THEORY OF QUADRATIC EQUATIONS AND FUNCTION
1) Determine the nature of the roots of the following equations:
a) 9x² - 12√2 x + 8= 0. Irrational & equal
b) x² - 5x + 6 = 0. Rational & unequal
c) 2x² + 4x - 3 = 0. Irrational & unequal
d) 3x² - 2 x + 5 = 0. Imaginary and unequal
2) For what value/s of m will the equation x² - 2(5+ 2m)x + 3(7+ 10m) = 0 have
a) equal roots. 2, 1/2
b) reciprocal roots. -2/3
3) Find the value of m, if the roots of x² - (5+ 2m)x + (10+ 2m) = 0 be equal in magnitude but opposite in sign. -5/2
4) Both the roots of the quadratic equation x² - (a+1) x + a + 4 = 0 are negative. Calculate the value of a. -4< a <-1
5) Form the quadratic equation whose roots are 3 and -5. x² + 2 x -15 = 0
6) Form the quadratic equations whose one root is
a) 2+ √3. x² - 4x + 1 = 0
b) 3- 4i. x² - 6x + 25 = 0
7) If α and β be the roots of the equation ax² + bx + c = 0, find the value of the following:
a) α²+ β². (b²- 2ac)/a²
b) α² - β². b√(b²- 4ac)/a²
c) α³ - β³. (3abc - b³)/a³
d) α⁴ - β⁴. (b⁴+ 2a²c² - 4ab²c)/a⁴
e) αβ⁻¹ - βα⁻¹. (b²- 2ac)/ac
f) α/β² + β/α². (3abc - b³)/ab²
8) If α and β be the roots of x²- px + q= 0, form the equation whose roots are αβ + α + β and αβ - α - β. x²- 2qx + (q²- p²)= 0
9) If the roots of the equation 3x²- 6x + 4=0 are α and β, find the value of (α/β + β/α) + 2(1/α + 1/β)+ 3αβ. 8
10) If αβ are the roots of the equation x²- 2x +3=0, find the equation whose roots are α/ β² and β/α². 9x²+ 10x +3=0
11) The roots of the equation px² - 2(p +2)x + 3p = 0 are α and β. If α - β= 2, calculate the values of α, β and p. -1 or 3, -3 or 1, -2/3
12) The ratio of the roots of the equation x²+ αx + α +2=0 is 2. Find the values of the parameter α. 6 or -3/2
13) If one root of the equation x²+ bx + 8 =0 be 4 and the roots of the equation x²+ bx + c =0 are equal, find the value of c. What will be the equation whose roots are inverse of roots of the first equation? 9, 8x²- 6x +1=0
14) Find the condition that the roots of the equation ax²+ bx + c =0 may differ by 5. b²- 4ac = 25a²
15) The ratio of the roots of ax²+ bx + c= 0 is 3:4. Prove that 12b²= 49ac.
16) If the roots of the equation px²+ qx + q= 0 be in the ratio m: n, show that
√(m/n) + √(n/m) + √(q/p)= 0.
17) If one root of ax²+ bx + c= 0 be the square of the other, prove that b³+ a²c + ac²= 3abc.
18) if the difference between the roots of ax²+ bx + c=0 be equal to the difference between the roots of px² + qx + r=0, show that p²(b²- 4ac)= a²(q²- 4pr).
19) Prove that if x²+ px + q= 0 and x²+ qx + p=0 have a common root, then either p= q or, p+ q+1= 0.
20) If the equation x²- 5x +6= 0 and x²+ mx +3=0 have a common root, find the value of m. -7/2 or -4
21) If the equation ax²+ bx + c=0 and bx²+ cx + a= 0 have a common root , prove that a³+ b³+ c³= 3abc.
22) Prove that if the equation x²+ bx + ca = 0 and x²+ cx + ab=0 (b≠ c) have only one root common, then the other roots will satisfy t²+ at + bc= 0
QUADRATIC EXPRESSION
1) Prove that expression 14x²- 9x +1 is positive for all real values of x, except when x lies between 1/7 and 1/2.
