Trigonometry · JEE Main previous year questions

2026 34 questions

1. Q52026 · 8 Apr 2026 · Shift 2

   Let $\alpha =3+4+8+9+13+14+\dots$ upto 40 terms. If $(\tan \beta {)}^{\frac{\alpha }{1020}}$ is a root of the equation $x^2+x-2=0$, $\beta \in \left(0,\frac{\pi }{2}\right)$, then ${\sin }^2\beta +3{\cos }^2\beta$ is equal to :

   1. A.$2$
   2. B.$\frac{7}{4}$
   3. C.$\frac{5}{2}$
   4. D.$\frac{3}{2}$
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   Answer: A

2. Q122026 · 8 Apr 2026 · Shift 2

   Let $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right)$ and $\beta =3{\cos }^{-1}\left(\frac{4}{9}\right)$, where inverse trigonometric functions take only the principal values. Given below are two statements : Statement I : $\cos (\alpha +\beta )>0$. Statement II : $\cos (\alpha )<0$. In the light of the above statements, choose the correct answer from the options given below :

   1. A.Both Statement I and Statement II are true
   2. B.Both Statement I and Statement II are false
   3. C.Statement I is true but Statement II is false
   4. D.Statement I is false but Statement II is true
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   Answer: A

3. Q222026 · 8 Apr 2026 · Shift 2· Numerical answer

   If ${\int }_{\pi /6}^{\pi /4}\left(\cot \left(x-\frac{\pi }{3}\right)\cot \left(x+\frac{\pi }{3}\right)+1\right)dx=\alpha {\log }_e(\sqrt{3}-1)$, then $9{\alpha }^2$ is equal to ____ .

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   Answer: 12

4. Q132026 · 6 Apr 2026 · Shift 1

   Let $0<\alpha <1$, $\beta =\frac{1}{3\alpha }$ and ${\tan }^{-1}(1-\alpha )+{\tan }^{-1}(1-\beta )=\frac{\pi }{4}$. Then $6(\alpha +\beta )$ is equal to:

   1. A.6
   2. B.7
   3. C.8
   4. D.9
   Show answer

   Answer: B

5. Q142026 · 6 Apr 2026 · Shift 1

   Given below are two statements : Statement I : 2, 6-diethylcyclohexanone and 6-methyl-2-n-propylcyclohexanone are metamers. Statement II : 2, 2, 6, 6 - tetramethylcyclohexanone exhibits keto-enol tautomerism. In the light of the above statements, choose the correct answer from the options given below

   1. A.Both Statement I and Statement II are true
   2. B.Both Statement I and Statement II are false
   3. C.Statement I is true but Statement II is false
   4. D.Statement I is false but Statement II is true
   Show answer

   Answer: B

6. Q242026 · 6 Apr 2026 · Shift 1· Numerical answer

   For the functions $f(\theta )=\alpha {\tan }^2\theta +\beta {\cot }^2\theta$, and $g(\theta )=\alpha {\sin }^2\theta +\beta {\cos }^2\theta$, $\alpha >\beta >0$, let ${\min }_{0<\theta <\frac{\pi }{2}}f(\theta )={\max }_{0<\theta <\pi }g(\theta )$. If the first term of a G.P. is $\left(\frac{\alpha }{2\beta }\right)$, its common ratio is $\left(\frac{2\beta }{\alpha }\right)$ and the sum of its first 10 terms is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to ____.

