Question 1: Palette 1/45

JEE Mains 2026 MCQ

Number of solutions of $\sqrt{3} \cos 2 \theta+8 \cos \theta+3 \sqrt{3}=0, \theta \in[-3 \pi, 2 \pi]$ is :

A

5

B

4

C

3

D

0

Question 2: Palette 2/45

JEE Mains 2026 MCQ

Let $S=\{\theta \in(-2 \pi, 2 \pi): \cos \theta+1=\sqrt{3} \sin \theta\}$.

Then $\sum\limits_{\theta \in \mathrm{S}} \theta$ is equal to :

A

$-\frac{2 \pi}{3}$

B

$-\frac{4 \pi}{3}$

C

$\frac{2 \pi}{3}$

D

$\frac{4 \pi}{3}$

Question 3: Palette 3/45

JEE Mains 2026 MCQ

The sum of all the integral values of $p$ such that the equation $3 \sin ^2 x+12 \cos x-3=p, x \in \mathbb{R}$, has at least one solution, is:

A

-54

B

-60

C

-75

D

-84

Question 4: Palette 4/45

JEE Mains 2026 MCQ

Let $S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a,\ a \in \mathbb{Z} \}$. Then $n(S)$ is equal to:

A

3

B

6

C

7

D

9

Question 5: Palette 5/45

JEE Mains 2025 MCQ

The number of solutions of the equation

$ \cos 2\theta \cos \frac{\theta}{2} + \cos \frac{5\theta}{2} = 2\cos^3 \frac{5\theta}{2} $ in $ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] $ is :

A

5

B

7

C

6

D

9

Question 6: Palette 6/45

JEE Mains 2025 MCQ

The number of solutions of the equation

$(4-\sqrt{3}) \sin x-2 \sqrt{3} \cos ^2 x=-\frac{4}{1+\sqrt{3}}, x \in\left[-2 \pi, \frac{5 \pi}{2}\right]$ is

A

4

B

3

C

6

D

5

Question 7: Palette 7/45

JEE Mains 2025 MCQ

$ \text { The number of solutions of the equation } 2 x+3 \tan x=\pi, x \in[-2 \pi, 2 \pi]-\left\{ \pm \frac{\pi}{2}, \pm \frac{3 \pi}{2}\right\} \text { is: } $

A

4

B

5

C

3

D

6

Question 8: Palette 8/45

JEE Mains 2025 MCQ

If $\theta \epsilon\left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right]$, then the number of solutions of $\sqrt{3} \operatorname{cosec}^2 \theta-2(\sqrt{3}-1) \operatorname{cosec} \theta-4=0$, is equal to :

A

7

B

10

C

6

D

8

Question 9: Palette 9/45

JEE Mains 2025 MCQ

If $\theta \in[-2 \pi, 2 \pi]$, then the number of solutions of $2 \sqrt{2} \cos ^2 \theta+(2-\sqrt{6}) \cos \theta-\sqrt{3}=0$, is equal to:

A

8

B

6

C

10

D

12

Question 10: Palette 10/45

JEE Mains 2025 MCQ

The sum of all values of $\theta \in[0,2 \pi]$ satisfying $2 \sin ^2 \theta=\cos 2 \theta$ and $2 \cos ^2 \theta=3 \sin \theta$ is

A

$\pi$

B

$\frac{5 \pi}{6}$

C

$\frac{\pi}{2}$

D

$4 \pi$

Question 11: Palette 11/45

JEE Mains 2024 MCQ

Let $|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]$. Then, the sum of all $\theta \in[0,2 \pi]$, where $\cos 3 \theta$ attains its maximum value, is :

A

$6 \pi$

B

$9 \pi$

C

$18 \pi$

D

$15 \pi$

Question 12: Palette 12/45

JEE Mains 2024 MCQ

The number of solutions of the equation $4 \sin ^2 x-4 \cos ^3 x+9-4 \cos x=0 ; x \in[-2 \pi, 2 \pi]$ is :