2) Show that for all real values of x, the expression 3x²- 12x +17 is positive.
3) Find for real values of x x, the maximum value of 3 - 20x - 25x². -2/5
4) Find for all real values of x, the minimum value of 3x²- 5x + 8. 5/6
5) Solve: x²+ 4x - 5> 0. x< -5 or x > 1
6) Determine the range of the values of x for which 2x²+ 3x - 9≤ 0. -3≤ x ≤ 3/2
7) Solve: (x +1)/(4x -1) < 0. -1< x <-1/4
8) Determine the range of values of x for which (x½+ x +1)/(x²+ 2) < 1/3, x being real. -1< x <-1/2
9) Find the range of real value of x for which (x -2)/(3x +4) < (x -4)/(3x -2). -4/3< x<2/3
10) Find all real values of x which satisfy
x²- 3x +2≥ 0 and x²- 3x - 4≤ 0. -1≤ x ≤ 1 or 2≤ x ≤ 4
11) Find the range of x in the equation (x²- 5x +7)/(x²- 7x + 12) < 1/2. 1< x <2 and 3<x<4
12) If x is real, prove that the value of the expression (x²- x +1)/(x²+ x +1) lies between 1/3 and 3.
13) If x be real, find the maximum and minimum value of (x +2)/(2x²+ 3x +6). 1/3 and -1/13
14) If 3p²= 5p +2 and 3q²= 5q + 2 where p≠ 1, obtain the equation whose roots are 3p - 2q and 3q - 2p. 3x²- 5x - 200= 0
15) find the quadratic equation one of whose roots is 2ab/{(a+ b) + √(a²+ b²)}. x²- 2(a + b)x + 2ab = 0
16) If 3b³+ 9a²c + ac²= 9abc, show that the square of one of the roots of the equation ax²+ bx + c= 0 is three times the other.
ARITHMETIC PROGRESSION
1)Determine the AP whose first term is 5 and common difference is -2. 5,3,1,-1
2) Find 10th term of the AP where first term is 5 and common difference is 2. 23
3) The 20th term of an AP is 79. If the first term is 3, find 10th term. 39
4) Which term of an AP 2+5+8+.... is 92? 31
5) Is 90 a term of the series 4+7+10+13+...? No
6) Determine the 2nd term and r-th term of the AP whose 6th term is 12 and 8th term is 22. -8, 5r - 18
7) Insert 5 Arithmetic means between 17 and 29. 19,21,23,25,27
8) Insert 4 Arithmetic means between (a- b)² and a²+ b²+ 8ab. a²+ b², (a+b)², a²+ b²+ 4ab, a²+ b²+ 6ab
9) Find the middle term/s of the following AP.
a) 5,8,11,....95. 50
b) 88,80,72,..... -64. 16,8
10) If a,b,c are in AP, show that 1/bc, 1/ca, 1/ab are also in AP.
11) If 1/(b + c), 1/(c + a), 1/(a+ b) are in AP. Show that a², b², c² are also in AP.
12) Find the value of k for which k²- 7k, k²+ 9 and 6 are in AP. -3 or -4
13) If a,b,c are in AP, show that (b + c)² - a², (c + a)² - b², (a+ b)² - c² are also in AP.
14) Find three numbers in AP whose sum is 15 and sum of their products in pair is 71. 3,5,7 or 7,5,3
15) Find the four terms in AP whose sum is 20 and the sum of whose squares is 120. 2,4,6,8 or 8,6,4,2
16) Find the sum of the following series:
a) 2+7+12+.... upto 20 terms. 990
b) (n -1)/2 + n/n + (n +1)/n +...... upto n terms. (3/2)(n -1)
17) Find the sum of series 7+10+13+.....+64. 710
18) The first and the last term of an AP having finite number of terms are respectively -2 and 124 and the sum of the AP is 6161. Find the number of terms in the AP. 101
19) Find the sum of all perfect squares between 200 to 800. 6699
20) Sum of n terms the series 3.7+5.10+7.13+....... (n/2)(4n²+17n +21)
21) 1/2.5 + 1/5.8+ 1/8.11+ ........ n/3(3n +2).
