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
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 f 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)
⁽ ⁾
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
θθ
θθθθθθθθ
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
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