f(x)= log{x³ + √(x²+1)} is entirely increasing.
4) Find the intervals of monotonocity of the function y= x² - log|x|, (x≠0). (- ∞, -1/√2) U (0,1/√2)
5) If a,b,c are real then
f(x)= x+ a² ab ac
ab x+ b² bc
ac bc x+ c² is decreasing in
a) ((-2/3)(a²+ b²+ c²),0)
b) (0, (2/3)(a²+ b²+ c²))
c) (0, (1/3)(a²+ b²+ c²)) d) no where a
6) Prove the following
i) y= eˣ + sinx is increasing in x ∈ R⁺.
ii) y= 2x - sinx - tan x is decreasing in x ∈ (0,π/2).
7) If f(x)= sinx + ln|secx + tanx| - 2x for x∈ (-π/2,π/2) then check the monotonicity of f(x). Increasing
8) The function f(x)= 2 log(x -2) - x²+ 4x +1 increases in the interval
a) (1,2) b) (2,3) c) (5/2,3) d) (2,4). B
9) Show that f(x)= sin⁻¹{x/√(1- x²) - ln x is decreasing in x ∈ [1/√3, √3]. Also find its range.
10) Let f(x)= x - 1/x. Find greatest and least value of f(x) for x ∈ (0,4). Not defined
11) Find the critical points and stationary point of the function f(x)= eˣ/x. 1
12) For x ∈ (0,π/2) prove that sinx < x < tanx.
13) For x∈ (0,1), show that x - x³/3 < tan⁻¹x < x - x³/6 and hence or otherwise find
lim ₓ→₀ [tan⁻¹x/x]
14) Find the larger of ln(1+ x) and (tan⁻¹x)/(1+ x), for x > 0. ln(1+ x)
15) Show that sinx < x - x³/6 + x²/120, for x > 0.
16) For x > 1, y = ln x satisfies
a) x - 1 > y b) x²- 1 > y c) y> x -1 d) (x -1)/x < y. Abd
17) prove that for any two numbers x₁ and x₂, (3e^x₁ + e^x₂)/4 > e^{(3x₁+x₂)/4}
18) In any triangle ABC show that sinA + sinB + sinC≤ 3√3/2
19) In any triangle ABC , prove that cosA + cosB + cosC≤ 3/2
ROLLE'S THEOREM
1) Verify Rolle's theorem for the function f(x)= x³- 3x²+ 2x in the interval [0,2].
2) Show that between any two roots of e⁻ˣ - cosx =0 there exists at least one root of sinx - e⁻ˣ = 0.
3) Verify Rolle's theorem for y= - x⁴⁾³ on the interval [-1,1].
4) Show that the between any two roots of tanx = 1 there points exist at least one root of tanx = -1.
5) Find c of the lagrange's mean value theorem for the function f(x)= 3x²+ 5x +7 in the interval [1,3]. 2
6) If f(x) is continuous and differentiable over [-2,5] and -4≤ 3 for all x in (-2,5), then the greatest possible value of f(5) - f(-2) is
a) 7 b) 9 c) 15 d) 21 d
7) If function f(x) and g(x) are continuous in [a, b] and differentiable in (a,b), show that there will be at least one point c, a < c < b such that
| f(a) f(b) = (b-a) |f(a) f'(c)
g(a) g(b)| g(a) g'(c)|
8) If f(x)= x⅖ in [a,b], then show that there exist at least one c in (a,b) such that a, c, b are in AP.
9) Using LMVT, Show that x/(1+ x) < ln(1+ x)< x for x> 0
10) If g(x)= f(x) + f(1- x) and f"(x)< 0; 0≤ x≤1, show that g(x) increasing in x ∈[0, 1/2] and decreasing in x ∈ [1/2,1]
11) prove that if 2a₀²< 15a, all roots of x⁵- a₀x⁴ + 3ax³ + bx² + cx + d=0 can not be real. it is given that a₀, a, b, c, d ∈ R.