22) Find the increasing AP, the sum of whose first three terms is 27 and the sum of their squares is 275. 5,9,13,17,...
23) Find four terms in AP if their sum and product are respectively 16 and 105. 4- √151, 4- √151/3, 4+ √151/3, 4+ √151 or reverse
24) Find the r-th term of an AP, the sum of whose first n terms in 2n + 3n². 6r -1
25) The sum of n terms of an AP is 3n²+ 5n. Find which term of the AP is 152. 25
26) The first and the last term of an AP are respectively -4 and 146, and the sum of AP= 7171. Find the number of terms of the AP and also its common difference. 101, 1.5
27) If pth, qth and r-th terms of an AP are a,b,c respectively, show that a(q - r)+ b(r - p)+ c(p - q)= 0.
28) If a,b,c respectively the sums of p, q and r terms of an equilateral AP. Show that
(a/p) (q - r)+ (b/q) (r - p)+ (c/r) (p - q)= 0.
29) The sum of p terms of a series in 2p²+ p. Show that the series is in AP.
30) If a,b,c are in AP. Show that
a) a²(b + c), b²(c + a), c²(a + b) are in AP.
b) a(b + c)/bc, b(c +a)/ca, c(a+ b)/ab are in AP.
31) If S₁, S₂, S₃ be the sums of n, 2n and 3n terms respectively of an AP, show that S₃ = 3(S₂ - S₁).
32) The sum of m terms of an AP is n and the sum of n terms is m. Find the sum of (m + n) terms. -(m + n)
33) Sum of n terms the series 1+3+6+10+...... (n/6) (n +1)(n +2).
34) The sum of the digits of a three digited number is 12. The digits are in AP, if the digits are reversed, then the number is diminished by 396. Find the number. 642
35) A man is employed in a company on Rs 8000 per month, and is promised an increment of Rs 25 per year. Find the total amount which he receives in 13 years and his pay in the last year. Rs 148200, Rs 1100
36) A man borrows Rs1200 at the total intrest of Rs 168. He repays the entire amount in 12 installments, each installment being less than the proceeding one by Rs 20. Find the first installment. Rs224
37) A sum of Rs 6240 is paid off in 30 installment, such that each installments is Rs 10 more that the preceding instalment. Calculate the value of the first installment. 63
38) A man arranges to pay off a debt of Rs 7200 by 20 installment which form an AP. When 15 of the instalment are paid, he finds that one third of his debt still remains unpaid. Find the attached amount of his 16th installment. Rs448
39) A circle is completely divided into n sectors in such a way that the angles of the sectors are in AP. If the smallest of these angles is 8° and the largest 72°, find n and the angle in the fourth sector. 9, 32°
40) The interior angles of a polygon are in AP. The smallest angle is 52° and the common difference is 8°. Find the number of sides of the polygon. 3.
41) The last term of an AP 25,8,11,....is x. The sum of the terms of the AP is 155. Find x.
42) a₁, a₂, a₃, a₄, a₅ are first five terms of an AP such that a₁+ a₃+ a₅ = -12 and a₁a₂a₃= 8. Find the first term and the common difference. 2, -3
43) Sum of the series 1/n + (n +1)/n + (2n +1)/n + ....+ (n²- n +1)/n. (n²- n+2)/2
44) If a₁, a₂,.....aₙ be in AP, show me that 1/a₁a₂ + 1/a₂a₃ + ....+ 1/aₙ₋₁aₙ = (n -1)/a₁aₙ.
GEOMETRIC PROGRESSION
1) Determine the GP whose first term is 3 and common ratio is -2. 3,-6,18,-36,...
