Trigonometry · JEE Main previous year questions

2026 34 questions

1. Q122026 · 23 Jan 2026 · Shift 2

   The least value of $\left({\cos }^2\theta -6\sin \theta \cos \theta +3{\sin }^2\theta +2\right)$ is

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

2. Q132026 · 23 Jan 2026 · Shift 2

   Let $\frac{\pi }{2}<\theta <\pi$ and $\cot \theta =-\frac{1}{2\sqrt{2}}$. Then the value of $\sin \left(\frac{15\theta }{2}\right)(\cos 8\theta +\sin 8\theta )+\cos \left(\frac{15\theta }{2}\right)(\cos 8\theta -\sin 8\theta )$ is equal to :

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

3. Q22026 · 22 Jan 2026 · Shift 1

   The number of solutions of ${\tan }^{-1}4x+{\tan }^{-1}6x=\frac{\pi }{6}$, where $-\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}$, is equal to :

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

4. Q212026 · 22 Jan 2026 · Shift 1· Numerical answer

   If $\frac{{\cos }^{2}48^{\circ }-{\sin }^{2}12^{\circ }}{{\sin }^{2}24^{\circ }-{\sin }^2{6}^{\circ }}=\frac{\alpha +\beta \sqrt{5}}{2}$, where $\alpha ,\beta \in \mathbb{N}$, then $\alpha +\beta$ is equal to ____

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   Answer: 4

5. Q232026 · 22 Jan 2026 · Shift 1· Numerical answer

   Let ABC be a triangle. Consider four points ${\mathrm{p}}_1,{\mathrm{p}}_2,{\mathrm{p}}_3,{\mathrm{p}}_4$ on the side AB , five points $p_5,p_6,p_7,p_8,p_9$ on the side $BC$, and four points $p_{10},p_{11},p_{12},p_{13}$ on the side AC . None of these points is a vertex of the triangle ABC . Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1,p_2,\dots ,p_{13}$, is ____

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   Answer: 660

6. Q32026 · 21 Jan 2026 · Shift 1

   Let a point A lie between the parallel lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ such that its distances from ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, respectively, is :

   1. A.$21\sqrt{3}$
   2. B.$12\sqrt{2}$
   3. C.$15\sqrt{6}$
   4. D.27
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   Answer: A

7. Q42026 · 21 Jan 2026 · Shift 1

   The value of $\mathrm{cosec}10^{\circ }-\sqrt{3}\sec 10^{\circ }$ is equal to :

   1. A.2
   2. B.6
   3. C.8
   4. D.4
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   Answer: D

8. Q182026 · 21 Jan 2026 · Shift 2

   For a triangle $ABC$, let $\vec{p}=\overrightarrow{BC},\vec{q}=\overrightarrow{CA}$ and $\vec{r}=\overrightarrow{BA}$. If $|\vec{p}|=2\sqrt{3},|\vec{q}|=2$ and $\cos \theta =\frac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$, then $|\vec{p}\times (\vec{q}-3\vec{r}){|}^2+3|\vec{r}{|}^2$ is equal to :

   1. A.200
   2. B.220
   3. C.410
   4. D.340
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   Answer: A

9. Q252026 · 21 Jan 2026 · Shift 2· Numerical answer

   Let the maximum value of ${\left({\sin }^{-1}x\right)}^2+{\left({\cos }^{-1}x\right)}^2$ for $x\in \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$ be $\frac{\mathrm{m}}{\mathrm{n}}{\pi }^2$, where $\gcd (\mathrm{m},\mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to ____.

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   Answer: 65