Trigonometric Ratios & Identities PYQs Set 17 (PDF)

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Set 17 of 18 · Questions 481-510 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 17 (Q481-510)
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481. $If sin 2 θ + sin 2 φ = \frac{1}{2} and cos 2 θ + cos 2 φ = \frac{3}{2}, then cos^{2} (θ - φ) =$ (1) $\frac{3}{8}$ (2) $\frac{5}{8}$ (3) $\frac{3}{4}$ (4) $\frac{5}{4}$ 482. $If \sum_{r = 1}^{13} {\frac{1}{s i n ( \frac{π}{4} + ( r - 1 ) \frac{π}{6} ) s i n ( \frac{π}{4} + \frac{r π}{6} )}} = a 3 + b, a, b \in Z, then a^{2} + b^{2} is equal to:$ (1) $10$ (2) $4$ (3) $2$ (4) $8$ 483. $The set of all values of λ for which the equation cos^{2} 2 x - 2 sin^{4} x - 2 cos^{2} x = λ$ (1) $[- 2, - 1]$ (2) $[- 2, - \frac{3}{2}]$ (3) $[- 1, \frac{1}{2}]$ (4) $[- \frac{3}{2}, - 1]$ 484. $If α, β and γ are angles that satisfy the following conditions, find the value of xyz . (1) tan (α) + tan (β) + tan (γ) = tan (α) \cdot tan (β) \cdot tan (γ) (2) x = cos (α) + isin (α) (3) y = cos (β) + isin (β) (4) z = cos (γ) + isin (γ)$ (1) $(1) , but not (-1)$ (2) $(-1) , but not (1)$ (3) $\pm 1$ (4) $0$ 485. $Which one of the following is possible?$ (1) $tan θ = 45$ (2) $cos θ = \frac{7}{3}$ (3) $sin θ = \frac{a ^{2} + b ^{2}}{a ^{2} - b ^{2}}, (a \neq = b)$ (4) $sec θ = \frac{4}{5}$ 486. $c os \frac{π}{12} =$ (1) $\frac{2 - 3}{2}$ (2) $\frac{2 + 3}{2}$ (3) $\frac{2 - 6}{4}$ (4) $\frac{2 + 6}{4}$ 487. $The value of 2sin(12°) - sin(72°) is :$ (1) $\frac{5 ( 1 - 3 )}{4}$ (2) $\frac{1 - 5}{8}$ (3) $\frac{3 ( 1 - 5 )}{2}$ (4) $\frac{3 ( 1 - 5 )}{4}$ 488. $cos 18^{\circ} =$ (1) $\frac{1}{8} (5 - 5)$ (2) $\frac{1}{2 2} 5 + 5$ (3) $\frac{5 - 1}{4}$ (4) $\frac{5 + 1}{4}$ 489. $The angle of elevation of a cloud C from a point P, 200 m above a still lake is 3 0^{\circ} . If the angle of depression of the image of C in the lake from the point P is 6 0^{\circ}, then P C (in m) is equal to:$ (1) $400$ (2) $4003$ (3) $100$ (4) $2003$ 490. $A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, withuniform speed. At that point, angle of depression of the boat with the man's eye is 30^{\circ} (Ignore man's height).After sailing for 20 seconds towards the base of the tower (which is at the level of water), the boat has reacheda point B, where the angle of depression is 45^{\circ} . Then the time taken (in seconds) by the boat from B to reachthe base of the tower is:$ (1) $10 (3 - 1)$ (2) $103$ (3) $10$ (4) $10 (3 + 1)$ 491. $In a △ ABC if ∠ A = 3 ∠ B, CA = 9 and BC = 16, then the length of AB is$ (1) $\frac{5}{3}$ (2) $\frac{7}{3}$ (3) $2$ (4) $\frac{35}{3}$ 492. $In a △ ABC, 2 Δ^{2} = \frac{a ^{2} b ^{2} c ^{2}}{a ^{2} + b ^{2} + c ^{2}}, then the triangle is$ (1) $equilateral$ (2) $isosceles$ (3) $right angled$ (4) $acute angled triangle$ 493. $If the angles of a △ ABC are in A.P., then$ (1) $c^{2} = a^{2} + b^{2} - ab$ (2) $a^{2} = b^{2} + c^{2} - ac$ (3) $b^{2} = a^{2} + c^{2} - ac$ (4) $b^{2} = a^{2} + c^{2}$ 494. $In a △ ABC, (a + b + c) (b + c - a) = λbc, then$ (1) $λ < - 6$ (2) $λ > 6$ (3) $0 < λ < 4$ (4) $λ > 4$ 495. $In a triangle ABC, if cos Acos B + sin Asin Bsin C = 1, then a : b : c =$ (1) $1 : 1 : 2$ (2) $1 : 1 : 1$ (3) $2 : 1 : 1$ (4) $1 : 2 : 1$ 496. $The value of the expression \frac{1 + sin 2 α}{cos ( 2 α - 2 π ) tan ( α - \frac{3 π}{4} )} - \frac{1}{4} sin 2 α (cot \frac{α}{2} + cot (\frac{3 π}{2} + \frac{α}{2})) is$ (1) $0$ (2) $1$ (3) $sin^{2} α 2$ (4) $sin^{2} α$ 497. $2 sin (\frac{π}{22}) sin (\frac{3 π}{22}) sin (\frac{5 π}{22}) sin (\frac{7 π}{22}) sin (\frac{9 π}{22}) is equal to$ (1) $\frac{3}{16}$ (2) $\frac{1}{16}$ (3) $\frac{1}{32}$ (4) $\frac{9}{32}$ 498. $In a △ ABC, \frac{a}{tan A} + \frac{b}{tan B} + \frac{c}{tan C} is equal to$ (1) $2 r$ (2) $r + 2 R$ (3) $2 r + R$ (4) $2 (r + R)$ 499. $In any triangle ABC, \frac{cos 2 A}{a ^{2}} - \frac{cos 2 B}{b ^{2}} =$ (1) $a^{2} - b^{2}$ (2) $\frac{1}{a ^{2}} - \frac{1}{b ^{2}}$ (3) $a^{2} + b^{2}$ (4) $\frac{1}{a ^{2}} + \frac{1}{b ^{2}}$ 500. $In a △ ABC, if a : b : c = 4 : 5 : 6, then \frac{1}{4 R} (r_{1} + r_{2} + r_{3}) is equal to$ (1) $\frac{71}{64}$ (2) $\frac{4}{5}$ (3) $\frac{81}{84}$ (4) $\frac{7}{9}$ 501. $In a △ ABC, if b = 10, a cos^{2} \frac{C}{2} + c cos^{2} \frac{A}{2} = 15, and the area of the triangle is 153 sq. units, then cot \frac{B}{2} =$ (1) $\frac{3}{2}$ (2) $\frac{1}{2}$ (3) $\frac{2}{3}$ (4) $\frac{5}{3}$ 502. $If a tan α + b tan β = (a + b) tan (\frac{α + β}{2}) and α - β \neq = 2 nπ then \frac{cos β}{cos α} =$ (1) $\frac{a}{b}$ (2) $\frac{a + b}{a - b}$ (3) $\frac{a ^{2} - b ^{2}}{a ^{2} + b ^{2}}$ (4) $\frac{b}{a}$ 503. $In △ ABC, \frac{1}{r _{1}^{2}} + \frac{1}{r _{2}^{2}} + \frac{1}{r _{3}^{2}} + \frac{1}{r ^{2}} =$ (1) $\frac{a ^{2} + b ^{2} + c ^{2}}{Δ ^{2}}$ (2) $\frac{a + b + c}{Δ ^{2}}$ (3) $\frac{s ^{2}}{Δ ^{2}}$ (4) $\frac{4 s ^{2}}{Δ ^{2}}$ 504. $In a △ ABC, if \frac{2 r _{2} r _{3}}{r _{2} - r _{1}} = r_{3} - r_{1}, then \frac{r _{1} ( r _{2} + r _{3} )}{r _{1} r _{2} + r _{2} r _{3} + r _{3} r _{1}} =$ (1) $\frac{a ^{2} + b ^{2} + c ^{2}}{Δ ^{2}}$ (2) $b - c$ (3) $1 2 R$ (4) $2 R$ 505. $In △ ABC, medians AD and BE are drawn. If AD = 4, ∠ DAB = \frac{π}{6} and ∠ ABE = \frac{π}{3}, then the area of △ ABC is$ (1) $\frac{8}{3} sq units$ (2) $\frac{16}{3} sq units$ (3) $\frac{32}{3 2} sq units$ (4) $\frac{64}{3} sq units$ 506. $In △ ABC, suppose the radius of the circle opposite to an ∠ A is denoted by r_{1}, similarly r_{2} \leftrightarrow ∠ B and r_{3} \leftrightarrow ∠ C . If r is the radius of inscribed circle, then, what is the value of \frac{ab - r _{1} r _{2}}{r _{3}} is equal to$ (1) $r_{1} r_{2} r_{3}$ (2) $r$ (3) $r_{1} r_{2} r_{3} 2$ (4) $\frac{r}{2}$ 507. $In a triangle, if b = 5, c = 6, tan \frac{A}{2} = \frac{1}{2}, then a =$ (1) $41$ (2) $21$ (3) $14$ (4) $22$ 508. $The trigonometric function y = tan x in the II quadrant$ (1) $decreases from 0 to \infty$ (2) $increase from 0 to \infty$ (3) $decreases from - \infty to 0$ (4) $increases from - \infty to 0$ 509. $If cot A = \frac{11}{60}, cos B = \frac{7}{25} and neither A nor B is in the first quadrant, then (A + \frac{B}{2}) lies in the quadrant$ (1) $I$ (2) $II$ (3) $III$ (4) $IV$ 510. $If x : y : z = tan (\frac{π}{15} + α) : tan (\frac{π}{15} + β) : tan (\frac{π}{15} + γ), then \frac{z + x}{z - x} sin^{2} (γ - α) + \frac{x + y}{x - y} sin^{2} (α - β) + \frac{y + z}{y - z} sin^{2} (β - γ) =$ (1) $sin^{2} θ$ (2) $cos^{2} θ$ (3) $0$ (4) $1$