2) Find the tenth term of the GP whose first term is 3 and common ratio is -2. -1536
3) The fifth term of a GP is 162 and the first term is 2. Find the common ratio. 3 or -3
4) The 10th term of a GP is -2560, and the first term is 5. Find the 5th and nth term of the GP. 5(-2)ⁿ⁻¹
5) Which term of the series 4+12+36+108+....is 2916? 7
6) Is 3000 a term of the GP 3,15,45,135,....? No
7) Determine first 3 terms and 9th term of a GP whose 4th term is 24 and 7th term is 192. 768
8) Insert 6 geometric means between 8 and 1/16.
9) Insert 5 geometric means between 3 and 81.
10) Find the sum of the following series:
a) 2+6+18+54+...to 10 terms. 59048
b) 1- 1/2 + 1/4 - 1/8 + ....to 12 terms. 4095/2048
11) Find the sum of the series 3+6+12+....+3072. 6141
12) Find the sum of first n terms of the following:
a) 4+44+444+.... (40/81)(10ⁿ -1) - 4n/9
b) 0.3+0.33+0.333+.... n/3 - 1/27(1- 1/10ⁿ)
13) Find the sum of the infinite GP 1, 1/2,1/4,1/8,.... 2
14) A man saves Rs100 in the first day and in each of the succeeding days he saved 9/10 th of what he saved in the previous day. Show that his total savings will not exceed Rs 1000 however long he may live.
15) If S₁, S₂, S₃,....Sₙ are the sums of infinite geometric series whose first terms are 1,2,3,....n and whose common ratios are 1/2,1/3,1/4,.....1/(n +1) respectively, then find the value of S₁²+ S₂²+ ....+ S½₂ₙ₋₁.
16) The side of a given square is equal to a. The midpoints of its sides are joined to form a new square. Again, the midpoints of the sides of this new square are joined to form another new square. This process is continued infinitely. Find the sum of the areas of the squares and the sum sum of the perimeter of the squares. 2a², 4√2(√2+1)a.
17) The sum of an infinite GP is 16 and the sum of the squares of its terms is 768/5. Find the common ratio and fourth terms. 1/4, 3/16
18) Find the sum of 1.3+ 2.3²+ 3.3²+....+n.3ⁿ. {(2n -1)3ⁿ⁺¹ +3}/4
19) Find the sum of n terms of the series 1+ 2. 1/5 + 3. 1/5² + 4. 1/5³+..... (25/16)(1- 5ⁿ) - 1/4.5ⁿ⁻¹.
20) If 3x +1, 7x and 10x + 8 be in GP. Find the value of x. 2 or -4/19
21) If a,b,c, d are in GP, show that a²- b², b²- c², c²- d² will also be in GP.
22) If a,b,c, d are in GP, show that (b - c)²+ (c - a)²+ (d - b)²= (a - d)².
23) The first three terms of a GP are x, x+ 3 and x+ 9. Find the value of x and the sum of first eight terms. 3, 765.
24) Show that.97777....= 44/45 using the summation formula of GP.
25) Calculate the least number of terms of the GP 5+10+20+... whose sum would exceed 1000000. 18
26) Show that in a GP the product of any two terms equidistant from the beginning and the end is equal the product of the first and the last terms.
27) If a,b,c are in GP and x, y be the arithmetic means between a, b and b,c respectively. Show that a/x + c/y= 2.
28) If 1/(x + y) , 1/2y, 1/(y+ z) are three consecutive terms of an AP, show that x, y, z are the three consecutive terms of a GP.
29) If the arithmetic mean between a and b is twice as large as their geometric mean. Show that the ratio between the numbers can be written as 2+ √3 : 2- √3.
30) If one arithmetic mean A and two geometric means G₁ and G₂ be inserted between any two numbers, then show that G₁³ + G₂³= 2AG₁G₂
31) If S be the sum, P the product and R the sum of reciprocals of first n terms in a GP. Prove P² Rⁿ = Sⁿ.
