Trigonometric Ratios & Identities PYQs Set 1 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 1 (Q1-30)
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1. $In a △ ABC, with usual notation, if a = 12, b = 16 and c = 20, then the ratio of the exradii of the triangle opposite to the angles in the order ∠ C, ∠ B and ∠ A is$ (1) $3 : 4 : 5$ (2) $6 : 3 : 2$ (3) $12 : 7 : 5$ (4) $2 : 3 : 5$ 2. $cos^{2} 5^{\circ} - cos^{2} 15^{\circ} - sin^{2} 15^{\circ} + sin^{2} 35^{\circ} + cos 15^{\circ} sin 15^{\circ} - cos 5^{\circ} sin 35^{\circ} =$ (1) $0$ (2) $1$ (3) $\frac{3}{2}$ (4) $2$ 3. $In △ ABC, if A is an acute angle, b = 6, c = 9 and sin A = \frac{2 14}{9}, then 3 a (cos B + cos C) =$ (1) $14$ (2) $20$ (3) $17$ (4) $23$ 4. $If △ ABC is such that ∠ A = 90^{\circ}, ∠ B \neq = ∠ C, then \frac{b ^{2} + c ^{2}}{b ^{2} - c ^{2}} sin (B - C) is equal to$ (1) $\frac{1}{3}$ (2) $\frac{1}{2}$ (3) $1$ (4) $\frac{3}{2}$ 5. $If (sin x - cos^{2} x) \geq (3 - 3 sin x + sin^{2} x) + 4 \| sin x - 1 \|, then x =$ (1) $(4 n + 1) \frac{π}{2}, n \in Z$ (2) $2 nπ + \frac{π}{3}, n \in Z$ (3) $nπ + \frac{π}{2}, n \in Z$ (4) $2 nπ + \frac{π}{6}, n \in Z$ 6. $Let S = (x \in (- π, π) : x \neq = 0, \pm \frac{π}{2}) The sum of all distinct solutions of the equation 3 sec x + cosec x + 2(tan x - cot x) = 0 in the set S is equal to$ (1) $- \frac{7 π}{9}$ (2) $- \frac{2 π}{9}$ (3) $0$ (4) $\frac{5 π}{9}$ 7. $If f (x) = \frac{cos ^{2} x + sin ^{4} x}{sin ^{2} x + cos ^{4} x} \forall x \in R, then f (2023) =$ (1) $1$ (2) $0$ (3) $2$ (4) $π$ 8. $If sin (A + B) sin (A - B) + cos (A + B) cos (A - B) = \frac{1}{2} and 0 < B < \frac{π}{2}, then B =$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{5 π}{12}$ 9. $If a = 2, b = 3, c = 4 in a triangle ABC, then cos C =$ (1) $\frac{1}{4}$ (2) $\frac{- 1}{4}$ (3) $\frac{1}{2}$ (4) $\frac{- 1}{2}$ 10. $2 cos^{- 1} x = sin^{- 1} (2 x 1 - x^{2}) is valid for all values of x satisfying$ (1) $0 \leq x \leq \frac{1}{2}$ (2) $-1\leqx\leq1$ (3) $0\leqx\leq1$ (4) $\frac{1}{2} \leq x \leq 1$ 11. $The period of \frac{sin x}{cos 3 x} + \frac{sin 3 x}{cos 9 x} + \frac{sin 9 x}{cos 27 x} + \frac{sin 27 x}{cos 81 x} is$ (1) $2 π 3$ (2) $\frac{π}{81}$ (3) $2 π$ (4) $π$ 12. $The value of cos \frac{π}{2 ^{2}} . cos \frac{π}{2 ^{3}} ...... cos \frac{π}{2 ^{10}} . sin \frac{π}{2 ^{10}} is :$ (1) $\frac{1}{1024}$ (2) $\frac{1}{2}$ (3) $\frac{1}{512}$ (4) $\frac{1}{256}$ 13. $The value of cos^{- 1} (cot (\frac{π}{2})) + cos^{- 1} (sin (\frac{2 π}{3})) is$ (1) $\frac{2 π}{3}$ (2) $\frac{π}{3}$ (3) $\frac{π}{2}$ (4) $π$ 14. $sin^{4} \frac{π}{8} + cos^{4} \frac{3 π}{8} - sin^{4} \frac{3 π}{8} + sin^{4} \frac{5 π}{8} + cos^{4} \frac{7 π}{8} - sin^{4} \frac{7 π}{8} =$ (1) $\frac{1}{4}$ (2) $\frac{1}{2}$ (3) $0$ (4) $\frac{3}{4}$ 15. $The number of integral values of k for which the