Trigonometric Ratios & Identities PYQs Set 14 (PDF)

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Set 14 of 18 · Questions 391-420 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 14 (Q391-420)
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391. $In any triangle ABC, r^{2} cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2} =$ (1) $Δ$ (2) $2Δ$ (3) $Δ^{2}$ (4) $5Δ$ 392. $The value of cos^{2} 10^{\circ} - cos 10^{\circ} \cdot cos 50^{\circ} + cos^{2} 50^{\circ} is$ (1) $\frac{3}{2} + cos 20^{\circ}$ (2) $\frac{3}{4}$ (3) $\frac{3}{4} (1 + cos 20^{\circ})$ (4) $\frac{3}{2}$ 393. $In a triangle ABC, if \frac{cos A}{a} = \frac{cos B}{b} = \frac{cos C}{c} and a = 2, then its area is$ (1) $23$ (2) $3$ (3) $\frac{3}{2}$ (4) $\frac{3}{4}$ 394. $s in 21^{\circ} cos 9^{\circ} - cos 84^{\circ} cos 6^{\circ} =$ (1) $1$ (2) $\frac{1}{4}$ (3) $\frac{1}{2}$ (4) $\frac{3}{2}$ 395. $Let α be the period of 3 sin \frac{πx}{3} - cos \frac{πx}{2} + tan \frac{πx}{4}, β be the period of sin^{2} (\frac{π}{7} + \frac{x}{4}) - sin^{2} (\frac{π}{7} - \frac{x}{4}), and γ be the period of cos^{4} x + sin^{4} x . Then \frac{αγ}{β} =$ (1) $\frac{3}{2}$ (2) $\frac{3}{4}$ (3) $3$ (4) $6$ 396. $If θ does not lie in the second quadrant and tan θ = \frac{- 3}{4}, then tan \frac{θ}{2} + sin 2 θ =$ (1) $\frac{97}{75}$ (2) $\frac{- 97}{75}$ (3) $\frac{- 47}{75}$ (4) $\frac{47}{75}$ 397. $sec^{2} x + 5 tan x + 5 =$ (1) $(tan x + 2) (tan x + 3)$ (2) $(tan x + 1) (tan x + 5)$ (3) $(tan x - 2) (tan x - 3)$ (4) $(sin x + 2) (sin x + 5)$ 398. $In △ ABC, if a = 7, b = 8 and c = 9 then \frac{1}{r _{1}^{2}} + \frac{1}{r _{2}^{2}} + \frac{1}{r _{3}^{2}} =$ (1) $\frac{97}{360}$ (2) $\frac{5}{72}$ (3) $\frac{169}{360}$ (4) $\frac{67}{72}$ 399. $For θ \in (0, \frac{π}{2}), sec h^{- 1} (cos θ) is equal to$ (1) $log (tan (\frac{π}{6} + \frac{θ}{2}))$ (2) $log (tan (\frac{π}{3} + \frac{θ}{2}))$ (3) $log (tan (\frac{π}{4} + \frac{θ}{2}))$ (4) $log (tan (\frac{π}{4} - \frac{θ}{2}))$ 400. $If tan hx = \frac{1}{2} then sinh 2 x - sec h 2 x =$ (1) $\frac{29}{15}$ (2) $\frac{11}{15}$ (3) $3$ (4) $\frac{- 13}{15}$ 401. $If \frac{1 + cos A}{1 - cos A} = \frac{x}{y}, then the value of tan A =$ (1) $\frac{2 xy}{x ^{2} - y ^{2}}$ (2) $\frac{2 xy}{y ^{2} - x ^{2}}$ (3) $\frac{x ^{2} + y ^{2}}{x ^{2} - y ^{2}}$ (4) $\frac{2 xy}{x ^{2} + y ^{2}}$ 402. $In a triangle ABC, (b + c) cos A + (c + a) cos B + (a + b) cos C =$ (1) $2 abc$ (2) $abc$ (3) $a + b + c$ (4) $a + b + c /2 abc$ 403. $If α and β are non-zero real numbers such that 2 (cosβ - cosα) + cosα cosβ = 1 . Then which of the following is true? [JEE (Adv) 2017]$ (1) $tan \frac{α}{2} = \pm 3 tan \frac{β}{2}$ (2) $tan \frac{β}{2} = \pm 3 tan \frac{α}{2}$ (3) $tan \frac{α}{2} = \pm tan \frac{β}{2}$ (4) $tan \frac{α}{2} = \pm 2 tan \frac{β}{2}$ 404. $A wire of length 44 cm is bent into an arc of a circle of radius 12 cm . The angle (in degrees) subtended by the arc at the centre of the circle is$ (1) $(\frac{11}{3})^{\circ}$ (2) $(\frac{660}{π})^{\circ}$ (3) $150^{\circ}$ (4) $(\frac{5}{3})^{\circ}$ 405. $If sin (\frac{π}{4} cos θ) = cos (\frac{π}{4} tan θ), then θ is equal to$ (1) $2 nπ + \frac{π}{4}$ (2) $2 nπ \pm \frac{π}{4}$ (3) $2 nπ - \frac{π}{4}$ (4) $nπ + \frac{π}{4}$ 406. $If tan β = \frac{tan α + tan γ}{1 + tan αtan γ}, then \frac{sin 2 α + sin 2 γ}{1 + sin 2 αsin 2 γ} is equal to$ (1) $sin 2 β$ (2) $cos 2 β$ (3) $tan 2 β$ (4) $sec 2 β$ 407. $If A + B + C = 