so by using mean value theorem compare which of the two is greatest lead the secondary activities of the function exist in satisfy if then show that for all 01
Function increasing and decreasing in decreasing function increasing function the function always increases always decreases never decreases sometimes increasing substan decreases the functions defined by decreasing for all decreasing and increasing increasing for all decreasing increasing in functions is increasing function is the function is mono technical decreasing for the value of a for which the function decreases for all real values LED quality expression which is positive increasing in decreasing increasing and decrease increasing decreasing which of the following property does have on the interval 06 exact continuous monotonic the length of a largest continuous interval in which the function is monotonic is the largest state of the real values for which the two set of the real values of x power which the function is positive number of solution of the equation roller stering in the indicated interval will not revalid for the following function consider the function is continuous role's theorem is not applicable in 23 roller free is not applicable in 23 is not a Rebel 237 is not applicable roll is theorem is applicable as satisfy all the condition Rolex is 12 if the function certificate then the value of age 123461 considered the function on the interval sector 6 the value of the satisfy the conclusion the value of theorem is BA functional for all 46 for all 24 then have a zero bit wing the theory best describe this age always theorem main value theorem maximum minimum value theorem intermediate value theorem considered then which of the following is correct cholesterol is applicable to both will not applicable with is applicable and rolls theorem is applicable both for real numbers then is increasing whenever is increasing is increasing wherever is decreasing is a decreasing when is decreasing nothing can be said in general then increases in the interval is increasing 12 is containers 13 does not exist as the maximum value no roots as a list one roots vanishes from some to function defined on an interval such that strictly decreasing while strictly increasing on them the product function is cleancreasing the product function decreasing on article increasing on is monotonic decreasing on
Differentiable function strictly increasing provided rainy season number of points provided vanises are discrete points through the number of this discovered points may not be finished provided Venice or discrete points on the number of these describes points must be infinite the function is it soon inverse decreases for all values of the graph entirely what the sex is bound for all x function is toy differentiable in 021 continuous 02 then which of the following is at definitely true considered the functions and the number of 01 at the derivatives benefits is 012 infinite increasing whenever is increasing increasing one hour is decreasing decreasing whenever is decreasing decreasing why are there in the interval both are increasing function both are decreasing function increasing function is an increasing function at least one real roots more than one real roots is a polynomial one real roots the functions increasing 032 decreasing increasing increasing decreasing suppose is differentiable for all such that as the value equals 3468 the function increasing on decreasing increasing decreasing decreasing increasing number of solutions are define the equation 123 number of solution satisfying the equation 123 decreases 023 decreases 231 023 231 cannot the fractional part integral part function respectively then which of the following statements sold goods for the function where Eva and increasing is odd and decreasing even and decreasing is odd and increasing number of roots of the equation 246 infinite equation no real roots to reality King groups exactly one negative group exactly one would one minus 1 and 1 the value of T for which the function decreases for all real x
Decoration as at least one route then the equation as a list one route then interval on which is applicable for dinantible in which is the applicable for decoration as at least one route 12 satisfy all the condition for always theorem to is 11 is decreasing function the greatest numbers is increaseing and decreasing good depend as the above function can be classified as injective but not adjective subjective but all injective neither injective notes adjective both in injective as well as adjective the graph is best represent as are the points at which is consequentaneous or Denver scratches integer functions equals to 2 equals to 3 greater than 3 greater than 2 but last than three considered the following functions this function has monotono city as given below decreasing increasing decreasing increasing a rectangle is found such that portion of the tangent to the curve into sed between the lines sports of the line interceptor between the curve and exacts to blood is given by 1020 12210201 area triangle roots to real to imagine Rose to complex two rational roots for reality King root 2 real considering rules and two irrational hits
