Trigonometry · JEE Main previous year questions

2023 30 questions

1. Q12023 · 13 Apr 2023 · Shift 1

   The area of the region enclosed by the curve $f(x)=\max \{\sin x,\cos x\}$, $-\pi \le x\le \pi$ and the $x$-axis is

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

2. Q142023 · 13 Apr 2023 · Shift 1

   $\underset{0\le x\le \pi }{\max }\left\{x-2\sin x\cos x+\frac{1}{3}\sin 3x\right\}=$

   1. A.$\frac{5\pi +2+3\sqrt{3}}{6}$
   2. B.$0$
   3. C.$\frac{\pi +2-3\sqrt{3}}{6}$
   4. D.$\pi$
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   Answer: A

3. Q242023 · 13 Apr 2023 · Shift 1· Numerical answer

   If $S=\left\{x\in \mathbb{R}:{\sin }^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right)-{\sin }^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi }{4}\right\}$, then $\sum _{x\in S}\left(\sin \left((x^2+x+5)\frac{\pi }{2}\right)-\cos \left((x^2+x+5)\pi \right)\right)$ is equal to ____________.

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   Answer: 4

4. Q232023 · 13 Apr 2023 · Shift 2· Numerical answer

   Let $f_n=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\sum _{k=1}^n{\sin }^{k-1}x\right)\left(\sum _{k=1}^n(2k-1){\sin }^{k-1}x\right)\cos xdx$, $n\in \mathbb{N}$. Then $f_{21}-f_{20}$ is equal to __________

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   Answer: 41

5. Q262023 · 13 Apr 2023 · Shift 2· Numerical answer

   For $x\in (-1,1]$, the number of solutions of the equation ${\sin }^{-1}x=2{\tan }^{-1}x$ is equal to ___________.

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   Answer: 2

6. Q72023 · 12 Apr 2023 · Shift 1

   If the point $\left(\alpha ,\frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x\cos \theta +y\sin \theta =7$, $\theta \in \left(0,\frac{\pi }{2}\right)$ between the co-ordinates axes, then $\alpha$ is equal to :

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

7. Q202023 · 12 Apr 2023 · Shift 1

   In a triangle $\mathrm{ABC}$, if $\cos A+2\cos B+\cos C=2$ and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then $\cos A-\cos C$ is equal to

   1. A.$\frac{3}{7}$
   2. B.$\frac{9}{7}$
   3. C.$\frac{10}{7}$
   4. D.$\frac{5}{7}$
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   Answer: C

8. Q202023 · 11 Apr 2023 · Shift 1

   The number of elements in the set $S=\{\theta \in [0,2\pi ]:3{\cos }^4\theta -5{\cos }^2\theta -2{\sin }^6\theta +2=0\}$ is :

   1. A.9
   2. B.8
   3. C.12
   4. D.10
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   Answer: A

9. Q202023 · 11 Apr 2023 · Shift 2

   The angle of elevation of the top $\mathrm{P}$ of a tower from the feet of one person standing due South of the tower is $45^{\circ }$ and from the feet of another person standing due west of the tower is $30^{\circ }$. If the height of the tower is $5\text{ meters}$, then the distance (in meters) between the two persons is equal to

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

10. Q52023 · 10 Apr 2023 · Shift 1

    If $f(x)=\frac{(\tan 1^{\circ })x+{\log }_e(123)}{x{\log }_e(1234)-(\tan 1^{\circ })}$, $x>0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is :

    1. A.2
    2. B.4
    3. C.0
    4. D.8
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    Answer: B

11. Q102023 · 10 Apr 2023 · Shift 1

    $96\cos \frac{\pi }{33}\cos \frac{2\pi }{33}\cos \frac{4\pi }{33}\cos \frac{8\pi }{33}\cos \frac{16\pi }{33}$ is equal to :

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

12. Q182023 · 10 Apr 2023 · Shift 2

    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 $\beta ={\sum }_{x\in S}{\tan }^2\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta -14{)}^2$ is equal to :

    1. A.16
    2. B.32
    3. C.8
    4. D.64
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    Answer: B

13. Q22023 · 8 Apr 2023 · Shift 1

    Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x},x\in [0,\pi ]-\left\{\frac{\pi }{4}\right\}$. Then $f\left(\frac{7\pi }{12}\right)f^{\prime \prime }\left(\frac{7\pi }{12}\right)$ is equal to

