Trigonometric Ratios & Identities PYQs Set 6 (PDF)

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Set 6 of 18 · Questions 151-180 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 6 (Q151-180)
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151. $If sin α - cos α = m and sin 2 α = n - m^{2}, where - 2 \leq m \leq 2, then n is equal to$ (1) $0$ (2) $1$ (3) $2$ (4) $-2$ 152. $A true statement among the following identities is$ (1) $cos 5 θ = 16 cos^{5} θ - 20 cos^{3} θ - 5 cos θ$ (2) $cos 5 θ = 20 cos^{3} θ - 16 cos^{5} θ + 5 cos θ$ (3) $cos 5 θ = 16 cos^{5} θ + 20 cos^{3} θ - 5 cos θ$ (4) $cos 5 θ = 16 cos^{5} θ - 20 cos^{3} θ + 5 cos θ$ 153. $In a triangle ABC, if a = 4, b = 5 and c = 7, then sin (\frac{A}{2}) =$ (1) $\frac{3}{35}$ (2) $\frac{35}{3}$ (3) $\frac{2}{35}$ (4) $\frac{1}{35}$ 154. $In △ ABC, if r_{1} = 2 r_{2} = 3 r_{3}, then$ (1) $b । c = 2 a$ (2) $a + b = 2 c$ (3) $a + c = 2 b$ (4) $Δ = abc$ 155. $cosec 48^{\circ} + cosec 96^{\circ} + cosec 192^{\circ} + cosec 384^{\circ} =$ (1) $-2$ (2) $-1$ (3) $0$ (4) $\frac{3}{2}$ 156. $Let α and β respectively be the maximum and the minimum values of the function f (θ) = 4 (sin^{4} (\frac{7 π}{2} - θ) + sin^{4} (11 π + θ)) - 2 (sin^{6} (\frac{3 π}{2} - θ) + sin^{6} (9 π - θ)), θ \in R . Then α + 2 β is equal to:$ (1) $4$ (2) $6$ (3) $5$ (4) $3$ 157. $The sum of all values of θ\in (0, \frac{π}{2}) satisfying s i n^{2} 2θ + co s^{4} 2θ = \frac{3}{4} is$ (1) $π$ (2) $\frac{π}{2}$ (3) $\frac{3 π}{8}$ (4) $\frac{5 π}{4}$ 158. $sin α + cos α = m \Rightarrow sin^{6} α + cos^{6} α =$ (1) $\frac{4 + 3 ( m ^{2} - 1 ) ^{2}}{4}$ (2) $\frac{4 - 3 ( m ^{2} - 1 ) ^{2}}{4}$ (3) $\frac{3 + 4 ( m ^{2} - 1 ) ^{2}}{4}$ (4) $\frac{4 - 3 ( m ^{2} + 1 ) ^{2}}{4}$ 159. $If 2 sin 2 θ = 3, then θ =$ (1) $15^{\circ}$ (2) $27^{\circ}$ (3) $30^{\circ}$ (4) $40^{\circ}$ 160. $If A and B (A > B) are acute angles, sin (A - B) = \frac{16}{65} and sin B = \frac{5}{13} then tan A + cot A =$ (1) $\frac{25}{12}$ (2) $\frac{12}{25}$ (3) $\frac{5}{12}$ (4) $\frac{12}{5}$ 161. $In a △ PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to$ (1) $π /4$ (2) $3 π /4$ (3) $5 π /6$ (4) $π /6$ 162. $If sin xcosh y = cos θ and cos xsinh y = sin θ, then sin^{2} x + cosh^{2} y =$ (1) $1$ (2) $2$ (3) $sin 2 θ$ (4) $cos 2 θ$ 163. $If cos (α + β) = \frac{3}{5}, sin (α - β) = \frac{5}{13} and 0 < α, β < \frac{π}{4} then tan (2 α) is equal to:$ (1) $\frac{63}{52}$ (2) $\frac{63}{16}$ (3) $\frac{21}{16}$ (4) $\frac{33}{52}$ 164. $In triangle ABC, if a = 7, b = 10, c = 11, then \frac{R}{r} =$ (1) $14$ (2) $77$ (3) $\frac{24}{11}$ (4) $\frac{55}{24}$ 165. $\sum sin^{2} (2 k + 1) \frac{π}{20} =$ (1) $5$ (2) $\frac{5}{2}$ (3) $3$ (4) $\frac{3}{2}$ 166. $In a