32) Find the sum of the infinite series: 1+ (1+ b)r + (1+ b + b²)r²+.... 1/{(1- r)(1- br)}
33) Find four positive integers x,y,z,w such that y,z,w are in AP; x,y,z are in GP and z+ w= 10, x+ y= 3. 1,2,4,6
34) S₁, S₂, S₃, ....Sₙ are the sums of n infinite GP. The first terms of these progressions are. 1, 2²-1, 2³ - 1,...., 2ⁿ- 1 and the common ratios 1/2, 1/2², 1/2³,....1/2ⁿ. Calculate the value of S₁ + S₂ +.... Sₙ. 2(2ⁿ -1)
35) If Sₙ be the sum of infinite GP series whose first term is n and the common ratio is 1/(n +1), find the sum S₁ + S₂ +.... Sₙ. n(n+3)/2
36) The sum of an infinite GP is 15 and the sum of their squares is 45. Find the series. 5+ 5.(2/3) + 5(2/3)² +....
37) The product of 3 numbers in GP is 729 and the sum of their squares is 819. Determine the numbers. 3,9,27 or -3,-9,-27
38) The product of 4 positive numbers in GP is 729 and the sum of two intermidiate terms is 12. Find the numbers. 27,9,3,1 or 1,3,9,27
39) The sum of four terms in GP is 60 and the arithmetic mean of the first and the last numbers is 18. Find the numbers. 32,16,8,4 or 4,8,12,32
40) Three numbers, whose sum is 70, are in GP. If each of the extremes is multiplied by 4 and the mean by 5, the numbers will be in AP. Find the numbers. 10,20,40
41) Three numbers are in AP and their sum is 21. If 1,5,15 be added to them respectively, they form a GP. Find the numbers. 5,7,9
42) 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 numbers b,c,18 are consecutive terms of a GP. 5,8,12
43) A bouncing tennis ball rebounds each time to a height equal to one half the height of the previous bounce. If it is dropped from a height of 16 metres, find the total distance it has travelled when it hits the ground for the 10th time. 767/16
44) A man borrows Rs 16,380 without intrest and repays the loan in 12 monthly installments, each installment (beginning with the second) being twice the preceding one. Find the amount of the last installment. Rs 8192
45) if a,b,c are in AP, a,x,b and b,y,c are in GP. Show that x², b², y² are in AP
46) a,b,c are in AP and b,c,a are in GP. Show that 1/c, 1/a, 1/b are in AP.
47) If p,q,r in AP, show that pth, qth and r-th terms of a GP are in GP.
48) Find the first term of a GP, whose sum to infinite terms is 8 and second term is 2. 4
49) If one geometric mean G and two arithmetic means a and b inserted between two quantities. Show that (2a - b)(2b - a)= G².
50) If S₁, S₂, ...., Sₚ are the sums of infinite GP whose first terms are 1,2,3,.... p and common ratios are 1/2,1/3,1/4,..... 1/(p+1) respectively, then show that S₁ + S₂ + S₃ + ..... Sₚ= p(p +3)/2.
51) If x= 1+ a + a²+ ...∞, y= 1+ b+ b²+.....∞, show that 1+ ab + a²b² +......∞= xy/(x + y -1) when |a|< 1, |b|< 1.
52) If 2/3= (x - 1/y)+ (x²- 1/y²) + ..... to ∞ and xy= 2, then find the values of x and y with the condition that |x|< 1.
53) a,b,c are three distinct real numbers and they are in GP. If a+ b + c= ab then show that x < -1 or x > 3.
PARTIAL FRACTIONS
Resolve:
1) (4x -3)/{(x -2)(2x +1)}.
2) (7x -11)/{(x³+ 2x²- x +2).
3) x⁴/(x³- x²- 4x +4).
4) (3x²+ 7x +8)/x(x +2)².
5) (x³+ x²+ x -1)/(x²-1)².
6) 1/{(x +1)(5- x²)}.
7) (x³- x²+ 3x -2)/{(x²+1)(x²+2)}.
8) (2x +7)/{(x³- 1)(x²+ x +1)}
MEASUREMENT OF ANGLES
1) Express: i) 50°. 0.87rad, 55ᵍ55' 55.5"
ii) 24° 13' 20"
in circular measure and also in centesimal measure. 0.42 rad, 26ᵍ 91' 11.1"
2) Express 12ᵍ 42' 20" is sexagesimal measure and also in circular measure.