A

0

B

3

C

1

D

2

Question 13: Palette 13/45

JEE Mains 2024 MCQ

If $2 \sin ^3 x+\sin 2 x \cos x+4 \sin x-4=0$ has exactly 3 solutions in the interval $\left[0, \frac{\mathrm{n} \pi}{2}\right], \mathrm{n} \in \mathrm{N}$, then the roots of the equation $x^2+\mathrm{n} x+(\mathrm{n}-3)=0$ belong to :

A

$(0, \infty)$

B

Z

C

$\left(-\frac{\sqrt{17}}{2}, \frac{\sqrt{17}}{2}\right)$

D

$(-\infty, 0)$

Question 14: Palette 14/45

JEE Mains 2024 MCQ

The sum of the solutions $x \in \mathbb{R}$ of the equation $\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6$ is

A

3

B

1

C

0

D

$-$1

Question 15: Palette 15/45

JEE Mains 2024 MCQ

If $\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$ is the solution of $4 \cos \theta+5 \sin \theta=1$, then the value of $\tan \alpha$ is

A

$\frac{10-\sqrt{10}}{12}$

B

$\frac{\sqrt{10}-10}{6}$

C

$\frac{\sqrt{10}-10}{12}$

D

$\frac{10-\sqrt{10}}{6}$

Question 16: Palette 16/45

JEE Mains 2024 MCQ

If $2 \tan ^2 \theta-5 \sec \theta=1$ has exactly 7 solutions in the interval $\left[0, \frac{n \pi}{2}\right]$, for the least value of $n \in \mathbf{N}$, then $\sum_\limits{k=1}^n \frac{k}{2^k}$ is equal to:

A

$\frac{1}{2^{14}}\left(2^{15}-15\right)$

B

$1-\frac{15}{2^{13}}$

C

$\frac{1}{2^{15}}\left(2^{14}-14\right)$

D

$\frac{1}{2^{13}}\left(2^{14}-15\right)$

Question 17: Palette 17/45

JEE Mains 2023 MCQ

The number of elements in the set

$S=\left\{\theta \in[0,2 \pi]: 3 \cos ^{4} \theta-5 \cos ^{2} \theta-2 \sin ^{6} \theta+2=0\right\}$ is :

A

9

B

8

C

12

D

10

Question 18: Palette 18/45

JEE Mains 2023 MCQ

Let $S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^{2} x}+9^{\tan ^{2} x}=10\right\}$ and $\beta=\sum_\limits{x \in S} \tan ^{2}\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta-14)^{2}$ is equal to :

A

16

B

32

C

8

D

64

Question 19: Palette 19/45

JEE Mains 2022 MCQ

The number of elements in the set $S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$ is :

A

1

B

3

C

0

D

infinite

Question 20: Palette 20/45

JEE Mains 2022 MCQ

Let $S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \pi}{6}\right)=-\frac{8}{\sqrt{3}}\right\}$. Then

A

$ S=\left\{\frac{\pi}{12}\right\} $

B

$ S=\left\{\frac{2 \pi}{3}\right\} $

C

$ \sum\limits_{\theta \in S} \theta=\frac{\pi}{2} $

D

$ \sum\limits_{\theta \in S} \theta=\frac{3\pi}{4} $

Question 21: Palette 21/45

JEE Mains 2022 MCQ

Let $S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .$ Then $n(s) + \sum\limits_{\theta \in S}^{} {\left( {\sec \left( {{\pi \over 4} + 2\theta } \right)\cos ec\left( {{\pi \over 4} + 2\theta } \right)} \right)} $ is equal to:

A

0

B

$-$2

C

$-$4

D

12

Question 22: Palette 22/45

JEE Mains 2022 MCQ

The number of solutions of $|\cos x|=\sin x$, such that $-4 \pi \leq x \leq 4 \pi$ is :

