Trigonometric Ratios & Identities PYQs Set 11 (PDF)

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Set 11 of 18 · Questions 301-330 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 11 (Q301-330)
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301. $If (1 + tan α) (1 + tan 4 α) = 2, α \in (0, \frac{π}{16}), then α is equal to$ (1) $\frac{π}{20}$ (2) $\frac{π}{30}$ (3) $\frac{π}{40}$ (4) $\frac{π}{60}$ 302. $The value of \frac{3 cosec 20 ^{\circ} - sec 20 ^{\circ}}{cos 20 ^{\circ} cos 40 ^{\circ} cos 60 ^{\circ} cos 80 ^{\circ}} is equal to$ (1) $16$ (2) $32$ (3) $64$ (4) $12$ 303. $The angle of △ ABC are in an arithmetic progression. If the larger sides a, b satisfy the relation \frac{3}{2} < \frac{b}{a} < 1, then the possible values of the smallest side are$ (1) $\frac{a \pm 4 b ^{2} - 3 a ^{2}}{2 a}$ (2) $\frac{a \pm 4 b ^{2} - 3 a ^{2}}{2 b}$ (3) $\frac{a \pm 4 b ^{2} - 3 a ^{2}}{2 c}$ (4) $\frac{a \pm 4 b ^{2} - 3 a ^{2}}{2}$ 304. $If sin^{4} α + 4 cos^{4} β + 2 = 42 sin αcos β; α, β \in [0, π] then cos (α + β) - cos (α - β) is equal to:$ (1) $0$ (2) $- 1$ (3) $2$ (4) $- 2$ 305. $A value of θ satisfying sin 5 θ - sin 3 θ + sin θ = 0 such that 0 < θ < \frac{π}{2} is$ (1) $\frac{π}{12}$ (2) $\frac{π}{6}$ (3) $\frac{π}{4}$ (4) $\frac{π}{2}$ 306. $cosh α + sinh α = e^{3} and sinh x = \frac{α}{α + 1}, then tanh x =$ (1) $\frac{α}{α + 2}$ (2) $\frac{α}{α - 3}$ (3) $\frac{α}{α + 4}$ (4) $\frac{2 α}{α - 1}$ 307. $If in a △ ABC, r_{1} = 2 r_{2} = 3 r_{3}, then the perimeter of the triangle is equal to$ (1) $3 a$ (2) $3 b$ (3) $3 c$ (4) $3 (a + b + c)$ 308. $A farmer owns a triangular plot in Guntur. He measures the lengths of the sides of his property as 4 cm . 5 cm and 7 cm . Then the area of land of the farmer in sq. cm is$ (1) $26$ (2) $46$ (3) $6$ (4) $86$ 309. $Which one of the following equations has no solution?$ (1) $cosec θ - sec θ = cosec θ \cdot sec θ$ (2) $cosec θ \cdot sec θ = 1$ (3) $cos θ + sin θ = 2$ (4) $3 sin θ - cos θ = 2$ 310. $If tan x = \frac{3}{4}, π < x < \frac{3 π}{2}, then the value of cos \frac{x}{2} is$ (1) $- \frac{1}{10}$ (2) $\frac{3}{10}$ (3) $\frac{1}{10}$ (4) $- \frac{3}{10}$ 311. $2 (sin^{6} θ + cos^{6} θ) - 3 (sin^{4} θ + cos^{4} θ) is equal to$ (1) $-1$ (2) $1$ (3) $0$ (4) $12$ 312. $The number of solutions in [0, 2 π] of the equation 1 6^{s i n^{2} x} + 1 6^{c o s^{2} x} = 10 is$ (1) $2$ (2) $4$ (3) $6$ (4) $8$ 313. $sinh (log (3 + 8)) =$ (1) $3^{\frac{3}{2}}$ (2) $2^{\frac{3}{2}}$ (3) $8^{\frac{2}{3}}$ (4) $3^{\frac{1}{2}}$ 314. $If \frac{1}{6} sin θ, cos θ and tan θ are in geometric progression, then the solution set of θ is$ (1) $2 nπ \pm (\frac{π}{6})$ (2) $2 nπ \pm (\frac{π}{3})$ (3) $nπ + 1 (- 1)^{n} (\frac{π}{3})$ (4) $nπ + (\frac{π}{3})$ 315. $1 + cos 10^{\circ} + cos 20^{\circ} + cos 30^{\circ} =$ (1) $4 sin 10^{\circ} sin 