3) Express 2π/5 radian in sexagesimal measure and also in centesimal measure.
4) Find the length of the arc of a circle of radius 5cm if it makes an angle of 60° at the centre (π= 3.14). 5.23cn
5) The area of a sector of a circle of radius 4cm is 8.2 square.cm. find the angle of the sector and also the length of the Arc of the sector. 4.1cm
6) A horse is running along a circular track of radius 126m. In 6 seconds it covers an which makes an angle of 50° at the centre. Find the speed of the horse in km per hour. (π= 22/7). 66kmph
7) A cyclist is moving at a speed of 10.56 kmph. The wheel is moving at a speed of 40 rotations per minute. Find the diameter of 10.56 kmph. The wheel is moving at a speed of 40 rotations per minute. Find the diameter of the wheel (π=22/7) 1.4m
TRIGONOMETRICAL RATIOS AND IDENTITIES
1) If cosθ = 12/13, find other t-ratios of θ.
2) Prove that for real values of x, sinθ= x + 1/x is an impossible equation.
3) Eliminate θ from the given equilation x = r cosθ and y = r sinθ. x²+ y²= r²
4) If cosecθ+ cotθ = 2+ √3, find the values of sinθ and cosθ. 1/2, √3/2
5) Prove:
a) (3- 4 sin²θ)/cos²θ= 3- tan²θ.
b) sinθ/(cotθ + cosecθ)= 2+ sinθ/(cotθ - cosecθ).
c) (tanθ + secθ -1)/(tanθ - secθ +1)= (1+ sinθ)/cosθ.
d) sec²x tan²y - tan²x sec²y = tan²y - tan²x.
e) sinθ(1+ tanθ)+ cosθ(1+ cotθ)= secθ+ cosecθ.
f) √{(1+ sinx)/(1- sinx)} - secx = secx - √{(1- sinx)/(1+ sinx)}.
g) If cosθ - sinθ= √2 sinθ show cosθ + sinθ= √2 cosθ.
h) If cosθ - sinθ= √2 sinθ show cosθ + sinθ= ± √2 cosθ.
i) If tan²x = 1+ 2 tan²y; show that cos²y = 2 cos²x.
j) Find tanx if 2 sin²x - 5 sinx cosx + 7 cos²x = 1.
k) If k tanθ= tan kθ, show that (sin²kθ)/sin²θ= = k²{1+ (k²-1) sin²θ}.
l) If a= b cosC + c cosB = c cosA + a cosC and c= a cosB + b cosA, prove that a/sinA = b/sinB = c/sinC.
m) If x= a secθ cosφ, y= b secθ sinφ and z= c tanθ, show that x²/a²+ y²/b²- z²/c²= 1.
n) If cosθ + secθ= √3, show that cos³θ+ sec³θ= 0.
o) If m= cosecA - sinA= secA - cosA then show that tanA = (n/m)¹⁾³.
p) show (1- sinA+ cosA)²= 2(1- sinA)(1+ cosA).
q) If (sinx - cosx)/(sinx + cosx)= tanθ, show that sinx + cosx =√2 cosθ.
r) If tanθ + sinθ= m and tanθ - sinθ= n, show that m²- n²= 4√(mn).
s) If sinθ+ cosecθ= 2 show that sinⁿθ + coseⁿθ= 2.
t) If tanθ= a/b, find the value of sinθ/cos⁸θ+ cosθ/sin⁸θ. (a⁹+ b⁹)(a²+ b²)⁷⁾²/(ab)⁸
u) If secx sec y + tan x tan y = sec z show that sec x tan y + tan x sec y = ± tan z.
v) If (sin⁴θ)/a + (cos⁴θ)/b = 1/(a + b), show that (sin⁸θ)/a³ + (cos⁸θ)/b³= 1/(a+ b)³.
TRIGONOMETRIC RATIOS OF SOME STANDARD ANGLES
1) Verify: sin60° = (2tan30°)/(1+ tan30°).
2) Show that (4/3) cot²30°+ 3 sin²60° - 2 cosec²60° - (3/4) tan²30° = 10/3.