A

4

B

6

C

8

D

12

Question 23: Palette 23/45

JEE Mains 2022 MCQ

Let for some real numbers $\alpha$ and $\beta$, $a = \alpha - i\beta $. If the system of equations $4ix + (1 + i)y = 0$ and $8\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)x + \overline a y = 0$ has more than one solution, then ${\alpha \over \beta }$ is equal to

A

$ - 2\sqrt 3 $

B

$2 - \sqrt 3 $

C

$2 + \sqrt 3 $

D

$ - 2 - \sqrt 3 $

Question 24: Palette 24/45

JEE Mains 2022 MCQ

The number of solutions of the equation

$\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x$, $x \in [ - 3\pi ,3\pi ]$ is :

A

8

B

5

C

6

D

7

Question 25: Palette 25/45

JEE Mains 2022 MCQ

Let $S = \left\{ {\theta \in [ - \pi ,\pi ] - \left\{ { \pm \,\,{\pi \over 2}} \right\}:\sin \theta \tan \theta + \tan \theta = \sin 2\theta } \right\}$.

If $T = \sum\limits_{\theta \, \in \,S}^{} {\cos 2\theta } $, then T + n(S) is equal to :

A

7 + $\sqrt 3 $

B

9

C

8 + $\sqrt 3 $

D

10

Question 26: Palette 26/45

JEE Mains 2021 MCQ

If n is the number of solutions of the equation
$2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{\pi \over 4} - x} \right) - 1} \right) = 1,x \in [0,\pi ]$ and S is the sum of all these solutions, then the ordered pair (n, S) is :

A

(3, 13$\pi$ / 9)

B

(2, 2$\pi$ / 3)

C

(2, 8$\pi$ / 9)

D

(3, 5$\pi$ / 3)

Question 27: Palette 27/45

JEE Mains 2021 MCQ

The number of solutions of the equation ${32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4}$ is :

A

3

B

1

C

0

D

2

Question 28: Palette 28/45

JEE Mains 2021 MCQ

The sum of solutions of the equation

${{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|$, $x \in \left( { - {\pi \over 2},{\pi \over 2}} \right) - \left\{ {{\pi \over 4}, - {\pi \over 4}} \right\}$ is :

A

$ - {{11\pi } \over {30}}$

B

${\pi \over {10}}$

C

$ - {{7\pi } \over {30}}$

D

$ - {\pi \over {15}}$

Question 29: Palette 29/45

JEE Mains 2021 MCQ

The sum of all values of x in [0, 2$\pi$], for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to :

A

8$\pi$

B

11$\pi$

C

12$\pi$

D

9$\pi$

Question 30: Palette 30/45

JEE Mains 2021 MCQ

The number of solutions of sin⁷x + cos⁷x = 1, x$\in$ [0, 4$\pi$] is equal to

A

11

B

7

C

5

D

9

Question 31: Palette 31/45

JEE Mains 2021 MCQ

The number of solutions of the equation x + 2tanx = ${\pi \over 2}$ in the interval [0, 2$\pi$] is :

A

4

B

3

C

2

D

5

Question 32: Palette 32/45

JEE Mains 2021 MCQ

The number of roots of the equation, (81)^sin2x + (81)^cos2x = 30 in the interval [ 0, $\pi$ ] is equal to :

A

2

B

3

C

4

D

8

Question 33: Palette 33/45

JEE Mains 2021 MCQ

All possible values of $\theta$ $\in$ [0, 2$\pi$] for which sin 2$\theta$ + tan 2$\theta$ > 0 lie in :

A

$\left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{11\pi } \over 6}} \right)$

B

$\left( {0,{\pi \over 2}} \right) \cup \left( {\pi ,{{3\pi } \over 2}} \right)$

C

$\left( {0,{\pi \over 2}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{7\pi } \over 6}} \right)$

D

$\left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{5\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{7\pi } \over 4}} \right)$

Question 34: Palette 34/45

JEE Mains 2019 MCQ

Let S be the set of all $\alpha $ $ \in $ R such that the equation, cos2x + $\alpha $sinx = 2$\alpha $– 7 has a solution. Then S is equal to :

