Trigonometric Equations PYQs Set 1 (PDF)

Set 1 of 2 · Questions 1-30 of 47

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Equations · Set 1 (Q1-30)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
1. $A value of θ \in (0, \frac{π}{3}), for which 1 + cos^{2} θ cos^{2} θ cos^{2} θ sin^{2} θ 1 + sin^{2} θ sin^{2} θ 4 cos 6 θ 4 cos 6 θ 1 + 4 cos 6 θ = 0 is:$ (1) $\frac{π}{9}$ (2) $\frac{π}{18}$ (3) $\frac{7 π}{24}$ (4) $\frac{7 π}{36}$ 2. $The number of solutions for the equation sin 2 x + cos 4 x = 2 is$ (1) $2$ (2) $Infinite$ (3) $0$ (4) $1$ 3. $The general solution of cot θ + tan θ = 2 is$ (1) $θ = \frac{nπ}{2} + (- 1)^{n} \frac{π}{8}$ (2) $θ = \frac{nπ}{2} + (- 1)^{n} \frac{π}{4}$ (3) $θ = \frac{nπ}{2} + (- 1)^{n} \frac{π}{6}$ (4) $θ = nπ + (- 1)^{n} \frac{π}{8}$ 4. $If α, - \frac{π}{2} < α < \frac{π}{2} is the solution of 4 cos θ + 5 sin θ = 1, then the value of tan α is$ (1) $\frac{10 - 10}{6}$ (2) $\frac{10 - 10}{12}$ (3) $\frac{10 - 10}{12}$ (4) $\frac{10 - 10}{6}$ 5. $The sum of all values of θ \in [0, 2 π] satisfying 2 sin^{2} θ = cos 2 θ and 2 cos^{2} θ = 3 sin θ is.$ (1) $4 π$ (2) $\frac{5 π}{6}$ (3) $π$ (4) $\frac{π}{2}$ 6. $The number of solutions of s i n^{7} x + co s^{7} x = 1 x \in [0, 4π], is equal to$ (1) $11$ (2) $7$ (3) $5$ (4) $9$ 7. $The general solution of \| sin x \| = cos x is (when n \in Z) given by$ (1) $nπ \pm \frac{π}{4}$ (2) $nπ - \frac{π}{4}$ (3) $nπ + \frac{π}{4}$ (4) $2 nπ \pm \frac{π}{4}$ 8. $Let f (θ) = 3 (sin^{4} (\frac{π}{2} - θ) + sin^{4} (3 π + θ)) - 2 (1 - sin^{2} 2 θ) and S = (θ ϵ (0, π) : f^{’} (θ) = - \frac{3}{2}) . If 4 β = \sum θ then f (β) is equal to$ (1) $\frac{11}{8}$ (2) $\frac{5}{4}$ (3) $\frac{9}{8}$ (4) $\frac{3}{2}$ 9. $Suppose θ ϵ (0, \frac{π}{4}) is a solution of 4 cos θ - 3 sin θ = 1 . Then cos θ is equal to :$ (1) $\frac{4}{( 3 6 + 2 )}$ (2) $\frac{6 + 6}{( 3 6 + 2 )}$ (3) $\frac{4}{( 3 6 - 2 )}$ (4) $\frac{6 - 6}{( 3 6 - 2 )}$ 10. $Number of solutions of 3 cos 2 θ + 8 cosθ + 33 = 0, θ \in [- 3 π, 2 π] is:$ (1) $0$ (2) $5$ (3) $3$ (4) $4$ 11. $If 0 \leq x < \frac{π}{2}, then the number of values of x for which sin x - sin 2 x + sin 3 x = 0, is;$ (1) $3$ (2) $1$ (3) $4$ (4) $2$ 12. $The general solution of 1 + sin^{2} x = 3 sin x \cdot cos x, tan x \neq = \frac{1}{2} is$ (1) $2 nπ - \frac{π}{4}, n \in Z$ (2) $2 nπ + \frac{π}{4}, n \in Z$ (3) $nπ + \frac{π}{4}, n \in Z$ (4) $nπ - \frac{π}{4}, n \in Z$ 13. $The sum of the solutions x \in R of the equation \frac{3 cos 2 x + cos ^{3} 2 x}{cos ^{6} x - sin ^{6} x} = x^{3} - x^{2} + 6 is$ (1) $0$ (2) $1$ (3) $-1$ (4) $3$ 14. $All the pairs (x, y) that satisfy the inequality 2^{s i n^{2} x - 2 s i n x + 5} \cdot \frac{1}{4 ^{s i n^{2} y}} \leq 1 also satisfy the equation:$ (1) $2 (sin x) = 3 sin y$ (2) $2 sin x = sin y$ (3) $sin x = 2 sin y$ (4) $sin x = ∣ sin y ∣$ 15. $If cos x = \| sin x \|, then the general solution is$ (1) $x = nπ \pm \frac{π}{4}, n \in Z$ (2) $x = nπ + (- 1)^{n} \frac{π}{4}, n \in Z$ (3) $x = 2 nπ \pm \frac{π}{4}, n \in Z$ (4) $x = (2 n + 1) π \pm \frac{π}{4}, n \in Z$ 16. $If the equation cos^{4} θ + sin^{4} θ + λ = 0 has real solutions for θ, then λ lies in the interval$ (1) $(- 1, - \frac{1}{2})$ (2) $(- \frac{3}{2}, - \frac{5}{4})$ (3) $(- \frac{1}{2}, - \frac{1}{4})$ (4) $(- \frac{5}{4}, - 1)$ 17. $The sum of all values of θ \in (0, \frac{π}{2}) satisfying sin^{2} 2 θ + cos^{4} 2 θ = \frac{3}{4} is:$ (1) $π$ (2) $\frac{5 π}{4}$ (3) $\frac{π}{2}$ (4) $\frac{3 π}{8}$ 18. $If 2 sin^{3} x + sin 2 x cos x + 4 sin x - 4 = 0 has exactly 3 solutions in the interval (0, \frac{nπ}{2}), n \in N, then the roots of the equation x^{2} + nx + (n - 3) = 0 belong to :$ (1) $(0, \infty)$ (2) $(- \infty, 0)$ (3) $(- \frac{17}{2}, \frac{17}{2})$ (4) $Z$ 19. $The number of solutions of the equation 4 sin^{2} x - 4 cos^{3} x + 9 - 4 cos x = 0; x \in [- 2 π, 2 π] is :$ (1) $1$ (2) $3$ (3) $2$ (4) $0$ 20. $If 2 tan^{2} θ - 5 sec θ = 1 has exactly 7 solutions in the interval (0, \frac{nπ}{2}), for the least value of n \in N then \sum_{k = 1}^{n} \frac{k}{2 ^{k}} is equal to :$ (1) $\frac{1}{2 ^{15}} (2^{14} - 14)$ (2) $\frac{1}{2 ^{14}} (2^{15} - 15)$ (3) $1 - \frac{15}{2 ^{13}}$ (4) $\frac{1}{2 ^{13}} (2^{14} - 15)$ 21. $Let S be the set of all α \in R such that the equation, cos 2 x + α sin x = 2 α - 7 has a solution. Then S is equal to :$ (1) $R$ (2) $[1, 4]$ (3) $[3, 7]$ (4) $[2, 6]$ 22. $The number of real solutions of the equation tan^{- 1} x (x + 1) + sin^{- 1} x^{2} + x + 1 = \frac{π}{2} is$ (1) $infinitely many$ (2) $one$ (3) $four$ (4) $two$ 23. $If (x) denotes the greatest integer \leq x, then the system of linear equations (sin θ) x + (- cos θ) y = 0 (cot θ) x + y = 0$ (1) $have infinitely many solutions if θ \in (\frac{π}{2}, \frac{2 π}{3}) and has a unique solution if θ \in (π, \frac{7 π}{6})$ (2) $has a unique solution if θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$ (3) $has a unique solution if θ \in (\frac{π}{2}, \frac{2 π}{3}) and have infinitely many solution if θ \in (π, \frac{7 π}{6})$ (4) $have infinitely many solution if θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$ 24. $The number of solutions of the equation 1 + sin^{4} x = cos^{2} 3 x, x \in (- \frac{5 π}{2}, \frac{5 π}{2}) is:$ (1) $3$ (2) $5$ (3) $7$ (4) $4$ 25. $Let S = (sin^{2} 2 θ : (sin^{4} θ + cos^{4} θ) x^{2} + (sin 2 θ) x + (sin^{6} θ + cos^{6} θ) = 0) has real roots} . If α and β be the smallest and largest elements of the set S, respectively, then 3 ((α - 2)^{2} + (β - 1)^{2}) equals_________$ 26. $The number of elements in the set (x \in (0, 180^{\circ}) : tan (x + 100^{\circ}) = tan (x + 50^{\circ}) tanxtan (x - 50^{\circ})) is __________.$ 27. $Let the set of all a \in R such that the equation cos 2 x + a sin x = 2 a - 7 has a solution be [p, q] and r = tan 9^{\circ} - tan 27^{\circ} - \frac{1}{cot 63 ^{\circ}} + tan 81^{\circ}, then pqr is equal to_____$ 28. $The number of solutions of the equation \| cot x \| = cot x + \frac{1}{sin x} in the interval [0, 2 π] is$ 29. $Let S be the sum of all solutions (in radians) of the equation s i n^{4} θ + co s^{4} θ - sin θ cos θ = 0 in [0, 4π] Then \frac{8 S}{π} is equal to .$ 30. $If the sum of solutions of the system of equations 2sin^{2} θ -cos2 θ = 0 and 2cos^{2} θ + 3sin θ = 0 in the interval [0,2 π] is k π, then k is equal to_______.$