Trigonometric Ratios & Identities PYQs Set 13 (PDF)

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Set 13 of 18 · Questions 361-390 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 13 (Q361-390)
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361. $If θ = \frac{π}{12} and x = log (cot (\frac{π}{4} + θ)), then cos hx =$ (1) $\frac{2}{3}$ (2) $\frac{- 2}{3}$ (3) $\frac{3}{2}$ (4) $\frac{- 3}{2}$ 362. $If α = log_{e} (2 + 3), then \frac{cosh α}{1 - tanh α} + \frac{sinh α}{1 - coth α} =$ (1) $4 + 23$ (2) $7 + 43$ (3) $\frac{3 + 1}{2}$ (4) $2 + 3$ 363. $If sin (5 x + \frac{π}{4}) = 0, then x is equal to$ (1) $\frac{- π}{20} + \frac{π}{2} n (n \in Z)$ (2) $\frac{π}{20} + \frac{π}{5} n (n \in Z)$ (3) $\frac{- π}{5} + \frac{π}{5} n (n \in Z)$ (4) $\frac{- π}{20} + \frac{π}{5} n (n \in Z)$ 364. $The value of tan \frac{π}{8} is equal to$ (1) $\frac{1}{2} `$ (2) $2 + 1$ (3) $\frac{1}{2 + 1}$ (4) $1 - 2$ 365. $The value of cos (\frac{7 π}{12}) is$ (1) $\frac{2 + 3}{4}$ (2) $\frac{2 - 3}{4}$ (3) $\frac{2 - 6}{4}$ (4) $\frac{2 + 6}{4}$ 366. $The value of tan 1^{\circ} tan 2^{\circ} tan 3^{\circ} .... tan 89^{\circ} is$ (1) $0$ (2) $1$ (3) $\frac{1}{2}$ (4) $-1$ 367. $If x = sin (2 tan^{- 1} 2) and y = sin (\frac{1}{2} tan^{- 1} \frac{4}{3}), then$ (1) $x > y$ (2) $x = y$ (3) $x = 0 = y$ (4) $x < y$ 368. $If sin (θ) + cosec (θ) = 2, then sin^{2020} (θ) + cosec^{2020} (θ) = ......$ (1) $2^{2020}$ (2) $2020.2^{2019}$ (3) $2^{2019}$ (4) $2$ 369. $Let x, y, z be real numbers and x \geq y \geq z \geq \frac{π}{12} . If x + y + z = \frac{π}{2}, then the minimum value of cos x \cdot sin y \cdot cos z is$ (1) $\frac{1}{2}$ (2) $\frac{1}{4}$ (3) $\frac{1}{6}$ (4) $\frac{1}{8}$ 370. $If sin 2 x = 4 cos x, then x =$ (1) $2 nπ \pm \frac{π}{2}, n \in Z$ (2) $\frac{nπ}{2} \pm \frac{π}{4}, n \in Z$ (3) $no value$ (4) $nπ + (- 1)^{n} \frac{π}{4}, n \in Z$ 371. $If 5(t a n^{2} x - co s^{2} x) = 2cos2x + 9, then the value of cos4x is$ (1) $\frac{- 3}{5}$ (2) $\frac{1}{3}$ (3) $\frac{2}{9}$ (4) $- \frac{7}{9}$ 372. $96 cos \frac{π}{33} cos \frac{2 π}{33} cos \frac{4 π}{33} cos \frac{8 π}{33} cos \frac{16 π}{33} is equal to$ (1) $3$ (2) $1$ (3) $4$ (4) $2$ 373. $\frac{1}{r ^{2}} + \frac{1}{r _{1}^{2}} + \frac{1}{r _{2}^{2}} + \frac{1}{r _{3}^{2}} equals$ (1) $\frac{a ^{2} + b ^{2} + c ^{2}}{Δ}$ (2) $\frac{a ^{2} + b ^{2} + c ^{2}}{Δ ^{2}}$ (3) $\frac{Δ ^{2}}{a ^{2} + b ^{2} + c ^{2}}$ (4) $\frac{Δ}{a ^{2} + b ^{2} + c ^{2}}$ 374. $If in a triangle ABC, a = 2, b = 3 and c = 4, then tan (\frac{A}{2}) =$ (1) $\frac{3}{15}$ (2) $\frac{4}{15}$ (3) $\frac{2}{15}$ (4) $\frac{1}{15}$ 375. $tan h (log x) =$ (1) $\frac{x + 1}{x - 1}$ (2) $\frac{x ^{2} + 1}{x ^{2} - 1}$ (3) $\frac{x ^{2} - 1}{x ^{2} + 1}$ (4) $2 x$ 376. $If A + B + C = \frac{π}{2}, then 2 cos (\frac{π}{4} - A) + 2 cos (\frac{π}{4} - B) + 2 cos (\frac{π}{4} - C) + 1 =$ (1) $42 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}$ (2) $4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}$ (3) $4 sin \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}$ (4) $42 sin \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}$ 377. $The value of sin (2 cos^{- 1} \frac{5}{3}) is$ (1) $\frac{5}{3}$ (2) $\frac{2 5}{3}$ (3) $\frac{4 5}{9}$ (4) $\frac{2 5}{9}$ 378. $The least value of (cos^{2} θ - 6 sinθ cosθ + 3 sin^{2} θ + 2) is$ (1) $-1$ (2) $1$ (3) $4 - 10$ (4) $4 + 10$ 379. $If cos^{- 1} x - cos^{- 1} \frac{y}{3} = α, where - 1 \leq x \leq 1, - 3 \leq y \leq 3, x \leq \frac{y}{3}, then for all x, y, 9 x^{2} - 6 xycos α + y^{2} is equal to$ (1) $sin^{2} α$ (2) $3 sin^{2} α$ (3) $9 sin^{2} α$ (4) $\frac{4}{9} sin^{2} α$ 380. $If in a triangle (1 - \frac{r _{1}}{r _{2}}) (1 - \frac{r _{1}}{r _{3}}) = 2, then the triangle is$ (1) $right angled triangle$ (2) $equilateral triangle$ (3) $∠ B = 60^{\circ}$ (4) $∠ C = 45^{\circ}$ 381. $sin \frac{2 π}{5} + sin \frac{4 π}{5} + sin \frac{6 π}{5} + sin \frac{8 π}{5} is equal to$ (1) $0$ (2) $1$ (3) $\frac{2}{2}$ (4) $\frac{1}{2}$ 382. $sin^{4} \frac{π}{8} + sin^{4} \frac{2 π}{8} + sin^{4} \frac{3 π}{8} + sin^{4} \frac{4 π}{8} + sin^{4} \frac{5 π}{8} + sin^{4} \frac{6 π}{8} + sin^{4} \frac{7 π}{8} =$ (1) $\frac{3}{2}$ (2) $\frac{5}{2}$ (3) $3$ (4) $\frac{7}{2}$ 383. $cos^{3} 110^{\circ} + cos^{3} 10^{\circ} + cos^{3} 130^{\circ} =$ (1) $\frac{3}{4}$ (2) $\frac{3}{8}$ (3) $\frac{3 3}{8}$ (4) $\frac{3 3}{4}$ 384. $The value of \sum_{k = 1}^{13} \frac{1}{sin ( \frac{π}{4} + \frac{( k - 1 ) π}{6} ) sin ( \frac{π}{4} + \frac{kπ}{6} )} is equal to$ (1) $3 - 3$ (2) $2 (3 - 3)$ (3) $2 (3 - 1)$ (4) $2 (2 + 3)$ 385. $If sinh x = - \frac{4}{3} then sinh 2 x + cosh 2 x =$ (1) $\frac{- 31}{41}$ (2) $\frac{- 20}{9}$ (3) $\frac{49}{41}$ (4) $\frac{1}{9}$ 386. $The degree measure of \frac{π}{32} is equal to$ (1) $5^{\circ} 30^{’} 20^{’ ’}$ (2) $5^{\circ} 37^{’} 20^{’ ’}$ (3) $5^{\circ} 37^{’} 30^{’ ’}$ (4) $4^{\circ} 30^{’} 30^{’ ’}$ 387. $The value of \frac{1 + tanh x}{1 - tan h x} is$ (1) $e^{x}$ (2) $e^{- 2 x}$ (3) $e^{2 x}$ (4) $e^{- x}$ 388. $If cosh 2 x = 199, then cot hx equals$ (1) $\frac{5}{3 11}$ (2) $\frac{5}{6 11}$ (3) $\frac{7}{3 11}$ (4) $\frac{10}{3 11}$ 389. $The positive integral value of n > 3 satisfying \frac{1}{sin ( \frac{π}{n} )} = \frac{1}{sin ( \frac{2 π}{n} )} + \frac{1}{sin ( \frac{3 π}{n} )}, is [JEE (Main) 2011]$ 390. $Let \frac{3 π}{4} < θ < π and 2 cotθ + \frac{1}{sin ^{2} θ} = k - cotθ, then k is equal to [JEE Main) 2014]$