Trigonometric Ratios & Identities PYQs Set 3 (PDF)

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Set 3 of 18 · Questions 61-90 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 3 (Q61-90)
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61. $If cos A + cos B + cos C = 0 = sin A + sin B + sin C, then cos (A - B) + cos (B - C) + cos (C - A) =$ (1) $0$ (2) $\frac{1}{2}$ (3) $\frac{3}{2}$ (4) $\frac{- 3}{2}$ 62. $If A > 0, B > 0 and A + B = \frac{π}{6}, then the minimum value of tanA + tanB, is [JEE (Main) 2016]$ (1) $2 - 3$ (2) $\frac{2}{3}$ (3) $3 - 2$ (4) $4 - 23$ 63. $If cos θ = \frac{- 3}{5} and π < θ < \frac{3 π}{2}, then tan \frac{θ}{2} + sin \frac{θ}{2} + 2 cos \frac{θ}{2} =$ (1) $-1$ (2) $1$ (3) $-2$ (4) $2$ 64. $The value of sin^{- 1} (cos \frac{53 π}{5}) is$ (1) $\frac{3 π}{5}$ (2) $\frac{- 3 π}{5}$ (3) $\frac{π}{10}$ (4) $\frac{- π}{10}$ 65. $The value of the expression tan (\frac{1}{2} cos^{- 1} \frac{2}{5}) is$ (1) $2 - 5$ (2) $5 - 2$ (3) $\frac{5 - 2}{2}$ (4) $5 - 2$ 66. $If sin θ - cos θ = \frac{1}{3}, then sin (2 θ) + cos (4 θ) + sin (6 θ) =$ (1) $\frac{37}{27}$ (2) $\frac{- 37}{27}$ (3) $\frac{- 43}{27}$ (4) $\frac{43}{27}$ 67. $In △ ABC, (a - b)^{2} cos^{2} \frac{c}{2} + (a + b)^{2} sin^{2} \frac{c}{2} =$ (1) $b^{2}$ (2) $c^{2}$ (3) $a^{2}$ (4) $a^{2} + b^{2} + c^{2}$ 68. $If x \neq = 0, then \frac{sin ( π + x ) cos ( \frac{π}{2} + x ) tan ( \frac{3 π}{2} - x ) cot ( 2 π - x )}{sin ( 2 π - x ) cos ( 2 π + x ) cosec ( - x ) sin ( \frac{3 π}{2} + x )} =$ (1) $0$ (2) $-1$ (3) $1$ (4) $2$ 69. $Let α, β be such that π < α - β < 3 π . If sinα + sinβ = - \frac{21}{65} and cosα + cosβ = - \frac{27}{65}, then the value of cos \frac{α - β}{2}, is [JEE (Main) 2004]$ (1) $- \frac{6}{65}$ (2) $\frac{3}{130}$ (3) $\frac{6}{65}$ (4) $- \frac{3}{130}$ 70. $In a triangle ABC, tan \frac{A}{2} tan \frac{B}{2} + tan \frac{B}{2} tan \frac{C}{2} + tan \frac{C}{2} tan \frac{A}{2} =$ (1) $0$ (2) $1$ (3) $\frac{1}{2}$ (4) $π$ 71. $If α = \frac{sin ^{3} x}{cos ^{2} x}, β = \frac{cos ^{3} x}{sin ^{2} x} and sin x + cos x = k, then αsin x + βcos x + 3 =$ (1) $\frac{2}{( k ^{2} - 1 ) ^{2}}$ (2) $\frac{4}{( k ^{2} - 1 ) ^{2}}$ (3) $\frac{k ^{2} - 1}{2}$ (4) $\frac{( k ^{2} - 1 ) ^{2}}{4}$ 72. $If cot α = 1 and sec β = - \frac{5}{3}, where π < α < \frac{3 π}{2} and \frac{π}{2} < β < π, then the value of tan (α + β) and the quadrant in which α + β lies, respectively are$ (1) $- \frac{1}{7} and I V^{t h} quadrant$ (2) $7 and I^{s t} quadrant$ (3) $- 7 and I V^{t h} quadrant$ (4) $\frac{1}{7} and I^{s t} quadrant$ 73. $The number of real roots of the equation sin (2 cos^{- 1} (cot (2 tan^{- 1} x))) = 0 that are greater than or equal to one