Trigonometric Ratios & Identities PYQs Set 16 (PDF)

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Set 16 of 18 · Questions 451-480 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 16 (Q451-480)
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451. $In a right-angled triangle, if the difference between the two acute angles is 60^{\circ}, then the ratio of the lengths of the hypotenuse and the perpendicular drawn to the hypotenuse from its opposite vertex is$ (1) $2 : 1$ (2) $4 : 1$ (3) $8 : 1$ (4) $3 : 1$ 452. $The area (in square units) of △ ABC if ∠ A = 75^{\circ}, ∠ B = 45^{\circ} and a = 2 (3 + 1) is$ (1) $6$ (2) $23$ (3) $6 - 23$ (4) $6 + 23$ 453. $The angle of elevation of a jet plane from a point A on the ground is 60^{\circ} . After a flight of 20 seconds at thespeed of 432 km / hour, the angle of elevation changes to 30^{\circ} . If the jet plane is flying at a constant height, then its height is:$ (1) $12003 m$ (2) $18003 m$ (3) $36003 m$ (4) $24003 m$ 454. $If cot (cos^{- 1} x) = sec (tan^{- 1} \frac{a}{b ^{2} - a ^{2}}), then x is equal to$ (1) $\frac{b}{2 b ^{2} - a ^{2}}$ (2) $\frac{a}{2 b ^{2} - a ^{2}}$ (3) $\frac{2 b ^{2} - a ^{2}}{a}$ (4) $\frac{2 b ^{2} - a ^{2}}{b}$ 455. $A spherical gas balloon of radius 16 meter subtends an angle 6 0^{0} at the eye of the observer A while the angle of elevation of its center from the eye of A is 7 5^{0} . Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is:$ (1) $8 (2 + 23 + 2)$ (2) $8 (6 + 2 + 2)$ (3) $8 (2 + 2 + 3)$ (4) $8 (6 - 2 + 2)$ 456. $Two poles, AB of length a metres and CD of length a + b(b \neq a) metres are erected at the same horizontal level with bases at B and D. If BD=x and , then:$ (1) $x^{2} + 2(a + 2b)x- b(a + b) = 0$ (2) $x^{2} + 2(a + 2b)x + a(a + b) = 0$ (3) $x^{2} - 2ax + b(a + b) = 0$ (4) $x^{2} - 2ax + a(a + b) = 0$ 457. $In triangle ABC, if a = 7, b = 8, tan C = \frac{3 5}{2} and C is an acute angle, then c =$ (1) $145$ (2) $5$ (3) $11$ (4) $9$ 458. $If sin^{4} α + 4 cos^{4} β + 2 = 42 sin α cos β; α, β \in [0, π], then cos (α + β) - cos (α - β) is equal to$ (1) $- 2$ (2) $0$ (3) $2$ (4) $-1$ 459. $Let a, b and c be the length of sides of a triangle ABC such that \frac{a + b}{7} = \frac{b + c}{8} = \frac{c + a}{9} . If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of \frac{R}{r} is equal to$ (1) $\frac{5}{2}$ (2) $2$ (3) $\frac{3}{2}$ (4) $1$ 460. $In a triangle ABC, if cos A + 2 cos B + cos C = 2 and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then cos A-cos C is equal to$ (1) $\frac{9}{7}$ (2) $\frac{10}{7}$ (3) $\frac{5}{7}$ (4) $\frac{3}{7}$ 461. $Assertion: coth x = \frac{1 - k}{1 + k} (0 < k < 2) . Reason: The graph of y = tanh x always lies between the lines y = - 1 and y = 1 . The correct option among the following is$ (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) $(A) is false but (R) is true$ 462. $If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are 30º, 45º and 60º respectively, then the ratio, AB : BC , is$ (1) $3 : 1$ (2) $3 : 2$ (3) $1 : 3$ (4) $2 : 3$ 463. $cos (2 cos^{- 1} \frac{1}{5} + sin^{- 1} \frac{1}{5}) =$ (1) $\frac{1}{5}$ (2) $\frac{- 2 6}{5}$ (3) $- \frac{1}{5}$ (4) $\frac{6}{5}$ 464. $The angle of elevation of the summit of a mountain from a point on the ground is 45^{\circ} . After climbing up one km towards the summit at an inclination of 30^{\circ} from the ground, the angle of elevation of the summit is found to be 60^{\circ} . Then the height (in