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   Answer: 1279

7. Q132026 · 6 Apr 2026 · Shift 2

   If $\sin ({\tan }^{-1}(x\sqrt{2}))=\cot ({\sin }^{-1}\sqrt{1-x^2})$, $x\in (0,1)$, then the value of $x$ is :

   1. A.$\frac{1}{2}$
   2. B.$\frac{1}{3}$
   3. C.$\frac{2}{3}$
   4. D.$\frac{5}{8}$
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   Answer: A

8. Q62026 · 5 Apr 2026 · Shift 1

   Let $\tan \mathrm{A},\tan \mathrm{B}$, where $\mathrm{A},\mathrm{B}\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, be the roots of the quadratic equation $x^2-2x-5=0$. Then $20{\sin }^2\left(\frac{\mathrm{A}+\mathrm{B}}{2}\right)$ is equal to:

   1. A.$10+\sqrt{10}$
   2. B.$10-2\sqrt{10}$
   3. C.$10-3\sqrt{10}$
   4. D.$10-\sqrt{10}$
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   Answer: C

9. Q242026 · 5 Apr 2026 · Shift 1· Numerical answer

   If $\frac{\pi }{4}+{\sum }_{p=1}^{11}{\tan }^{-1}\left(\frac{2^{p-1}}{1+2^{2p-1}}\right)=\alpha$ , then $\tan \alpha$ is equal to ____.

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   Answer: 2048

10. Q232026 · 5 Apr 2026 · Shift 2· Numerical answer

    If $S=\left\{\theta \in [-\pi ,\pi ]:\cos \theta \cos \frac{5\theta }{2}=\cos 7\theta \cos \frac{7\theta }{2}\right\}$, then $n(S)$ is equal to ____.

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    Answer: 19

11. Q232026 · 4 Apr 2026 · Shift 1· Numerical answer

    If $A=\frac{\sin 3^{\circ }}{\cos 9^{\circ }}+\frac{\sin 9^{\circ }}{\cos 27^{\circ }}+\frac{\sin 27^{\circ }}{\cos 81^{\circ }}$ and $B=\tan 81^{\circ }-\tan 3^{\circ }$, then $\frac{B}{A}$ is equal to _________.

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    Answer: 2

12. Q242026 · 4 Apr 2026 · Shift 1· Numerical answer

    Let $\overrightarrow{a_k}=(\tan {\theta }_k)\hat{i}+\hat{j}$ and $\overrightarrow{b_k}=\hat{i}-(\cot {\theta }_k)\hat{j}$, where ${\theta }_k=\frac{2^{k-1}\pi }{2^n+1}$, for some $n\in \mathbb{N}$, $n>5$. Then the value of $\frac{\sum _{k=1}^n|\overrightarrow{a_k}{|}^2}{\sum _{k=1}^n|\overrightarrow{b_k}{|}^2}$ is _________.

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    Answer: 3

13. Q122026 · 4 Apr 2026 · Shift 2

    ${\max }_{0\le x\le \pi }\left[16\sin \left(\frac{x}{2}\right){\cos }^3\left(\frac{x}{2}\right)\right]$ is equal to:

    1. A.$\frac{3\sqrt{3}}{2}$
    2. B.$3\sqrt{3}$
    3. C.$4\sqrt{3}$
    4. D.$6\sqrt{3}$
    Show answer

    Answer: B

14. Q112026 · 2 Apr 2026 · Shift 1

    If $\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)=\mathrm{K}$, then the value of $\sin \left(\frac{10\mathrm{K}\pi }{3}\right)$ is :

    1. A.$\frac{\sqrt{3}+1}{2\sqrt{2}}$
    2. B.$\frac{\sqrt{3}-1}{\sqrt{2}}$
    3. C.$\frac{\sqrt{3}}{2}$
    4. D.$\frac{1}{2}$
    Show answer

    Answer: A

15. Q122026 · 2 Apr 2026 · Shift 1

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

    1. A.3
    2. B.6
    3. C.7
    4. D.9
    Show answer

    Answer: D

16. Q132026 · 2 Apr 2026 · Shift 2

    Let $\mathrm{P}=\{\theta \in [0,4\pi ]:{\tan }^2\theta \ne 1\}$ and $\mathrm{S}=\{a\in \mathbb{Z}:2({\cos }^8\theta -{\sin }^8\theta )\sec 2\theta =a^2,\theta \in \mathrm{P}\}$. Then $n(\mathrm{S})$ is :