Find the interval of the monotational for the following function represent to solution the number line values of X satisfying then equality so that the function decreases everywhere find the trouble of the monotonicity of the function find the interval of the city of the function find the value who is the function is decreasing increasing find the range of the value A for who is the function is monotonic and find the set of the values for which is invertible find the range of the values find the set of all values of parameter a for who is the function increases for all and has no critical points for all find the greatest in the list value of the following function the given interval if the exist Prove the following differentiable on support show that for using monotonic City prove that identify which is greater verify always being positive integer check the validity of them for the function if the equation has a positive road prove that the equation also a positive root smaller than differentiable function for such that so that the exist number of satisfying are continuous in andreavable in then show that there is a value of a line between a and b such that the functions taken equal values of the end of the 2011 next year that the disability of the concept does not beneficial any point of the interval and explain the deviation from the relation 0182.0 graph given by prove that there exist a point on the prob to insect that tangent is parallel find the coordinates with the head of language formula Prove the inqualities for the condition the continuous and differentiable on then show that show that the functions cannot have more than 2 real roots is available industry of the behaviour of the following function custard their graphs investigate the behaviour of the functions and construct graph how many solution does they question process
Let Gaya the differences function for all prove that increase is increases find all the values of parameter for which the function increases no critical point prove that is differentiable on any Riyal there is an such that be continuous on and differentiable so that the existing such that prove that in equality is using determine which of the two numbers is a great time identify which is greater function suppose that on the interval 24 the function is the currency of 215 find the bounding function 24 using prove that all positive then show that show that for all interval which is increasing find the minimum value of differentiable function is double differential functions that prove that the exist some 33 such that
For all consider the following statements both the sin x and the process decreasing function in interval if a differentiable function decreases in interval ab then available also decreases in AB which of the following is true both are wrong both are correct but is not correct explanation is correct is correct explanation is correct is wrong then decreases increasing decreasing increasing decreasing so that the equation as a unique root in the interval 121 and length of the longest interval in who is the function increasing age using the relation or otherwise prove that differentiable function so there is exist so that their exist in the applicable using rolls during prove that at least one route between prove that using the inqualities if any used is quickly increasing as a local area at least one person such that where then the maximum number of zeros in the interval for each real there is a point such that if a continuous function defined on the real line positive and negative values in the equation as a route for example if it is not continuous function is positive but some point and its minimum negative the setup all values of Cape for which has the two distinct root is let the function be given by even and strictly increasing Audi strictly decreasing increasing or odd but strictly increasing let be a non constunt why the principle concept depends on such that then benefits at least twice for the function for at least one in the interval for all in the intervalistically decreasing in the interval the real valued function defined on the interval then which of the following statement is true
MAXIMUM AND MINIMUM
Discuss the local maximum and local minimum values of is increasing is continuous does not exist as the maximum value local maximum minimum at respective than order is find the point of local Maxima find the global maximum and global minimum find the local maximum value a functions can find the local maximum value of identify a point of maximum minimum find point of focal maximum and minimum and define the point of a local maximum volume square institute of side length and then holding up the flash find the side of the square base cut off a conical vessel is to be prepared out of circular sheet of Gold of unit radius how much Central area is to be removed from the ship so that vessel as maximum volume find the two positive numbers X and Y such that same is 60 and his maximum size in metres find the