Trigonometric Ratios & Identities PYQs Set 12 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 12 (Q331-360)
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331. $c os 1^{\circ} \cdot cos 2^{\circ} \cdot cos 3^{\circ} \dots cos 179^{\circ} =$ (1) $0$ (2) $1$ (3) $- \frac{1}{2}$ (4) $-1$ 332. $If tan A = \frac{2}{3}, then sin 4 A =$ (1) $\frac{8}{27}$ (2) $\frac{120}{169}$ (3) $\frac{144}{169}$ (4) $\frac{16}{27}$ 333. $If cotx = \frac{5}{12} for some x \in (π, \frac{3 π}{2}), then sin 7 x (cos \frac{13 x}{2} + sin \frac{13 x}{2}) + cos 7 x (cos \frac{13 x}{2} - sin \frac{13 x}{2}) is equal to$ (1) $\frac{4}{26}$ (2) $\frac{6}{26}$ (3) $\frac{5}{13}$ (4) $\frac{1}{13}$ 334. $tan \frac{π}{5} + 2 tan \frac{2 π}{5} + 4 cot \frac{4 π}{5} is equal to$ (1) $cot \frac{π}{5}$ (2) $cot \frac{2 π}{5}$ (3) $cot \frac{3 π}{5}$ (4) $cot \frac{4 π}{5}$ 335. $The value of cos^{2} 45^{\circ} - sin^{2} 15^{\circ} is$ (1) $\frac{3 + 1}{2 2}$ (2) $\frac{3}{4}$ (3) $\frac{3 - 1}{2 2}$ (4) $\frac{3}{2}$ 336. $If cos x + cos y + cos α = 0 and sin x + sin y + sin α = 0, then cot (\frac{x + y}{2}) is equal to$ (1) $sin α$ (2) $cos α$ (3) $tan α$ (4) $cot α$ 337. $In a triangle ABC, if a = 3, b = 7, c = 8 then sin \frac{B}{2} tan \frac{C - A}{2} =$ (1) $\frac{15 3}{22 7}$ (2) $\frac{5 2}{11 7}$ (3) $\frac{5 3}{11}$ (4) $\frac{5 3}{22}$ 338. $In any triangle ABC, a (b cos C - c cos B) =$ (1) $b - c$ (2) $b + c$ (3) $b^{2} - c^{2}$ (4) $b^{2} + c^{2}$ 339. $The value of sin \frac{5 π}{12} sin \frac{π}{12} is$ (1) $0$ (2) $\frac{1}{2}$ (3) $1$ (4) $\frac{1}{4}$ 340. $Let a, b and c be the lengths of the sides of triangle with its opposite angles A, B and C respectively. If ∠ C = 60^{\circ}, then the value of \frac{c ( a + b ) + ( a ^{2} + b ^{2} )}{( b + c ) ( c + a )} is$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $1$ (4) $3$ 341. $In △ ABC if x = tan (\frac{B - C}{2}) tan \frac{A}{2}, y = tan (\frac{C - A}{2}) tan \frac{B}{2}, and z = tan (\frac{A - B}{2}) tan \frac{C}{2}, then (x + y + z) is equal to$ (1) $xyz$ (2) $- xyz$ (3) $2 xyz$ (4) $\frac{1}{2} xyz$ 342. $The value of sin 18^{\circ} is$ (1) $\frac{5 + 1}{4}$ (2) $\frac{5 - 1}{4}$ (3) $\frac{4}{5 + 1}$ (4) $\frac{4}{5 - 1}$ 343. $If cos θ = \frac{- 3}{5} and π < θ < 3 π /2, then tan (\frac{θ}{2}) is equal to$ (1) $2$ (2) $-2$ (3) $1$ (4) $-1$ 344. $If cos^{3} θ + cos^{3} (\frac{2 π}{3} + θ) + cos^{3} (\frac{4 π}{3} + θ) = acos 3 θ then a is equal to$ (1) $\frac{1}{4}$ (2) $\frac{3}{4}$ (3) $\frac{5}{4}$ (4) $\frac{7}{4}$ 345. $Let f (θ) = sinθ (sinθ + sin 3 θ) . Then, f (θ) [JEE (Main) 2000]$ (1) $\geq0 only when θ\geq0$ (2) $\leq0 for all real θ$ (3) $\geq0 for all real θ$ (4) $\leq0 only when θ\leq0$ 346. $If in a △ ABC, s (s - a) = (s - b) (s - c), then$ (1) $∠ A = \frac{π}{4}$ (2) $∠ B = \frac{π}{3}$ (3) $∠ A = \frac{π}{2}$ (4) $∠ B = \frac{π}{2}$ 347. $In △ ABC, if \frac{1}{a + b} + \frac{1}{c + a} = \frac{3}{a + b + c}, then sin A is equal to$ (1) $1$ (2) $\frac{1}{2}$ (3) $\frac{3}{2}$ (4) $\frac{4}{5}$ 348. $If \frac{π}{2} \leq x \leq \frac{3 π}{4}, then cos^{- 1} (\frac{12}{13} cos x + \frac{5}{13} sin x) is equal to$ (1) $x - tan^{- 1} \frac{4}{3}$ (2) $x + tan^{- 1} \frac{4}{5}$ (3) $x - tan^{- 1} \frac{5}{12}$ (4) $x + tan^{- 1} \frac{5}{12}$ 349. $\frac{sin 70 ^{\circ} + cos 40 ^{\circ}}{cos 70 ^{\circ} + sin 40 ^{\circ}} =$ (1) $1$ (2) $\frac{1}{3}$ (3) $3$ (4) $\frac{1}{2}$ 350. $Assertion (A): In △ABC, if r = 6, r_{2} = 36, R = 15 then c^{2} + a^{2} = b^{2} Reason (R): In △ABC, if r : R : r_{2} = 1 : 2.5 : 6 then B=90\circThe correct option among the following is$ (1) $Both (A) and (R) are true. (R) is a correct explanation of (A)$ (2) $Both (A) and (R) are true, but (R) is not a correct explanation of (A)$ (3) $(A) is true and (R) is false$ (4) $(A) is false and (R) is true$ 351. $sin^{2} 17.5^{\circ} + sin^{2} 72.5^{\circ} is equal to$ (1) $cos^{2} 90^{\circ}$ (2) $tan^{2} 45^{\circ}$ (3) $cos^{2} 30^{\circ}$ (4) $sin^{2} 45^{\circ}$ 352. $If A = 2 tan^{- 1} (\frac{1 + x}{1 - x}) and B = cos^{- 1} (\frac{1 - x ^{2}}{1 + x ^{2}}), where x \in (0, 1), then A - B =$ (1) $π /2$ (2) $4 tan^{- 1} x$ (3) $π /4$ (4) $tan^{- 1} x$ 353. $The value of e^{log_{10} tan 1^{\circ} + log_{10} tan 2^{\circ} + log_{10} tan 3^{\circ} + \dots . + log_{10} tan 89^{\circ}} is$ (1) $\frac{1}{e}$ (2) $0$ (3) $1$ (4) $3$ 354. $If sin (θ - α), sin θ and sin (θ + α) are in H.P., then the value of cos 2 θ is$ (1) $1 + 4 cos^{2} \frac{α}{2}$ (2) $1 - 4 cos^{2} \frac{α}{2}$ (3) $- 1 - 4 cos^{2} \frac{α}{2}$ (4) $- 1 + 4 cos^{2} \frac{α}{2}$ 355. $For x \in (0, π), the equation sinx + 2 sin 2x - sin 3x = 3 has$ (1) $infinitely many solutions$ (2) $three solutions$ (3) $one solution$ (4) $no solution$ 356. $If tan θ \cdot tan (120^{\circ} - θ) tan (120^{\circ} + θ) = \frac{1}{3}, then θ is equal to$ (1) $\frac{nπ}{3} + \frac{π}{18}, n \in Z$ (2) $\frac{nπ}{3} + \frac{π}{12}, n \in Z$ (3) $\frac{nπ}{12} + \frac{π}{12}, n \in Z$ (4) $\frac{nπ}{3} + \frac{π}{6}, n \in Z$ 357. $If A + B + C = π, cos B = cos Acos C, then tan Atan C =$ (1) $0$ (2) $1$ (3) $2$ (4) $\frac{1}{2}$ 358. $The number of values of θ in the interval (- \frac{π}{2}, \frac{π}{2}) such that θ \neq \frac{nπ}{5} for n = 0, \pm1, \pm 2 and tanθ = cot 5θ as well as sin 2θ = cos 4θ is$ 359. $Let a, b, c be three non-zero real numbers such that the equation 3 a cos x + 2b sinx = c, x \in (- \frac{π}{2}, \frac{π}{2}), has two distinct real roots α and β with α + β = \frac{π}{3} . Then, the value of \frac{b}{a} is ______.$ 360. $Let S = {θ ϵ (0, 2 π) : tan (π cos θ) + tan (π sin θ) = 0} Then \sum sin^{2} (θ + \frac{π}{4})$