equation 7 cos x + 5 sin x = 2 k + 1 has a solution, is$ (1) $4$ (2) $6$ (3) $8$ (4) $10$ 16. $\frac{1}{cos 290 ^{\circ}} + \frac{1}{3 sin 250 ^{\circ}} =$ (1) $\frac{3}{4}$ (2) $\frac{4}{3}$ (3) $\frac{2}{3}$ (4) $\frac{3}{2}$ 17. $If cos θ - sin θ = 5 sin θ, then cos θ + 4 sin θ =$ (1) $5 cos θ$ (2) $5 sin θ$ (3) $5 sin θ$ (4) $5 cos θ$ 18. $In a △ ABC, with usual notation, if r = r_{1} - r_{2} - r_{3}, then 2 R =$ (1) $a$ (2) $b + c$ (3) $c$ (4) $c + a$ 19. $If e^{(cos^{2} x + cos^{4} x + cos^{6} x + \dots \infty) log_{e} 2} satisfies the equation t^{2} - 9 t + 8 = 0, then the value of \frac{2 sin x}{sin x + 3 cos x} (0 < x < \frac{π}{2}) is:$ (1) $\frac{3}{2}$ (2) $23$ (3) $\frac{1}{2}$ (4) $3$ 20. $If sin hx = tan A, then \| tan hx \| =$ (1) $\| sin A \|$ (2) $\| cos A \|$ (3) $\| sec A \|$ (4) $\| cosec A \|$ 21. $If sinh u = tan θ, then cosh u is equal to$ (1) $- sec θ$ (2) $sec θ$ (3) $sin θ$ (4) $cot θ$ 22. $If tan A = \frac{1}{x ( x ^{2} + x + 1 )}, tan B = \frac{x}{x ^{2} + x + 1} and tan C = (x^{- 3} + x^{- 2} + x^{- 1})^{\frac{1}{2}}, 0 < A, B, C < \frac{π}{2}, then A + B is equal to :$ (1) $C$ (2) $π - C$ (3) $2 π - C$ (4) $\frac{π}{2} - C$ 23. $If sin^{- 1} (\frac{2 a}{1 + a ^{2}}) + cos^{- 1} (\frac{1 - a ^{2}}{1 + a ^{2}}) = tan^{- 1} (\frac{2 x}{1 - x ^{2}}) where a, x \in (0, 1) then the value of x is$ (1) $\frac{2 a}{1 + a ^{2}}$ (2) $0$ (3) $\frac{2 a}{1 - a ^{2}}$ (4) $\frac{a}{2}$ 24. $The range of the function, f (x) = log_{5} (3 + cos (\frac{3 π}{4} + x) + cos (\frac{π}{4} + x) + cos (\frac{π}{4} - x) - cos (\frac{3 π}{4} - x)) is$ (1) $(0, 5)$ (2) $(- 2, 2)$ (3) $(\frac{1}{5}, 5)$ (4) $(0, 2)$ 25. $If L = sin^{2} (\frac{π}{16}) - sin^{2} (\frac{π}{8}) and M = cos^{2} (\frac{π}{16}) - sin^{2} (\frac{π}{8}) then$ (1) $L = - \frac{1}{2 2} + \frac{1}{2} cos \frac{π}{8}$ (2) $M = \frac{1}{2 2} + \frac{1}{2} cos \frac{π}{8}$ (3) $M = \frac{1}{4 2} + \frac{1}{4} cos \frac{π}{8}$ (4) $L = \frac{1}{4 2} - \frac{1}{4} cos \frac{π}{8}$ 26. $Let f_{k} (x) = \frac{1}{k} (sin^{k} x + cos^{k} x) For k = 1, 2, 3, \dots Then for all x \in R the value of f_{4} (x) - f_{6} (x) is equal to:$ (1) $\frac{1}{12}$ (2) $\frac{1}{4}$ (3) $\frac{- 1}{12}$ (4) $\frac{5}{12}$ 27. $In a triangle ABC, if (b + c)^{2} sin^{2} \frac{A}{2} + (b - c)^{2} cos^{2} \frac{A}{2} = K (1 - cos 2 A),then K =$ (1) $R^{2}$ (2) $2 R^{2}$ (3) $R$ (4) $2 R$ 28. $If A, B, C are the angles of △ ABC, then cot A \cdot cot B + cot B \cdot cot C + cot C \cdot cot A =$ (1) $0$ (2) $1$ (3) $2$ (4) $- 1$ 29. $If A + B + C = \frac{3 π}{2} then 4 sin A sin B sin C + cos 2 A + cos 2 B + cos 2 C =$ (1) $- sin (A + B + C)$ (2) $cos (A + B + C)$ (3) $sin (A + B + C)$ (4) $2 - cos (A + B + C)$ 30. $The number of integral values of ’ k ’ for which the equation 3 sin x + 4 cos x = k + 1 has a solution, k \in R is$