coordinates of the points on the curve which attell least distance from the line find the minimum value where find the coordinate of the point on the line which is at the maximum distance from the line idhar circular cylinder is inscribed in a given cone find the dimension of the cylinder such that its volume is maximum and all regular square pyramids a volume 36 find the dimension the parameter having list into lateral surface area then find the minimum value of all closed right circular cylinder of a given volume of 100 CVC centimetres find the dimensions of the cylinder which has minimum surface area the point of influxe for the curve 1100 1000 find the unplugged in the points also draw the graph giving the importance of maximum in a computer concavity find the point of influence for the curve find the interval for in a which cities concave upward concave download and all the values of a for which the function processes critical points divide 64 * 2 parts of the sum of the cubes of two parts is minimum the three sides of the trapezium are equally its been from long find the area of the trapezium and it is maximum so that the triangle of the maximum area that can be inscribed in a given circle is an equilateral triangle
Online 00 on the functions text maximum value of the point 0 13 12 14 the value of a so that the sum of the squares of the roots of the wholesale assume the list valued to 031 the slab of the tanks into the curve maximum when X = 13 12 12 the real number X1 added to its reciprocal give the minimum value of the sum attacks equals to 1122 if the functions where attention maximum and minimum at P and a few respectively such that 1223 is polynomial in the real variable has another maximum nor minimum only one minimum for all as a exactly one exactly 2 local Maxima at local minimum does not have any local extreme has a global minimum two sides of a triangular length if the triangle is to have a maximum area then the length of the median from the vertex containing the sides the difference between the greatest and the least value of equation of the straight line passing through some of the positive intercept on the coordinator series triangle text on the curve the maximum area of the triangle is a solid time will a break is to be made from fit of a clay the brick master be three times as long as its wide the with a brick for which to will a minimum surface area is then let be a twice continuously differentiable positive functional open interval function are defined then which of the following is concave words is concave towards does not have a critical points can give upwards a for the point 13 set value for which the function processing negative point of inflection which of the following statement is true for the general to the functions if the derived as two distinct and cube as one local maximum 1 local minimum if the derivatives exactly one real root of the exactly one relative extinction derivatives in I just Max 3 month values 1 to let me the set of real values of parameter for which the functions as exactly one local exactly one local minimum in the Saturdays the value of a for who is the function as a local minimum a continuous functions local maximum then maybe non zero in real numbers
The setup values of people who is the points of extreme of the functions 3533 13 74 94 134 52 the functions is defined where peak you are positively there as a maximum value for X = minimum the function satisfy the equality is the name of line and the function is such that it is defined on the interval 11 to the increasing function it is not function the point zero zero kodinar the point on the graph on the function area of the triangle made by tangent the coordinate Axis the greatest area is the list value of a for which equation at least one solution on the interview 3579 reader of the following mathematical statement carefully differentiable function to maximum anti-derabad is a periodic function is also a periodic function If as a period then for any has a maximum at then is increasing and decreasing in for now indicate the correct alternative exactly one statement is correct exactly two statement is correct all of the post it Sarkar the lateral age of the regular rectangular pyramids of long the lateral age of Methane angle with the plane of the base the value for who is the volume of the pyramid is greater is are two points on a circle centre in radius the angle being then the radius of the circle scribe in the triangle is maximum then in a regular triangular prism the distance from the centre of one base to one another vertices of other base is the attitude of the prism for who is the volume is greatest BF polynomial real variable with neither maximum nor a minimum only one maximum only one minimum only one maximum only one minimum local maximum local minimum local maximum 1 local minimum the coordinates of the point in the parabola which is the minimum distance from the circle 24