triangle ABC, let ∠ C = \frac{π}{2} . If r and R are respectively inradius and circumradius of ABC, then R + r =$ (1) $\frac{a - b}{2}$ (2) $\frac{a + b}{2}$ (3) $a + b$ (4) $a - b$ 167. $If 1 + sin θ + sin^{2} θ + \dots . upto \infty = 23 + 4, then θ =$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{3 π}{4}$ 168. $sin 22 \frac{1 ^{\circ}}{2} =$ (1) $\frac{2 + 2}{4}$ (2) $\frac{2 + 2}{4}$ (3) $\frac{2 - 2}{4}$ (4) $\frac{2 - 2}{4}$ 169. $Let θ, φ \in [0, 2π] be such that 2cosθ(1 - sinφ) = sin2θ (tan \frac{θ}{2} + cot \frac{θ}{2}) cosφ - 1, tan(2π - θ) > 0 and -1 < sinθ < - \frac{3}{2} . Then φ cannot satisfy$ (1) $0 < φ < \frac{π}{2}$ (2) $\frac{π}{2} < φ < \frac{4 π}{3}$ (3) $\frac{4 π}{3} < φ < \frac{3 π}{2}$ (4) $\frac{3 π}{2} < φ < 2π$ 170. $In △ ABC, if ∠ C = \frac{π}{3}, then \frac{3}{a + b + c} - \frac{1}{a + c} equals$ (1) $\frac{1}{a + b}$ (2) $\frac{1}{b + c}$ (3) $\frac{1}{2 a + b}$ (4) $\frac{1}{b + 2 c}$ 171. $The value of cos^{2} 10^{\circ} - cos 10^{\circ} cos 50^{\circ} + cos^{2} 50^{\circ} is:$ (1) $\frac{3}{4} + cos 20^{\circ}$ (2) $3/4$ (3) $\frac{3}{2} (1 + cos 20^{\circ})$ (4) $3/2$ 172. $Principal solutions of the equation sin 2 x + cos 2 x = 0, where π < x < 2 π are$ (1) $\frac{7 π}{8}, \frac{11 π}{8}$ (2) $\frac{9 π}{8}, \frac{13 π}{8}$ (3) $\frac{11 π}{8}, \frac{15 π}{8}$ (4) $\frac{15 π}{8}, \frac{19 π}{8}$ 173. $If a, b, c are the sides of a triangle A B C and b a c 111 a c b = 0, then 2 (cos A + cos B + cos C) =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 174. $Let \| cos θcos (60 - θ) cos (60 + θ) \| \leq \frac{1}{8}, θ ϵ [0, 2 π] . Then, the sum of all θ ϵ [0, 2 π], where cos 3 θ attains its maximum value, is :$ (1) $15 π$ (2) $18 π$ (3) $6 π$ (4) $9 π$ 175. $If cos θ \neq = 0 and sec θ - 1 = (2 - 1) tan θ, then θ =$ (1) $nπ + \frac{π}{8}, n \in Z$ (2) $2 nπ + \frac{π}{4} or 2nπ,n\inZ$ (3) $2 nπ + \frac{π}{8}, n \in Z$ (4) $2 nπ - \frac{π}{4}, or 2nπ,n\inZ$ 176. $sin^{2} (3^{\circ}) + sin^{2} (6^{\circ}) + sin^{2} (9^{\circ}) + \dots + sin^{2} (84^{\circ}) + sin^{2} (87^{\circ}) + sin^{2} (90^{\circ}) =$ (1) $\frac{31}{2}$ (2) $\frac{39}{2}$ (3) $\frac{59}{2}$ (4) $36$ 177. $The possible value of sin^{6} (θ) + cos^{6} (θ) - 3 cos^{4} (θ) is$ (1) $2$ (2) $-2$ (3) $-3$ (4) $3$ 178. $If 5 (tan^{2} x - cos^{2} x) = 2 cos 2 x + 9, then the value of 9 cos 4 x, is [JEE (Main) 2017]$ 179. $Let S = (- π, \frac{π}{2}) - (- \frac{π}{2}, - \frac{π}{4}, - \frac{3 π}{4}, \frac{π}{4}) Then the number of elements in the set A = {θ \in S : tan θ (1 + 5 tan (2 θ)) = 5 - tan (2 θ)} is____$ 180. $Let \frac{2 sin α}{1 + cos 2 α} = \frac{1}{7} and \frac{1 - cos 2 β}{2} = \frac{1}{10} Where α, β \in (0, \frac{π}{2}) Then tan (α + 2 β) is equal to$