Trigonometric Ratios & Identities PYQs Set 9 (PDF)

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Set 9 of 18 · Questions 241-270 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 9 (Q241-270)
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241. $The value of cot \frac{π}{24} is:$ (1) $2 + 3 + 2 - 6$ (2) $2 + 3 + 2 + 6$ (3) $2 - 3 - 2 + 6$ (4) $32 - 3 - 6$ 242. $If A = s i n^{2} x + co s^{4} x, then for all real x$ (1) $\frac{3}{4} \leq A \leq 1$ (2) $\frac{13}{16} \leq A \leq 1$ (3) $1 \leq A \leq 2$ (4) $\frac{3}{4} \leq A \leq \frac{13}{16}$ 243. $If α + β = \frac{π}{2} and β + γ = α, then tanα equals [JEE (Main) 2001]$ (1) $2 (tanβ + tanγ)$ (2) $tanβ + tanγ$ (3) $tanβ + 2 tanγ$ (4) $2 tanβ + tanγ$ 244. $If tan θ_{1} = kcot θ_{2}, then \frac{cos ( θ _{1} + θ _{2} )}{cos ( θ _{1} - θ _{2} )} =$ (1) $\frac{1 + k}{1 - k}$ (2) $\frac{1 - k}{1 + k}$ (3) $\frac{k + 1}{k - 1}$ (4) $\frac{k - 1}{k + 1}$ 245. $Let cos (α + β) = \frac{4}{5} and let sin (α - β) = \frac{5}{13}, where 0 < α, β \leq \frac{π}{4} . Then, tan 2 α = [JEE (Main) 2010]$ (1) $19/12$ (2) $20/7$ (3) $25/16$ (4) $56/33$ 246. $The number of real values of x \in \| 0, 2 π \| - (\frac{π}{2}, \frac{3 π}{2}) satisfying the equation \| cos x \|^{2 sin^{2} x - 3 sin x + 1} = 1, is$ (1) $3$ (2) $4$ (3) $5$ (4) $6$ 247. $If cot^{- 1} \frac{3}{4} + sin^{- 1} \frac{5}{13} = sin^{- 1} (k), then k =$ (1) $\frac{63}{65}$ (2) $\frac{12}{13}$ (3) $\frac{65}{68}$ (4) $\frac{5}{12}$ 248. $If cosec θ - cot θ = 2017, then quadrant in which θ lies is$ (1) $I$ (2) $IV$ (3) $III$ (4) $II$ 249. $Let A and B denote the statements: A : cosα + cosβ + cosγ = 0; B : sinα + sinβ + sinγ = 0 If cos (α - β) + cos (β - γ) + cos (γ - α) = - \frac{3}{2}, then [JEE (Main) 2009]$ (1) $A is true and B is false$ (2) $A is false and B is true$ (3) $both A and B are true$ (4) $both A and B are false$ 250. $cos^{4} \frac{π}{24} - sin^{4} \frac{π}{24} =$ (1) $\frac{2 - 3}{2}$ (2) $\frac{2 + 3}{2}$ (3) $\frac{2 - 6}{4}$ (4) $\frac{2 + 6}{4}$ 251. $If sin x = - \frac{3}{5}, where π < x < \frac{3 π}{2}, then 80 (tan^{2} x - cos x) is equal to$ (1) $108$ (2) $109$ (3) $18$ (4) $19$ 252. $In △ ABC, if r = 1, R = 4 and △ = 8, then \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} =$ (1) $8$ (2) $\frac{1}{4}$ (3) $\frac{1}{8}$ (4) $\frac{1}{16}$ 253. $Let \frac{π}{2} < θ < π and cotθ = - \frac{1}{2 2} . Then the value of sin (\frac{15 θ}{2}) (cos 8 θ + sin 8 θ) + cos (\frac{15 θ}{2}) (cos 8 θ - sin 8 θ) is equal to$ (1) $- \frac{2}{3}$ (2) $\frac{2 - 1}{3}$ (3) $\frac{2}{3}$ (4) $\frac{1 - 2}{3}$ 254. $In a triangle ABC, if \frac{r}{r _{1}} = \frac{1}{2}, then 4 tan \frac{A}{2} (tan \frac{B}{2} + tan \frac{C}{2}) =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 255. $cos^{2} (x) + cos^{2} (x + \frac{π}{3}) + cos^{2} (x - \frac{π}{3}) =$ (1) $\frac{3}{2}$ (2) $\frac{1}{2}$ (3) $\frac{- 3}{2}$ (4) $\frac{- 1}{2}$ 256. $In any triangle ABC, the simplified form of \frac{cos 2 A}{a ^{2}} - \frac{cos 2 B}{b ^{2}} is$ (1) $a^{2} - b^{2}$ (2) $1 a^{2} - b^{2}$ (3) $\frac{1}{a ^{2}} - \frac{1}{b ^{2}}$ (4) $a^{2} + b^{2}$ 257. $The value of tan (1^{\circ}) + tan (89^{\circ}) is$ (1) $\frac{2}{sin ( 2 ^{\circ} )}$ (2) $\frac{1}{sin ( 2 ^{\circ} )}$ (3) $\frac{1}{sin ( 1 ^{\circ} )}$ (4) $\frac{2}{sin ( 1 ^{\circ} )}$ 258. $Let f : (- 1, 1) \to R be such that f (cos 4 θ) = \frac{2}{2 - sec ^{2} θ} for θ \in (0, π /4) \cup (π /4, π /2) . Then the values of f (\frac{1}{3}) are [JEE (Main) 2012]$ (1) $1 \pm \frac{3}{2}$ (2) $1 \pm \frac{2}{3}$ (3) $1 \pm \frac{1}{3}$ (4) $1 \pm \frac{1}{2}$ 259. $In △ ABC, if a = 2, B = tan^{- 1} \frac{1}{2}, and C = tan^{- 1} \frac{1}{3}, then (A, b) =$ (1) $(\frac{π}{4}, \frac{2 2}{5})$ (2) $(\frac{3 π}{4}, \frac{2}{5})$ (3) $(\frac{π}{4}, \frac{2}{5})$ (4) $(\frac{3 π}{4}, \frac{2 2}{5})$ 260. $If xcos θ = ycos (θ + \frac{2 π}{3}) = zcos (θ + \frac{4 π}{3}), then \frac{1}{x} + \frac{1}{y} + \frac{1}{z} =$ (1) $1$ (2) $2$ (3) $0$ (4) $3$ 261. $The general solution of the equation tan^{2} x = 1 is$ (1) $nπ + \frac{π}{4}$ (2) $nπ - \frac{π}{4}$ (3) $nπ \pm \frac{π}{4}$ (4) $2 nπ \pm \frac{π}{4}$ 262. $If 2 sin (θ + \frac{π}{3}) = cos (θ - \frac{π}{6}), then tan θ =$ (1) $3$ (2) $- \frac{1}{3}$ (3) $\frac{1}{3}$ (4) $- 3$ 263. $\frac{tan ^{- 1} ( 3 ) - sec ^{- 1} ( - 2 )}{cosec ^{- 1} ( - 2 ) + cos ^{- 1} ( - \frac{1}{2} )} =$ (1) $\frac{4}{5}$ (2) $- \frac{4}{5}$ (3) $\frac{3}{5}$ (4) $0$ 264. $If y = sin^{2} θ + cos^{4} θ, then for all real values of θ [JEE (Main) 2011]$ (1) $y \in [1, 2]$ (2) $y \in [13/16, 1]$ (3) $y \in [3/4, 13/16]$ (4) $y \in [3/4, 1]$ 265. $α = sin 36° is a root of which of the following equation$ (1) $10 x^{4} - 10 x^{2} -5 = 0$ (2) $16 x^{4} + 20 x^{2} -5 = 0$ (3) $16 x^{4} - 20 x^{2} +5 = 0$ (4) $16 x^{4} - 10 x^{2} +5 = 0$ 266. $If sin x + sin y = p, cos x + cos y = q, then sec (x + y) =$ (1) $\frac{2 pq}{p ^{2} + q ^{2}}$ (2) $\frac{p ^{2} + q ^{2}}{q ^{2} - p ^{2}}$ (3) $\frac{2 pq}{p ^{2} + q ^{2}}$ (4) $\frac{p + q}{p ^{2} + q ^{2}}$ 267. $In a triangle ABC s (\frac{r _{1} - r}{a} + \frac{r _{2} - r}{b} + \frac{r _{3} - r}{c}) =$ (1) $\frac{1}{r _{1}} + \frac{1}{r _{2}} + \frac{1}{r _{3}}$ (2) $r_{1} + r_{2} + r_{3}$ (3) $r_{1} r_{2} r_{3}$ (4) $\frac{1}{r} - \frac{1}{r _{1}} + \frac{1}{r _{2}} + \frac{1}{r _{3}}$ 268. $If the angles of a triangle are in the ratio 3 : 4 : 5, then the sides are in the ratio$ (1) $2 : 6 : 3 + 1$ (2) $2 : 6 : 3 + 1$ (3) $2 : 3 : 3 + 1$ (4) $3 : 4 : 5$ 269. $If the area of triangle ABC is b^{2} - (c - a)^{2}, then tan B =$ (1) $1$ (2) $\frac{13}{15}$ (3) $\frac{1}{4}$ (4) $\frac{8}{15}$ 270. $If \| sin α - cos α \| = \frac{3}{4}, then \| sec 2 α - tan 2 α \| =$ (1) $\frac{12}{17}$ (2) $\frac{4}{23}$ (3) $\frac{3}{23}$ (4) $\frac{7}{23}$