4 points lie in the order of the parabola and the coordinates 23 11 27 the basis of ever information as the following the value of roots of the equation are the value of function at minimum than area of the collateral and coordinates are for the function as the following is the point of inflection then is a point of minimum than is equals to graph is considered in them is equals to a graph is name the list value is consider the function has a local minimum at where sufficient small than has local minimum the largest form in the sequence 7th term the function attends local minimum at 7 the function attends global maximum consider an acute angles triangle minimum value if a continuous curve is concave upward then centroid of the triangle inscribed in the curve always lies of the thrice variable function such that where also the question of a no common books the equation has at least 5 real roots equation rules real distinguish them as at least distinct suppose is a real valued polynomial function of degree 65 the following condition as minimum value at as a maximum value at on the basis of evil information answer the following question number of solution of the equation 1234 range of if the area bounded by where A and B are relatively prime then the value of
Find the points of the local maximum minimum the following functions find all possible real values of such that has the smallest value of a cube vanises at relative minimum maximum at then find the cubic find the absolute maximum value of the following functions and why we do real variation such that find the minimum value of any triangular sheet of poster as its area 18 the marginal top in the bottom at 75 and at the side 50 what are the dimensions of the poster of the area of the printer the space is maximum as it turning value 21 find AB so that the timing values in maximum the flower bed is be in the shape of a circular structure of radius Central angle if the area is fixed in the perimeter is minimum find what are the dimension of the rectangular plot of the great stadium which can be layed out within a triangle of the 36 and altitude 12 assume that one side of the rectangle lies on the base of the triangle for the given surface of a right circular code when the volume is maximum prove that the semi vertical origin is at all the line tangent to the graph of the line find the equation of the tangent lines of minimum and maximum slope suppose is a function satisfying the following condition as a minimum value of 50 to ab or some constraints determine the constant a b and function consider the functions find the X and Y intersective the exist derivatives and the interval on which is an increasing and the interval on which decreasing relative maximum and minimum points any influx in point the function defined for all real numbers as the following properties some constant find the interval on which is increasing and decreasing and any local maximum or minimum balance the graph is concave down and concave the function of the circle cuts access another circle centre line segments find the maximum area of the triangle investigat for the maximum minimum for the functions the graph of the derivative of the continuous function mujhe local minimum is concave up as inflation point number of critical points find the values the graph of the derivatives of a continuous function in soon with what interval is increasing or decreasing at what values of extras have a local maximum state the expordinates of the points of inflection assuming that sketch a graph of Window perimeter including the base of the arch is in the form of rectangular surrounded by semicircle the semi circular motion is switched with the coloured glass while the rectangular Potter is treated with clear glass the clear glass translate three times of much light of per square metre as the colour glass does what is the ratio for the side of the rectangle to windows distance from the origin is meaning
Consider the following functions find whether continuous Arnold find the minimum the maximum at the exist does exist find inclison point of the grap consider the functions find the zeros infection point if anyone the graph local maximum minimum Ascent as the graphs case the graph and a computer the value of the defaulter girl given two points 204 in the line find the coordinates of the point on the line so that the perimeter of the easiest find the set of the values for the cubic A3 distinct solutions the sum of the length of the hypotenuse and other side of the right angle triangle is given so that the area of the triangle is maximum in the angle between the sides is proved that among all Triangles with the given perimeter the equilateral triangle of the maximum area the value of a for which have a positive point of maximum lies in the integral find the value of use calculus to prove that inqualities you may use inquality to prove that find the maximum perimeter of the triangle base and having vertical angle what is the radius of the smallest circular disc large enough to cover every acute isosceles triangle of the given perimeter is Sameer is in the sea at a distance from the closest point on a state show the house of the swimmer is on the share at a distance from he can swim at the speed and work