Trigonometric Ratios & Identities PYQs Set 2 (PDF)

Back to Trigonometric Ratios & Identities

Set 2 of 18 · Questions 31-60 of 519

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 2 (Q31-60)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
31. $cos \frac{2 π}{7} + cos \frac{4 π}{7} + cos \frac{6 π}{7} + cos \frac{7 π}{7} is equal to$ (1) $\frac{1}{2}$ (2) $1$ (3) $\frac{- 1}{2}$ (4) $\frac{- 3}{2}$ 32. $In triangle ABC, if A = \frac{π}{3} and B = \frac{π}{4} then \frac{a ^{2} - b ^{2}}{c ^{2}} =$ (1) $2 - 3$ (2) $2 + 3$ (3) $2 - 1$ (4) $2 + 1$ 33. $If cosech x = \frac{4}{5}, then cosh x =$ (1) $\frac{41}{21}$ (2) $\frac{41}{19} .$ (3) $\frac{41}{25}$ (4) $\frac{41}{16}$ 34. $The length of the sides of a triangle are 13, 14 & 15 . If R and r respectively denote circumradius and inradius of this triangle, then 8 R - r =$ (1) $41$ (2) $51$ (3) $61$ (4) $71$ 35. $If sec (x) = cos h (θ), then find tan h^{2} (\frac{θ}{2})$ (1) $sec^{2} (\frac{x}{2})$ (2) $tan^{2} (\frac{x}{2})$ (3) $tan2x3$ (4) $sech^{2} (\frac{x}{2})$ 36. $If the value of \frac{3 cos 36 ^{\circ} + 5 sin 18 ^{\circ}}{5 cos 36 ^{\circ} - 3 sin 18 ^{\circ}} is \frac{a 5 - b}{c}, where a, b, c are natural numbers and gcd (a, c) = 1, then a + b + c is equal to :$ (1) $40$ (2) $52$ (3) $50$ (4) $54$ 37. $In a triangle ABC, r_{1} cot \frac{A}{2} + r_{2} cot \frac{B}{2} + r_{3} cot \frac{C}{2} =$ (1) $s$ (2) $2 s$ (3) $3 s$ (4) $\frac{s}{2}$ 38. $In a triangle ABC, (b + c) sin \frac{A}{2} =$ (1) $a cos A$ (2) $a cos (\frac{B - C}{2})$ (3) $a sin (\frac{B + C}{2})$ (4) $a sin (\frac{B - C}{2})$ 39. $Suppose θ and φ (\neq = 0) are such that sec (θ + φ), secθ and sec (θ - φ) are in A.P. If cosθ = kcos (\frac{φ}{2}) for some k, then k is equal to [JEE (Main) 2012]$ (1) $\pm 1$ (2) $\pm 2$ (3) $\pm 2$ (4) $\pm \frac{1}{2}$ 40. $The value of tan^{- 1} (\frac{1}{8}) + tan^{- 1} (\frac{1}{2}) + tan^{- 1} (\frac{1}{5}) is$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{2}$ 41. $cos θ (cosec θ - sec θ) - cot θ =$ (1) $-1$ (2) $1$ (3) $0$ (4) $cos^{2} θ - tan^{2} θ$ 42. $The value of sin (\frac{5 π}{24}) \cdot cos (\frac{π}{24}) is$ (1) $\frac{1 + 2}{4}$ (2) $1 + 2$ (3) $\frac{1 - 2}{4}$ (4) $1 - 2$ 43. $If tan \frac{θ}{2} = cosec θ - sin θ, then tan^{2} \frac{θ}{2} =$ (1) $2$ (2) $- 2 + 5$ (3) $-2$ (4) $5$ 44. $The value of cos \frac{π}{2 ^{2}} \cdot cos \frac{π}{2 ^{3}} \cdot \dots \cdot \cdot cos \frac{π}{2 ^{10}} \cdot sin \frac{π}{2 ^{10}}$ (1) $\frac{1}{512}$ (2) $\frac{1}{1024}$ (3) $\frac{1}{256}$ (4) $\frac{1}{2}$ 45. $The angles A, B and C of a triangle ABC are in A.P. If b : c = 3 : 2, then the angle A is$ (1) $75^{\circ}$ (2) $45^{\circ}$ (3) $30^{\circ}$ (4) $15^{\circ}$ 46. $For any θ \in (\frac{π}{4}, \frac{π}{2}) the expression 3 (sin θ - cos θ)^{4} + 6 (sin θ + cos θ)^{2} + 4 sin^{6} θ equals:$ (1) $13 - 4 cos^{2} θ + 6 sin^{2} θ cos^{2} θ$ (2) $13 - 4 cos^{6} θ$ (3) $13 - 4 cos^{2} θ + 6 cos^{4} θ$ (4) $13 - 4 cos^{4} θ + 2 sin^{2} θ cos^{2} θ$ 47. $\frac{3 x + 1}{( x - 1 ) ( x + 3 )} = \frac{A}{x - 1} + \frac{B}{x + 3}, then sin^{- 1} \frac{A}{B} =$ (1) $\frac{π}{4}$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{6}$ 48. $16sin(20°) sin(40°) sin(80°) is equal to :$ (1) $3$ (2) $23$ (3) $3$ (4) $43$ 49. $In △ ABC, with usual notations, 2 acsin (\frac{1}{2} (A - B + C)) is equal to$ (1) $a^{2} + b^{2} - c^{2}$ (2) $c^{2} + a^{2} - b^{2}$ (3) $b^{2} - c^{2} - a^{2}$ (4) $c^{2} - a^{2} - b^{2}$ 50. $The value of (sin 70^{\circ}) (cot 10^{\circ} cot 70^{\circ} - 1) is$ (1) $2/3$ (2) $1$ (3) $0$ (4) $3/2$ 51. $tan h^{- 1} \frac{1}{2} + cot h^{- 1} 3 =$ (1) $log 6$ (2) $log 6$ (3) $- log 6$ (4) $- log 6$ 52. $In a △ ABC, if 3 a = b + c, then cot \frac{B}{2} cot \frac{C}{2} =,$ (1) $1$ (2) $2$ (3) $\frac{1}{3}$ (4) $\frac{1}{2}$ 53. $The value of x satisfying the equation 3 cosec x = 4 sin x are$ (1) $\frac{π}{6}, \frac{π}{3}$ (2) $\pm \frac{π}{6}$ (3) $\pm \frac{π}{3}$ (4) $\frac{π}{3}, \frac{π}{4}$ 54. $The sides of a triangle are 6 + 12, 48 and 24 . The tangent of the smallest angle of the triangle is$ (1) $2 - 1$ (2) $\frac{1}{3}$ (3) $1$ (4) $3$ 55. $If cos α + 2 cos β + 3 cos γ = 0, sin α + 2 sin β + 3 sin γ = 0 and α + β + γ = π, then sin 3 α + 8 sin 3 β + 27 sin 3 γ =$ (1) $9$ (2) $-18$ (3) $0$ (4) $3$ 56. $The sides of a triangle inscribed in a given circle subtend angles α, β, γ at the centre. The minimum value of the AM of cos (α + \frac{π}{2}), cos (β + \frac{π}{2}) and cos (γ + \frac{π}{2}), is equal to$ (1) $\frac{3}{2}$ (2) $\frac{- 3}{2}$ (3) $\frac{- 2}{3}$ (4) $2$ 57. $If y = \frac{2 ( tan α + cot α )}{1 + tan ^{2} α} + \frac{1}{sin ^{2} α} when α \in (\frac{3 π}{4}, π) then find$ (1) $4$ (2) $2$ (3) $3$ (4) $-4$ 58. $In a triangle ABC, (tan \frac{A}{2} tan \frac{B}{2} tan \frac{C}{2})^{2} \leq$ (1) $\frac{1}{27}$ (2) $\frac{1}{18}$ (3) $\frac{1}{9}$ (4) $\frac{1}{3}$ 59. $The maximum value of (cos α_{1}) (cos α_{2}) \dots (cos α_{n}), under the restrictions 0 \leq α_{1}, α_{2}, \dots, α_{n} \leq \frac{π}{2}, and (cot α_{1}) (cot α_{2}) \dots (cot α_{n}) = 1 is [JEE (Main) 2001]$ (1) $\frac{1}{2 ^{n /2}}$ (2) $\frac{1}{2 ^{n}}$ (3) $\frac{1}{2 n}$ (4) $1$ 60. $The principal solutions of the equation sec x + tan x = 2 cos x are$ (1) $\frac{π}{6}, \frac{5 π}{6}$ (2) $\frac{π}{6}, \frac{π}{20}$ (3) $\frac{π}{6}, \frac{2 π}{3}$ (4) $\frac{π}{6}, \frac{π}{12}$