3) Find the value of: {sin(π/3)+ sin(π/6)}/{cos(π/3)+ cos(π/6)} + {sin(π/3) - sin(π/6)}/{cos(π/3) - cos(π/6)}. 0
4) If α and β positive acute angles such that sin(2α - β)= 1/√2 and tan(α +β)= 1. Find α and β. 30°, 15°
5) For a triangle ABC, the angle A is obtuse and sin(B + C)= √3/2 and tan(B - C)= 1/√3. Find the angles of the triangle. 120°,45°,15°.
6) Solve for θ if 0≤θ ≤ 90°.
a) cosθ + √3 sinθ = 2. 60°
b) cotθ + tanθ = 2 secθ. 30°
TRIGONOMETRICAL RATIOS OF ALLIED ANGLES
1) The value of:
a) sin(-30°). -1/2
b) cos(-200°). - cos20°
c) sin210°. -1/2
d) tan(-150°). 1/√3
e) sec150°. -2/√3
f) cot240°. 1/√3
g) sin(-225°). 1/√2
h) cos390°. √3/2
i) cot315°. -1
j) sec510°. -2/√3
k) sin(-300°). √3/2
l) sin480°. √3/2
m) tan(1860°). √3
n) sin315°. -1/√2
o) tan(-570°). -1/√3
p) cos330°. √3/2
2) Find the value of sin135° cos315° + sin420° cos330°. 5/4
3) If tanθ = -5/11, find the values of sin θ and cosθ.
4) For any quadrilateral ABCD, show that cos(1/2) (A+ B)+ cos(1//2)(C + D)= 0.
5) If sinθ + cosθ= 0 and θ lies in the second quadrant, find the value of θ, cosecθ, and cotθ. -1, √2, -1
6) Find the value of θ between 0° and 360° if sin²θ= 3/4. 60°, 120°, 240°, 300°
7) If 0°<θ <360°, find the value of θ in the equation cotθ + tanθ = 2 secθ. 30°, 150°
8) Prove that tan(π/12) tan(5π/12)tan(7π/12) tan(11π/12)= 1.
9) Prove that sin²(π/8) + sin²(3π/8)+ sin²(5π/8) + sin²(7π/8)= 2.
10) Prove that the value of
{sin³(2π- x) cos³(2π- x)}/{cos²(3π/2 + x) sin³(2π + x)} . {tan(π- x)/{cosec²(π - x)}. sec²(π+ x)}/sinx is independent of x.
11) 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)]- 2[sin⁶(π/2 + x)+ sin⁶(5π - x)] is independent of x.
12) If tan25°= a, show that {(tan155° - tan 115°)/(1+ tan155°. Tan115°)}= (1- a²)/2a.
13) If sec(x - y)= 1, sec(x + y)= 2/√3, find positive magnitude of x and y.
14) Find the value of this sin{nπ + (-1)ⁿ π/3} when n is any positive integer. √3/2, √3/2
15) Show that sin θ+ sin(π+ θ)+ sin(2π+ θ)+ sin(3π+ θ)+ .....to n terms is either 0 or sin θ according to n is even or odd. 0, sin θ
COMPOUND ANGLES
1) Find the values of
a) sin75°. (√3+1)/2√2
b) cos105°. -(√3-1)/2√2
c) sin15°. (√3-1)/2√2
d) cos15°. (√3+1)/2√2
e) cot15°. 2√3
2) Show that: tan(A - B) tan(A - B)= (sin²A - sin²B)/(cos²A - sin²B).
3) show that: cot(A+ B) cot(A - B)= (cos²A - sin²B)/(sin²A - sin²B).
4) show that cos75° = (√3-1)/2√2.
5) show: cos80°40'. cos39°20' - sin80°40' . sin39°20' = -1/2.
6) Prove: cot2A + tanA = cosec2A.
7) If tan θ= (a sinx + b sin y)/(a cos x + b cos y) show that a sin( θ- x)+ b sin( θ- y)= 0.