    1. A.0
    2. B.1
    3. C.2
    4. D.3
    Show answer

    Answer: A

17. Q182026 · 28 Jan 2026 · Shift 1

    If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{{\sin }^2\mathrm{C}}{{\sin }^2\mathrm{A}}=1$, $\mathrm{A},\mathrm{B},\mathrm{C}\in \left(0,\frac{\pi }{2}\right)$, then

    1. A.$\tan \mathrm{A},\tan \mathrm{C},\tan \mathrm{B}$ are in A.P.
    2. B.$\tan \mathrm{A},\tan \mathrm{C},\tan \mathrm{B}$ are in G.P.
    3. C.$\tan A,\tan B,\tan C$ are in G.P.
    4. D.$\tan A,\tan B,\tan C$ are in A.P.
    Show answer

    Answer: B

18. Q232026 · 28 Jan 2026 · Shift 1· Numerical answer

    If $k=\tan \left(\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2}{\sin }^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation ${\sin }^{-1}(kx-1)={\sin }^{-1}x-{\cos }^{-1}x$ is ____ .

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    Answer: 1

19. Q42026 · 28 Jan 2026 · Shift 2

    Considering the principal values of inverse trigonometric functions, the value of the expression $\tan \left(2{\sin }^{-1}\left(\frac{2}{\sqrt{13}}\right)-2{\cos }^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is equal to :

    1. A.$\frac{33}{56}$
    2. B.$-\frac{33}{56}$
    3. C.$-\frac{16}{63}$
    4. D.$\frac{16}{63}$
    Show answer

    Answer: A

20. Q192026 · 28 Jan 2026 · Shift 2

    The sum of all the elements in the range of $f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x),x\ne \frac{n\pi }{2},n\in \mathbb{Z}$, where $\text{Sgn}(t)=\{\begin{array}{ll}1,& \text{if }t>0\\ -1,& \text{if }t<0\end{array}$ is :

    1. A.4
    2. B.0
    3. C.2
    4. D.-2
    Show answer

    Answer: C

21. Q62026 · 24 Jan 2026 · Shift 1

    The value of $\frac{\sqrt{3}\mathrm{cosec}20^{\circ }-\sec 20^{\circ }}{\cos 20^{\circ }\cos 40^{\circ }\cos 60^{\circ }\cos 80^{\circ }}$ is equal to

    1. A.32
    2. B.64
    3. C.12
    4. D.16
    Show answer

    Answer: B

22. Q82026 · 24 Jan 2026 · Shift 1

    If $\cot x=\frac{5}{12}$ for some $x\in \left(\pi ,\frac{3\pi }{2}\right)$, then $\sin 7x\left(\cos \frac{13x}{2}+\sin \frac{13x}{2}\right)+\cos 7x\left(\cos \frac{13x}{2}-\sin \frac{13x}{2}\right)$ is equal to

    1. A.$\frac{5}{\sqrt{13}}$
    2. B.$\frac{6}{\sqrt{26}}$
    3. C.$\frac{4}{\sqrt{26}}$
    4. D.$\frac{1}{\sqrt{13}}$
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    Answer: D

23. Q72026 · 23 Jan 2026 · Shift 1

    The vertices B and C of a triangle ABC lie on the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}$. The coordinates of A and $B$ are $(1,6,3)$ and $(4,9,\alpha )$ respectively and $C$ is at a distance of 10 units from $B$. The area (in sq. units) of $△ABC$ is :

    1. A.$20\sqrt{13}$
    2. B.$5\sqrt{13}$
    3. C.$15\sqrt{13}$
    4. D.$10\sqrt{13}$
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    Answer: B

24. Q92026 · 23 Jan 2026 · Shift 1

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

    1. A.5
    2. B.4
    3. C.3
    4. D.0
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    Answer: A

25. Q222026 · 23 Jan 2026 · Shift 1· Numerical answer

    Let the area of the region bounded by the curve $y=\max \{\sin x,\cos x\}$, lines $x=0$, $x=\frac{3\pi }{2}$, and the $x$-axis be $A$. Then, $A+A^2$ is equal to ____ .

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    Answer: 12