km) of the summit from the ground is:$ (1) $\frac{1}{3 + 1}$ (2) $\frac{1}{3 - 1}$ (3) $\frac{3 + 1}{3 - 1}$ (4) $\frac{3 - 1}{3 + 1}$ 465. $Let S = {θ \in [- 2 π, 2 π] : 2 cos^{2} θ + 3 sin θ = 0} Then the sum of the elements of S is:$ (1) $\frac{13 π}{6}$ (2) $\frac{5 π}{3}$ (3) $2 π$ (4) $π$ 466. $The value of 36(4 co s^{2} 9° - 1) (4 co s^{2} 27° - 1) (4 co s^{2} 81° - 1) (4 co s^{2} 243° - 1) is$ (1) $54$ (2) $18$ (3) $27$ (4) $36$ 467. $For a triangle ABC, the value of cos 2 A + cos 2 B + cos 2 C is least. It its inradius is 3 and incentre is M, then which of the following is NOT correct?$ (1) $Perimeter of ∆ABC is 183$ (2) $sin 2 A + sin 2 B + sin 2 C = sin A + sin B + sin C$ (3) $MA . MB = - 18$ (4) $Area of ∆ABC is \frac{27 3}{2}$ 468. $Let C_{(} α, β_{)} be the circumcentre of the triangle formed by the lines 4x + 3y = 69, 4y- 3x = 17, and x + 7y = 61. Then (α - β)^{2} + α + β is equal to$ (1) $18$ (2) $17$ (3) $15$ (4) $16$ 469. $A pole stands vertically inside a triangular park ABC . Let the angle of elevation of the top of the pole fromeach corner of the park be \frac{π}{3} . If the radius of the circumcircle ot Δ ABC is 2, then the height of the pole isequal to:$ (1) $\frac{2 3}{3}$ (2) $23$ (3) $3$ (4) $\frac{1}{3}$ 470. $Let PQR be a triangle of area Δ with a = 2, b = \frac{7}{2} and c = \frac{5}{2}, where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then \frac{2 s in P - s in 2 P}{2 s in P + s in 2 P} equals$ (1) $\frac{3}{4∆}$ (2) $\frac{45}{4∆}$ (3) $(\frac{3}{4Δ})^{2}$ (4) $(\frac{45}{4Δ})^{2}$ 471. $Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let \frac{π}{8} and θ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan^{2} θ is equal to$ (1) $\frac{3 - 2 2}{2}$ (2) $\frac{3 + 2}{2}$ (3) $\frac{3 - 2 2}{4}$ (4) $\frac{3 - 2}{4}$ 472. $The value of \sum_{k = 1}^{13} \frac{1}{s i n ( \frac{π}{4} + ( k - 1 ) \frac{π}{6} ) s i n ( \frac{π}{4} + \frac{k π}{6} )} is equal [JEE (Adv.) 2016]$ (1) $3 - 3$ (2) $2 (3 - 3)$ (3) $2 (3 - 1)$ (4) $2 (2 + 3)$ 473. $For 0 < θ < \frac{π}{2}, the solution(s) of \sum_{m = 1}^{6} csc (θ + \frac{( m - 1 ) π}{4}) csc (θ + \frac{mπ}{4}) = 42 is(are)$ (1) $\frac{π}{4}$ (2) $\frac{π}{6}$ (3) $\frac{π}{12}$ (4) $\frac{5 π}{12}$ 474. $The range of sec^{- 1} x is$ (1) $[0, π]$ (2) $[0, π] - {\frac{π}{2}}$ (3) $(\frac{- π}{2}, \frac{π}{2})$ (4) $(0, π)$ 475. $For a triangle with the sides of lengths 6, 5 and 9, the radius of the incircle is$ (1) $3$ (2) $2$ (3) $5$ (4) $\frac{3}{2}$ 476. $If sum of all the solutions of the equation 8 cos x (cos (\frac{π}{6} + x) cos (\frac{π}{6} - x) - \frac{1}{2}) = 1 in [0, π] is k π, then k is equal to :$ (1) $\frac{8}{9}$ (2) $\frac{20}{9}$ (3) $\frac{2}{3}$ (4) $\frac{13}{9}$ 477. $tan 2 α \cdot tan (3 0^{\circ} - α) + tan 2 α \cdot tan (6 0^{\circ} - α) + tan (6 0^{\circ} - α) \cdot tan (3 0^{\circ} - α) is equal to$ (1) $tan 3 α$ (2) $tan^{2} 2 α - tan^{2} 60^{\circ}$ (3) $1$ (4) $0$ 478. $In ∆ ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of ∆ ABC is 30 cm² and R and r are respectively the radii of circumcircle and incircle of ∆ ABC then the value of 2R + r (in cm) is equal to ___$ 479. $Let the plane x + 3y-2z + 6 = 0 meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle ABC is (α, β, \frac{6}{7}), then 98 (α + β)^{2} is equal to__________.$ 480. $The number of distinct solutions of the equation \frac{5}{4} cos^{2} 2 x + cos^{4} x + sin^{4} x + cos^{6} x + sin^{6} x = 2 in the interval [0, 2 π] is$