Trigonometric Ratios & Identities PYQs Set 10 (PDF)

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Set 10 of 18 · Questions 271-300 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 10 (Q271-300)
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271. $The value of tan 67 \frac{1}{2}^{\circ} + cot 67 \frac{1}{2}^{\circ} is$ (1) $22$ (2) $2 - 2$ (3) $2$ (4) $32$ 272. $If e^{it} = cos t + isin t and e^{- it} = cos t - isin t then cosh (x + iy) - cosh (x - iy) =$ (1) $2 sinh xsinh y$ (2) $2 isin hxcos y$ (3) $2 cosh xcos y$ (4) $2 isinh xsin y$ 273. $The value of \frac{sin θ + sin 3 θ}{cos θ + cos 3 θ} is$ (1) $cos 2 θ$ (2) $cot 2 θ$ (3) $tan 2 θ$ (4) $cosec θ + sin θ$ 274. $cos^{2} 76^{\circ} + sin^{2} 46^{\circ} + sin 76^{\circ} cos 46^{\circ} =$ (1) $\frac{3}{4}$ (2) $1$ (3) $\frac{5}{4}$ (4) $2$ 275. $Let cos(α + β) = \frac{4}{5} and let sin(α - β) = \frac{5}{13}, where 0 \leq α, β \leq \frac{π}{4} .Then tan 2α =$ (1) $\frac{56}{33}$ (2) $\frac{19}{12}$ (3) $\frac{20}{7}$ (4) $\frac{25}{16}$ 276. $cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{3 π}{7} cos \frac{π}{14} cos \frac{3 π}{14} cos \frac{5 π}{14} =$ (1) $\frac{1}{16} (sin \frac{π}{7} + sin \frac{2 π}{7} + sin \frac{3 π}{7})$ (2) $\frac{1}{8} (sin \frac{2 π}{7} + sin \frac{3 π}{7} - sin \frac{π}{7})$ (3) $\frac{1}{32} (sin \frac{2 π}{7} + sin \frac{3 π}{7} - sin \frac{π}{7})$ (4) $\frac{1}{32} (sin \frac{π}{7} - sin \frac{2 π}{7} + sin \frac{3 π}{7})$ 277. $If sin θ = sin α, then$ (1) $\frac{θ + α}{2} is any multiple of \frac{π}{2} and \frac{θ - α}{2} is any odd multiple of π.$ (2) $\frac{θ + α}{2} is any odd multiple of \frac{π}{2} and \frac{θ - α}{2} is any multiple of π.$ (3) $\frac{θ + α}{2} is any multiple of \frac{π}{2} and \frac{θ - α}{2} is any even multiple of π.$ (4) $\frac{θ + α}{2} is multiple of \frac{π}{2} or \frac{θ - α}{2} is a multiple of π$ 278. $In △ ABC, A = 75^{\circ} and B = 45^{\circ}, then the value of b + c 2 is equal to$ (1) $a$ (2) $3 a$ (3) $2 a$ (4) $4 a$ 279. $The value of sin^{- 1} (\frac{2 2}{3}) + sin^{- 1} (\frac{1}{3}) is equal to$ (1) $\frac{π}{2}$ (2) $\frac{2 π}{3}$ (3) $\frac{π}{6}$ (4) $\frac{π}{4}$ 280. $If cosec hx = \frac{4}{5}, then sin h x =$ (1) $\frac{4}{5}$ (2) $\frac{5}{4}$ (3) $\frac{2}{3}$ (4) $\frac{2}{5}$ 281. $The value of cos (\frac{2 π}{7}) + cos (\frac{4 π}{7}) + cos (\frac{6 π}{7}) is equal to :$ (1) $-1$ (2) $- \frac{1}{2}$ (3) $- \frac{1}{3}$ (4) $- \frac{1}{4}$ 282. $Given both θ and φ are acute angles and sinθ = \frac{1}{2}, cos φ = \frac{1}{3}, then the value of θ + φ belongs to [JEE (Main) 2004]$ (1) $(π /3, π /2]$ (2) $(π /2, 2 π /3]$ (3) $(2 π /3, 5 π /6]$ (4) $(5 π /6, π]$ 283. $The value of x in (0, \frac{π}{2}) satisfying the equation (sin x) (cos x) = \frac{1}{4} is$ (1) $\frac{π}{6}$ (2) $\frac{π}{3}$ (3) $\frac{π}{8}$ (4) $\frac{π}{12}$ 284. $3 cosec 20^{\circ} - sec 20^{\circ} =$ (1) $2$ (2) $4$ (3) $3$ (4) $1$ 285. $If sin x - sin y = \frac{1}{2} and