8) If cosA = 3/5 and sinB= 5/13 (A, B < 90°) show that cos(A - B)= 56/65.
9) show that: cos²x + cos(60° - x)+ cos²(60°+ x)= 3/2.
10) Show that sin²A + sin²(120°- A)+ sin²(120°+ A)= 3/2.
11) Show that cos²(A - 129°)+ cos²A + cos²(A + 120°)= 3/2.
12) Find the quadrant in which the angle (x + y) terminate if sin x = 3/5; cos y = -5/13, x being in the first and y being in the second quadrant. sin(x + y) is positive and cos(x + y) is negative, x + y is in the second quadrant
13) Express cos θ+ √3 sin θ in the form r cos(x - θ) and determine the value of r and x. 2, 60°
14) Find the maximum and minimum values of 5 cos θ+ 12 sin θ + 12. 25,-1
15) If A+ B = 225°, show that tanA + tanB = 1- tanA tanB.
16) If A + B + C=90°, show that tanB tanC + tanC tanA + tanA tanB = 1.
17) An angle θ is divided into two parts x and y, such that tan x : tan y = K: 1, show that sin(x - y)= {(K -1) sin θ}/(K +1).
18) If sinx sin y - cos x cos y +1= 0, show that 1+ cot x tan y = 0.
19) If m tan( θ- 30°)= n tan( θ- 120°), show that cos2θ = (m + n)/2(m - n).
20) Show that {cos x + cos(y + z)}{cos x + cos(y - z)}= cos²x + cos²y + cos²z + 2 cosx cos y cos z -1.
21) If cos(x - y)= -1, show that cosx + cos y = 0 and sin x + sin y = 0 (x, y have real valus).
22) If cos(b - y) + cos(y - a) + cos(a - b)= -3/2, then show that cos a + cos b + cos y = 0 and sin a + sin b sin y = 0.
23) Let 0< x < π, 0< y <π and cos x + cos y - cos(x + y) = 3/2. Show that x= y =π/3.
24) Given that tan x= 3/4, cos y= -12/13 and x and y are in the same quadrant and lie between 0° and 360°. Find without use of tables the value of
a) sin(x - y). 16/65
b) cos(x/y). ±√1/26
25) If sin(x + y)= 4/5, cos(x - y)= 12/13 and 0< x<π/4, 0< y < π/4, find tan2x. 33/56
26) If x and y are the solutions of a tan θ+ b sec θ= c show that tan(x + y)= 2ac/(a²- c²).
27) If x and y are two distinct values of θ (0≤ x<2π, 0≤ y ≤2π) satisfying the equation sin( θ+x)= (1/2) sin2x. Show that (sin x + sin y)/+cos x + cos y)= cot x.
TRANSFORMATION OF PRODUCTS, SUMS AND DIFFERENCES
1)
θθ θ θ
α β γ
LIMITS
1) lim ₓ→₂ [(x²-4)/{√(3x -2) - √(x +2)}]. 8
2) lim ₓ→₀ [{√(2- x) - √(2+ x)}/x]. -1/√2
3) lim ₓ→₀{(xᵐ - aᵐ)/(xⁿ - aⁿ)}. (m/n) aᵐ⁻ⁿ
4) lim ₓ→₃{(x⁵- 243)/(x²-9)}. 135/2
5) lim ₓ→ₐ {√(x+2)³ - √(a +2)³}/(x - a). (3/2) √(a +2)
6) lim ₓ→₂ {xⁿ - 2ⁿ)/(x -2)}= 80 and n ∈N, find n. 5
7) lim ₓ→₁ {(1- x⁻¹⁾³)/(1- x⁻²⁾³)}. 1/2
8) lim ₓ→₁ {(x⁴-1)/(x -1)}= lim ₓ→ₖ{(x³- k³)/(x²- k²)}. Find the value of k. 8/3
9) lim ₓ→₀ {(1+ x)⁶ - }/{(1+ x)² -1}. 3
₁ ₂ ₃ ₄₅₆₇₈₉ ₀ ₙₐₓ→
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