60^{\circ}, then cos (30^{\circ} - A) + cos (30^{\circ} - B) + cos (30^{\circ} - C) + sin (A + B + C) =$ (1) $4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}$ (2) $4 sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}$ (3) $4 cos \frac{A}{2} cos \frac{B}{2} sin \frac{C}{2}$ (4) $4 cos \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}$ 408. $The general solution of cos (x) - sin (x) = 0 is$ (1) $nπ - \frac{π}{4}, n \in Z$ (2) $2 nπ + \frac{π}{4}, n \in Z$ (3) $nπ + \frac{π}{4}, n \in Z$ (4) $2 nπ - \frac{π}{4}, n \in Z$ 409. $If (1 + tan 1^{\circ}) (1 + tan 2^{\circ}) \dots (1 + tan 45^{\circ}) = 2^{n}, then n =$ (1) $0$ (2) $32$ (3) $23$ (4) $2$ 410. $If a, b and c are the sides of aABC and exradii r_{1}, r_{2} and r_{3} are 12,6 and 4 respectively. Then, a + 2 b + 3 c is equal to$ (1) $24$ (2) $44$ (3) $30$ (4) $54$ 411. $If 1 + sin x + sin^{2} x + \dots . upto \infty = 4 + 23, 0 < x < π and x \neq = \frac{π}{2}, then x =$ (1) $\frac{π}{3}, \frac{2 π}{3}$ (2) $\frac{π}{6}, \frac{π}{3}$ (3) $\frac{π}{3}, \frac{5 π}{6}$ (4) $\frac{2 π}{3}, \frac{π}{6}$ 412. $The number of solutions of sin x + sin 3 x + sin 5 x = 0 in the interval (\frac{π}{2}, \frac{3 π}{2}) is$ (1) $2$ (2) $3$ (3) $4$ (4) $5$ 413. $In a triangle ABC, if a^{2} - b^{2} - c^{2} = bc (λ^{2} - 2 λ - 1), then$ (1) $0\leqλ\leq4$ (2) $-1\leqλ\leq2$ (3) $-1\leqλ\leq3$ (4) $0\leqλ\leq3$ 414. $Let \frac{s in A}{s in B} = \frac{s i n _{(} A - C _{)}}{s i n _{(} C - B _{)}}, where A,B,C are angles of a triangle ABC. If the lengths of the sides opposite these angles are a,b,c respectively, then :$ (1) $b^{2} - a^{2} = a^{2} + c^{2}$ (2) $b^{2}, c^{2}, a^{2} are in A.P.$ (3) $c^{2}, a^{2}, b^{2} are in A.P.$ (4) $a^{2}, b^{2}, c^{2} are in A.P.$ 415. $Assertion : If A = 15^{\circ}, B = 17^{\circ} and C = 13^{\circ}, then cot 2 A + cot 2 B + cot 2 C = cot 2 Acot 2 Bcot 2 C Reason : In a △ PQR, tanP2tanQ2+tanQ2tanR2+tanP2tanR2=1 The correct option among the following is$ (1) $(A) is true, (R) is true and (R) is the correct explanation for (A)$ (2) $(A) is true, (R) is true but (R) is not the correct explanation for (A)$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 416. $Let S = (θ \in (0, \frac{π}{2}) : \sum sec (θ + (m - 1) \frac{π}{6}) sec (θ + \frac{mπ}{6}) = - \frac{8}{3}) Then$ (1) $S = (\frac{π}{12})$ (2) $S = (\frac{2 π}{3})$ (3) $\sum θ = \frac{π}{2}$ (4) $\sum θ = \frac{3 π}{4}$ 417. $The value of the sin 1^{\circ} + sin 2^{\circ} + \dots .. + sin 359^{\circ} is equal to$ (1) $0$ (2) $1$ (3) $-1$ (4) $180$ 418. $Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middlepoint of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary,then the height of the shorter pole (in meters) is:$ (1) $25$ (2) $203$ (3) $30$ (4) $253$ 419. $ABCD is a trapezium such that AB and CD are parallel and BC ⊥ CD. If ∠ADB = θ , BC = p and CD = q, then AB is equal to :$ (1) $\frac{( p ^{2} + q ^{2} ) s in θ}{p cos θ + q s in θ}$ (2) $\frac{p ^{2} + q ^{2} cos θ}{p cos θ + q s in θ}$ (3) $\frac{p ^{2} + q ^{2}}{p ^{2} cos θ + q ^{2} s in θ}$ (4) $\frac{( p ^{2} + q ^{2} ) s in θ}{( p cos θ + q s in θ ) ^{2}}$ 420. $In a triangle ABC with fixed base BC, the vertex A moves such that cos B + cos C = 4 sin2 \frac{A}{2} . If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C respectively, then$ (1) $b + c = 4a$ (2) $b + c = 2a$ (3) $locus of points A is an ellipse$ (4) $locus of point A is a pair of straight lines$