cos x - cos y = 1, then tan (x + y) =$ (1) $\frac{3}{8}$ (2) $- \frac{3}{8}$ (3) $\frac{4}{3}$ (4) $- \frac{4}{3}$ 286. $The solution set of the trigonometric equation tan θ + 5 cot θ = sec θ is$ (1) $(\frac{θ}{θ} = 2 nπ \pm \frac{π}{3}, n \in Z)$ (2) $(\frac{θ}{θ} = nπ + (- 1)^{n} \frac{π}{2}, n \in Z)$ (3) $(\frac{θ}{θ} = nπ + \frac{π}{6}, n \in Z)$ (4) $ϕ$ 287. $In △ ABC, if 8 R^{2} = a^{2} + b^{2} + c^{2}, then the triangle is a/an$ (1) $right angled triangle$ (2) $equilateral triangle$ (3) $scalene triangle$ (4) $obtuse angled triangle$ 288. $c os \frac{π}{7} - cos \frac{2 π}{7} + cos \frac{3 π}{7} - cos \frac{4 π}{7} + cos \frac{5 π}{7} - cos \frac{6 π}{7} =$ (1) $0$ (2) $\frac{3}{2}$ (3) $\frac{3}{4}$ (4) $1$ 289. $If 0 \leq θ \leq 2 π, 0 \leq α \leq 2 π and sec^{2018} θ + cosec^{2018} α = 2, then the value of cos^{2020} θ + sin^{2022} α =$ (1) $\frac{3}{2}$ (2) $\frac{1}{2 ^{2020}}$ (3) $1$ (4) $2$ 290. $If cos α + cos β + cos γ = 0 and sin α + sin β + sin γ = 0 then cos 2 α + cos 2 β + cos 2 γ =$ (1) $\frac{3}{2}$ (2) $cos^{2} \frac{α}{2} + cos^{2} \frac{β}{2} + cos^{2} \frac{γ}{2}$ (3) $3 sin (α + β + γ)$ (4) $cos (α + β) + cos (β + γ) + cos (γ + α)$ 291. $The value of \frac{sin 85 ^{\circ} - sin 35 ^{\circ}}{cos 65 ^{\circ}} =$ (1) $2$ (2) $-1$ (3) $1$ (4) $0$ 292. $If the solution of the equation log_{cosx} cot x + 4 log_{sin x} tan x = 1, x ϵ (0, \frac{π}{2}), is sin^{- 1} (\frac{α + β}{2}), where α, β are integers, then α + β is equal to$ (1) $3$ (2) $5$ (3) $6$ (4) $4$ 293. $The perimeter of a △ ABC is 6 times the arithmetic mean of the values of the sine of its angles. If its side BC is of unit length, then ∠ A =$ (1) $\frac{π}{6}$ (2) $\frac{π}{3}$ (3) $\frac{π}{2}$ (4) $π$ 294. $The expression \frac{tanA}{1 - cotA} + \frac{cotA}{1 - tanA} can be written as [JEE (Main) 2013]$ (1) $sinA cosA + 1$ (2) $secA cosecA + 1$ (3) $tanA + cotA$ (4) $secA + cosecA$ 295. $In a △ ABC, if ∠ A = 90^{\circ}, then cos^{- 1} (\frac{R}{r _{2} + r _{3}}) is equal to$ (1) $90^{\circ}$ (2) $30^{\circ}$ (3) $60^{\circ}$ (4) $45^{\circ}$ 296. $If 0 < θ < 2 π, then the intervals of values of θ for which 2 sin^{2} θ - 5 sinθ + 2 > 0, is [JEE (Main) 2006]$ (1) $(0, \frac{π}{6}) \cup (\frac{5 π}{6}, 2 π)$ (2) $(\frac{π}{8}, \frac{5 π}{6})$ (3) $(0, \frac{π}{8}) \cup (\frac{π}{6}, \frac{5 π}{6})$ (4) $(\frac{41 π}{48}, π)$ 297. $In a triangle ABC, m ∠ A, m ∠ B, m ∠ C are in A.P. and lengths of two larger sides are 10 units, 9 units respectively, then the length (in units) of the third side is$ (1) $5 + 6$ (2) $5 - 1$ (3) $6 + 1$ (4) $5 + 1$ 298. $cos (\frac{π}{7}) cos (\frac{2 π}{7}) cos (\frac{4 π}{7}) =$ (1) $\frac{- 1}{8}$ (2) $\frac{1}{8}$ (3) $- \frac{3 3}{8}$ (4) $1$ 299. $The value of sin 10^{\circ} \cdot sin 30^{\circ} \cdot sin 50^{\circ} \cdot sin 70^{\circ} is$ (1) $\frac{1}{8}$ (2) $\frac{3}{16}$ (3) $\frac{3}{16}$ (4) $\frac{1}{16}$ 300. $The value of tan 9° - tan 27° - tan 63° + tan 81° is_____.$