20^{\circ} sin 30^{\circ}$ (2) $4 cos 5^{\circ} cos 10^{\circ} cos 15^{\circ}$ (3) $4 cos 10^{\circ} cos 20^{\circ} cos 30^{\circ}$ (4) $4 sin 5^{\circ} sin 10^{\circ} sin 15^{\circ}$ 316. $If cot θ = - \frac{2}{3} and θ does not lie in the 4^{th} quadrant, then \frac{( 5 sin θ + cos θ ) ^{2}}{tan θ + cot θ} =$ (1) $-13$ (2) $-6$ (3) $- \frac{1734}{169}$ (4) $13$ 317. $If \frac{tan ( A - B )}{tanA} + \frac{sin ^{2} C}{sin ^{2} A} = 1, A, B, C \in (0, \frac{π}{2}), then$ (1) $tanA, tanC, tanB are in A.P.$ (2) $tanA, tanB, tanC are in G.P.$ (3) $tanA, tanC, tanB are in G.P.$ (4) $tanA, tanB, tanC are in A.P.$ 318. $In a triangle ABC, if \frac{a}{tan A} = \frac{b}{tan B} = \frac{c}{tan C} then cos^{2} A + cos^{2} B + cos^{2} C =$ (1) $2$ (2) $\frac{3}{4}$ (3) $\frac{3 + 1}{2}$ (4) $\frac{2 3 - 1}{2}$ 319. $If S is the circumcentre of a triangle ABC, a = 5, b = 6, c = 9 and SB = \frac{27}{4 2} then sin 2 C =$ (1) $\frac{4 2}{9}$ (2) $\frac{4 2}{27}$ (3) $\frac{- 4 2}{27}$ (4) $\frac{- 4 2}{9}$ 320. $If 3 sin^{4} x + 2 cos^{4} x = \frac{6}{5} and x is an acute angle, then tan 2 x =$ (1) $\frac{2 6}{5}$ (2) $26$ (3) $\frac{3 2}{5}$ (4) $\frac{2 3}{5}$ 321. $If tan^{- 1} 2 x + tan^{- 1} 3 x = \frac{π}{4}, then x =$ (1) $-1$ (2) $\frac{1}{3}$ (3) $\frac{1}{6}$ (4) $\frac{1}{2}$ 322. $If acos 2 θ + bsin 2 θ = c has α and β as its roots, then the value of tan α + tan β is$ (1) $\frac{2 b}{c + a}$ (2) $\frac{2 a}{b + c}$ (3) $\frac{b}{c + a}$ (4) $\frac{a}{b + c}$ 323. $If α \leq 2 sin^{- 1} x + cos^{- 1} x \leq β, then$ (1) $α = \frac{- π}{2}, β = \frac{3 π}{2}$ (2) $α = 0, β = 2 π$ (3) $α = \frac{- π}{2}, β = \frac{π}{2}$ (4) $α = 0, β = π$ 324. $If sin x + sin^{2} x = 1, x \in (0, \frac{π}{2}), then (cos^{12} x + tan^{12} x) + 3 (cos^{10} x + tan^{10} x + cos^{8} x + tan^{8} x) + (cos^{6} x + tan^{6} x) is equal to:$ (1) $4$ (2) $1$ (3) $3$ (4) $2$ 325. $Let ACB be a triangle with right-angle at C . Let AB = 29 units, BC = 21 units and ∠ ABC = θ . Then, cos^{2} θ - sin^{2} θ is equal to$ (1) $1$ (2) $\frac{41}{841}$ (3) $\frac{40}{441}$ (4) $\frac{41}{800}$ 326. $In △ ABC, if cot \frac{A}{2} : cot \frac{B}{2} : cot \frac{C}{2} = 3 : 7 : 9, then a : b : c =$ (1) $8 : 6 : 5$ (2) $5 : 6 : 8$ (3) $10 : 8 : 5$ (4) $5 : 8 : 10$ 327. $If tan A + cot A = 2, then the value of tan^{4} A + cot^{4} A =$ (1) $1$ (2) $4$ (3) $5$ (4) $2$ 328. $The number of ordered pairs (α, β), where α, β \in (- π, π) satisfying cos (α - β) = 1 and cos (α + β) = \frac{1}{e}, is [JEE (Main) 2005]$ 329. $In a triangle ABC, BC = 7, AC = 8, AB = α \in N and cos A = \frac{2}{3} . If 49 cos (3 C) + 42 = \frac{m}{n}, where gcd (m, n) = 1, then m + n is equal to______$ 330. $The positive integer value of n > 3 satisfying the equation \frac{1}{sin ( \frac{π}{n} )} = \frac{1}{sin ( \frac{2 π}{n} )} + \frac{1}{sin ( \frac{3 π}{n} )} is$