A

[2, 6]

B

[3, 7]

C

[1, 4]

D

R

Question 35: Palette 35/45

JEE Mains 2019 MCQ

If [x] denotes the greatest integer $ \le $ x, then the system of linear equations [sin $\theta $]x + [–cos$\theta $]y = 0, [cot$\theta $]x + y = 0

A

has a unique solution if $\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right)$ and have infinitely many solutions if $\theta \in \left( {\pi ,{{7\pi } \over 6}} \right)$

B

have infinitely many solutions if $\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right)$ and has a unique solution if $\theta \in \left( {\pi ,{{7\pi } \over 6}} \right)$

C

have infinitely many solutions if $\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right) \cup \left( {\pi ,{{7\pi } \over 6}} \right)$

D

has a unique solution if $\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right) \cup \left( {\pi ,{{7\pi } \over 6}} \right)$

Question 36: Palette 36/45

JEE Mains 2019 MCQ

The number of solutions of the equation
1 + sin⁴ x = cos²3x, $x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} \right]$ is :

A

5

B

3

C

7

D

4

Question 37: Palette 37/45

JEE Mains 2019 MCQ

Let S = {$\theta $ $ \in $ [–2$\pi $, 2$\pi $] : 2cos²$\theta $ + 3sin$\theta $ = 0}. Then the sum of the elements of S is

A

$\pi $

B

2$\pi $

C

${{13\pi } \over 6}$

D

${{5\pi } \over 3}$

Question 38: Palette 38/45

JEE Mains 2019 MCQ

The sum of all values of $\theta $ $ \in $$\left( {0,{\pi \over 2}} \right)$ satisfying
sin² 2$\theta $ + cos⁴ 2$\theta $ = ${3 \over 4}$ is -

A

${{5\pi } \over 4}$

B

${\pi \over 2}$

C

$\pi $

D

${{3\pi } \over 8}$

Question 39: Palette 39/45

JEE Mains 2019 MCQ

If  0 $ \le $ x < ${\pi \over 2}$,  then the number of values of x for which sin x $-$ sin 2x + sin 3x = 0, is :

A

3

B

1

C

4

D

2

Question 40: Palette 40/45

JEE Mains 2018 MCQ

If sum of all the solutions of the equation

$8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi \over 6} - x} \right) - {1 \over 2}} \right) = 1$

in [0, $\pi $] is k$\pi $, then k is equal to

A

${{20} \over 9}$

B

${2 \over 3}$

C

${{13} \over 9}$

D

${{8} \over 9}$

Question 41: Palette 41/45

JEE Mains 2018 MCQ

The number of solutions of sin3x = cos 2x, in the interval $\left( {{\pi \over 2},\pi } \right)$ is :

A

1

B

2

C

3

D

4

Question 42: Palette 42/45

JEE Mains 2016 MCQ

The number of x $ \in $ [0,   2$\pi $ ] for which

$\left| {\sqrt {2{{\sin }^4}x + 18{{\cos }^2}x} - \sqrt {2{{\cos }^4}x + 18{{\sin }^2}x} } \right| = 1$ is :

A

2

B

4

C

6

D

8

Question 43: Palette 43/45

JEE Mains 2016 MCQ

If $0 \le x < 2\pi $, then the number of real values of $x$, which satisfy the equation $\,\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is:

A

7

B

9

C

3

D

5

Question 44: Palette 44/45

JEE Mains 2006 MCQ

The number of values of $x$ in the interval $\left[ {0,3\pi } \right]\,$ satisfying the equation $2{\sin ^2}x + 5\sin x - 3 = 0$ is

A

4

B

6

C

1

D

2

Question 45: Palette 45/45

JEE Mains 2002 MCQ

The number of solution of $\tan \,x + \sec \,x = 2\cos \,x$ in $\left[ {0,\,2\,\pi } \right]$ is

A

2

B

3

C

0

D

1