at a speed what point on the show should be land so that he reach is house in the shortest possibility find the interval on which should lie in the order of the exactly one minimum exactly one maximum with the vertices that triangle a parallelogram with the vertices in the line segments using calculus show the maximum area of the parallelogram find the point on the curve that is for this from the point zero to determine the points in the maximum of the functions where the constant let the square of unit area considered any quadrilateral which Rizwan vertex and his side if ABCD remove the length of the side of the quadrilateral prove that find the coordinate of the all points on the ellipse for which the area of triangle with maximum but you know the origin in the foot of the perpendicular to the tangent at
Local maximo mein no local maximum local minimum nostrument the minimum value is the range of straight line with negative slope passes through the point 82 enquired the positive coordinates Axis at point P and Q find the absolutely minimum value as varies where is the origin the medium value more than the maximum value being a real for every the value is greater than or equals to for a circle find the value for which theory and closed by the tangents drawn from the point 68 to the circle and the chord of the contact is maximum ab polynomial or degree college local maximum local minimum at the distance between 12 and where point of local minimum is increasing local minimum the value local minimum at a local minimum local Maxima local minimum local Maxima new local minimum the total number of local Maxima local function the minimum value is 33 matrices of real numbers where is symmetrics to symmetric where is transpose of the matrix and possible value are and integers satisfying must be less than the possible values consider the function define which of the following is true which of the following is true and as a local minimum and as a local maximum is increasing has neither local maximum local minimum is decreasing 11 and other local maximum which of the following is true is positive and negative as a negati when a positive changes sign on a boat and does not change sign the maximum value of the functions on the set let be a polynomial degree 4 having X-Men 1 2 in the value of be real valued functions define on the integral they not respecting maximum minimum 01 then let be a function defined the setup all real numbers such that 2010 2009 2011 2012 is a function defined with values in interval such that the number of points at which
ᵢⱼ ∀
θ θ
θθθθθθθ θ
ₑₑₑₑₑₑₑ
INDIFINITE INTEGRAL
1)
Cos³x/(sin²x + sinx) dx. ln|sinx|- sinx
2) ∫ (x² -1) dx/{(x⁴ + 3x² +1)(tan⁻¹(x + 1/x). ln|tan⁻¹(x + 1/x)|
3) ∫ x²dx/(9+ 16x⁶). (1/36) tan⁻¹(4x³/3)
4) ∫ cos³x dx. Sinx - (1/3) sin³x
5) ∫ dx/√{(x - a)(x - b)}. 2sin⁻¹√{(x - a)/(b - a)}
6) ∫ √{(1- √x)/(1+ √x)} . (1/x) dx. -2ln|{1+√(1- x)}/√x|+ 2cos⁻¹√x
7) ∫ √{(x -3)/(2-x)} dx. √{(x -2)(3-x)}} - 2sin⁻¹√(3- x)
8) ∫ dx/{x √(x² +4)}. (1/2) ln|{√(x¹+4) -2}/x|
9) ∫ cos√x dx. 2[√x sin√x + cos√x]
10) ∫ xdx/(1+ sinx). ,-x(1- sinx)/cosx + ln|1+ sinx|
11) ∫ xeˣ dx. xeˣ- eˣ
12) ∫ x³ sin(x)² dx. (1/2){- x² cosx² + sinx²)
13) ∫ eˣ{(1- x)/(1+ x²)} dx. eˣ/(1+ x²)
14) ∫ eˣ{tan⁻¹x + 1/(1+ x²)) dx. eˣ tan⁻¹x
15) ∫ xₑx²(sin²x + cosx²) dx. (1/2) ₑx¹ sin(x²)
16) ∫ (x + sinx)/(1+ cosx) dx. x tan(x/2)
17) ∫ (tan(eˣ)+ xeˣ sec²(eˣ)) dx. x tan(eˣ)
18) ∫ (ln x +1) dx. x ln x
19) ∫ sin³x cos⁵x dx. (Sin⁴x/4 - 2 (sin⁶x)/6 + (sin⁸x)/8
20) ∫ sin²x cos⁴x dx. (Sin6x)/192 - (sin4x)/64 + (sin2x)/64 + x/16
21) ∫ √sinx/cos⁹⁾²x dx. (2/3) tan³⁾²x + (2/7) tan⁷⁾²x
22) ∫ sin²x/cos⁴x dx. (tan³x)/3
23) ∫ √sinx/cos⁵⁾²x dx. (2/3) Tan³⁾²x
24) ∫ sin²x cos⁵x dx. (sin³x)/3 -(2sin⁵x)/5+ (sin⁷x)/7
25) ∫ dx/(2+ sin²x). (1/√6) tan⁻¹(√3 tanx)/√2
26) ∫ dx/(2 sinx + 3 cosx)². -1/2(2 tanx +3)
27) ∫ dx/(1+ 4 sin²x). (1/√5) tan⁻¹(√5 tanx)
28) ∫ dx/(3 sin²x + sinx cosx +1). (2/√15) tan⁻¹{(8 tanx+1)/√15}
29) ∫ dx/(3sinx + 4 cosx). (1/5 ln|(1+ 2 tan(x/2))/(4- 2 tan(x/2))
30) ∫ dx/(3+ sinx). (1/√2) tan⁻¹((3tan(x/2)-1)/2√2
31) ∫ dx/(1+ 4sinx + 3 cosx). (1/√2) tan⁻¹((3tan(x/2)+1)/2√2
32) ∫ (2+ 3 cosθ)/(sinθ + 2 cosθ +3) dθ. (60/5)+ (3/5) (ln|sinθ + 2 cosθ+3| - (8/5) tan⁻¹((tan(θ/2)+1)/2|
33) ∫ sinx/(sinx + cosx). x/2 (1/2) ln|sinx + cosx|
34) ∫ (3 sinx + 2 cosx)/(3 cosx + 2 sinx) dx. 12x/13 - (5/13) lmpn|3 cosx + 2 sinx|
35) ∫xdx/{(x -2)(x +5)}. (2/7) ln|x -2)|+ 5(5/7) ln|(x +5)|
36) ∫ x⁴/{(x +2)(x² +1)} dx. x²/2 - 2x + (2/5) tan⁻¹x + (16/5) ln|x +2| - (1/10) ln|x²+1|
37) ∫(3x +2)/{(x +1)(x +3)}. (-1/2) Ln|x +1|+ (7/2) ln|x+3|
38) ∫ (x² -1)/{(x +1)(x +2)² dx. ln|x+2|+ 3/(x -2)
39) ∫ dx/(2x²+ x -1). (1/3) Ln|{(2x-1)/2(x+1)}
40) ∫ (3x +2)/(4x² + 4x +5) dx. (3/8) Ln|4x²+ 4x +5|+ (1/8) tan⁻¹(x + 1/2)
41) ∫ dx/(x² + x+ 1). (2/√3)tan⁻¹{2x + 1)/2√3}
42) ∫ (5x +4)dx/√(x² + 4x+ 1). 5√(x²+4x+1) - 6 ln[(x +2)+ √(x²+ 4x +1]
43) ∫ 2x²/(x⁴+1) dx.
44) ∫ 2/(x⁴+1) dx.
45) ∫ 4dx/(sin⁴x + cos⁴x). (2√2)tan⁻¹{(tanx - 1/tanx)}/√2}
46) ∫ dx/(x⁴ + 5x² +1). (1/2)[(1/√7)tan⁻¹{(x - 1/x)/}/√7} - (1/√3) tan⁻¹{(x + 1/x)}/√3]
47) ∫ ²⁴²⁴²²²²²ⁿ²ⁿ⁽ⁿ⁻¹⁾ⁿⁿⁿ¹⁾ⁿⁿⁿ¹⁾ⁿ²²³¹⁾³³⁵¹⁾³⁴³⁵⁵³⁾⁵⁸⁸⁻¹²³² ∫ ⁻¹ θ
No comments:
Post a Comment