are$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 74. $cos 12^{\circ} \cdot co s 24^{\circ} \cdot co s 36^{\circ} \cdot co s 48^{\circ} \cdot co s 72^{\circ} \cdot co s 84^{\circ} =$ (1) $\frac{1}{32}$ (2) $\frac{1}{16}$ (3) $\frac{1}{64}$ (4) $\frac{1}{128}$ 75. $If cos A = \frac{- 60}{61} and tan B = - \frac{7}{24} and neither A nor B is the second quadrant, then the angle A + \frac{B}{2} lies in the quadrant$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 76. $If \sum_{r = 1}^{50} tan^{- 1} \frac{1}{2 r ^{2}} = p, then tanp is$ (1) $\frac{100}{101}$ (2) $\frac{51}{50}$ (3) $\frac{50}{51}$ (4) $\frac{101}{102}$ 77. $In a △ ABC, if a = 13, b = 14 and c = 15, then sin \frac{A}{2} =$ (1) $\frac{1}{5}$ (2) $\frac{1}{7}$ (3) $\frac{3}{4}$ (4) $\frac{4}{5}$ 78. $The number of solutions of sec xcos 5 x + 1 = 0 in the interval [0, 2 π] is$ (1) $5$ (2) $8$ (3) $10$ (4) $12$ 79. $Let α, β and γ be such that 0 < α < β < γ < 2 π . For any x \in R if cos (x + α) + cos (x + β) + cos (x + γ) = 0, then tan (γ - α) =$ (1) $- 3$ (2) $0$ (3) $1$ (4) $3$ 80. $If y = log_{e} tan (\frac{π}{4} + \frac{x}{2}), then tan h (\frac{y}{2}) =$ (1) $cot \frac{x}{2}$ (2) $tan x$ (3) $cot hx$ (4) $tan \frac{x}{2}$ 81. $If sin θ + cos θ = \frac{1}{2}, then 16 (sin (2 θ) + cos (4 θ) + sin (6 θ)) is equal to:$ (1) $23$ (2) $-27$ (3) $-23$ (4) $27$ 82. $Value of cos^{3} \frac{π}{8} cos \frac{3 π}{8} + sin^{3} \frac{π}{8} sin \frac{3 π}{8} is$ (1) $\frac{1}{2 2}$ (2) $\frac{1}{2}$ (3) $\frac{1}{2}$ (4) $- \frac{1}{2}$ 83. $The value of sin (2 sin^{- 1} 0.8) is equal to$ (1) $0.48$ (2) $sin 1.2^{\circ}$ (3) $sin 1.6^{\circ}$ (4) $0.96$ 84. $If tanh (x) = \frac{1}{3}, then tan h (3 x) is$ (1) $\frac{8}{9}$ (2) $\frac{7}{9}$ (3) $1$ (4) $\frac{2}{3}$ 85. $The general solution of cos 2 x - 2 tan x + 2 = 0 is$ (1) $(2 n + 1) \frac{π}{3}, n \in Z$ (2) $(n + 1) \frac{π}{3}, n \in Z$ (3) $nπ + \frac{π}{3}, n \in Z$ (4) $nπ + \frac{π}{4}, n \in Z$ 86. $co t \frac{π}{16} \cdot co t \frac{2 π}{16} \cdot co t \frac{3 π}{16} \cdot co t \frac{4 π}{16} \cdot co t \frac{5 π}{16} \cdot co t \frac{6 π}{16} \cdot co t \frac{7 π}{16} is equal to$ (1) $0$ (2) $1$ (3) $\frac{1}{2}$ (4) $2$ 87. $1 + sec^{2} x sin^{2} x =$ (1) $sin 2 x$ (2) $sin^{2} x$ (3) $tan^{2} x$ (4) $sec^{2} x$ 88. $If 2 tan^{- 1} (cos x) = tan^{- 1} (2 cosec x), then sin x + cos x =$ (1) $22$ (2) $2$ (3) $\frac{1}{2}$ (4) $\frac{1}{2}$ 89. $The maximum value of the expression \frac{1}{sin ^{2} θ + 3 sinθ cosθ + 5 cos ^{2} θ}, is[JEE (Main) 2010]$ 90. $Let cos (α + β) = - \frac{1}{10} and sin (α - β) = \frac{3}{8}, where 0 < α < \frac{π}{3} and 0 < β < \frac{π}{4} . If tan 2 α = \frac{3 ( 1 - r 5 )}{11 ( s + 5 )}, r, s \in N, then r + s is equal to____.$