    1. A.$\frac{2}{3\sqrt{3}}$
    2. B.$\frac{2}{9}$
    3. C.$\frac{-1}{3\sqrt{3}}$
    4. D.$\frac{-2}{3}$
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    Answer: B

14. Q12023 · 8 Apr 2023 · Shift 2

    Let $A=\left\{\theta \in (0,2\pi ):\frac{1+2i\sin \theta }{1-i\sin \theta }\text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is :

    1. A.$3\pi$
    2. B.$\pi$
    3. C.$2\pi$
    4. D.$4\pi$
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    Answer: D

15. Q122023 · 8 Apr 2023 · Shift 2

    The value of $36\left(4{\cos }^2{9}^{\circ }-1\right)\left(4{\cos }^{2}27^{\circ }-1\right)\left(4{\cos }^{2}81^{\circ }-1\right)\left(4{\cos }^{2}243^{\circ }-1\right)$ is :

    1. A.$18$
    2. B.$36$
    3. C.$54$
    4. D.$27$
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    Answer: B

16. Q82023 · 6 Apr 2023 · Shift 1

    Let $I(x)=\int \frac{x^2(x{\sec }^{2}x+\tan x)}{(x\tan x+1{)}^2}dx$. If $I(0)=0$, then $I\left(\frac{\pi }{4}\right)$ is equal to :

    1. A.${\log }_e\frac{(\pi +4{)}^2}{32}-\frac{{\pi }^2}{4(\pi +4)}$
    2. B.${\log }_e\frac{(\pi +4{)}^2}{16}-\frac{{\pi }^2}{4(\pi +4)}$
    3. C.${\log }_e\frac{(\pi +4{)}^2}{16}+\frac{{\pi }^2}{4(\pi +4)}$
    4. D.${\log }_e\frac{(\pi +4{)}^2}{32}+\frac{{\pi }^2}{4(\pi +4)}$
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    Answer: A

17. Q242023 · 6 Apr 2023 · Shift 2· Numerical answer

    The value of $\tan 9^{\circ }-\tan 27^{\circ }-\tan 63^{\circ }+\tan 81^{\circ }$ is ________.

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    Answer: 4

18. Q22023 · 1 Feb 2023 · Shift 1

    Let $S$ be the set of all solutions of the equation ${\cos }^{-1}(2x)-2{\cos }^{-1}\left(\sqrt{1-x^2}\right)=\pi ,x\in \left[-\frac{1}{2},\frac{1}{2}\right]$. Then ${\sum }_{x\in S}2{\sin }^{-1}\left(x^2-1\right)$ is equal to :

    1. A.$\pi -2{\sin }^{-1}\left(\frac{\sqrt{3}}{4}\right)$
    2. B.$\pi -{\sin }^{-1}\left(\frac{\sqrt{3}}{4}\right)$
    3. C.$-\frac{2\pi }{3}$
    4. D.$0$
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    Answer: D

19. Q32023 · 31 Jan 2023 · Shift 1

    If ${\sin }^{-1}\frac{\alpha }{17}+{\cos }^{-1}\frac{4}{5}-{\tan }^{-1}\frac{77}{36}=0,0<\alpha <13$, then ${\sin }^{-1}(\sin \alpha )+{\cos }^{-1}(\cos \alpha )$ is equal to :

    1. A.$16$
    2. B.$\pi$
    3. C.$16-5\pi$
    4. D.$0$
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    Answer: B

20. Q152023 · 31 Jan 2023 · Shift 1

    Let $y=f(x)={\sin }^3\left(\frac{\pi }{3}\left(\cos \left(\frac{\pi }{3\sqrt{2}}{\left(-4x^3+5x^2+1\right)}^{\frac{3}{2}}\right)\right)\right)$. Then, at $x=1$,

    1. A.$2y^{\prime }+\sqrt{3}{\pi }^{2}y=0$
    2. B.$y^{\prime }+3{\pi }^{2}y=0$
    3. C.$\sqrt{2}{y}^{\prime }-3{\pi }^{2}y=0$
    4. D.$2y^{\prime }+3{\pi }^{2}y=0$
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    Answer: D

21. Q42023 · 31 Jan 2023 · Shift 2

    Let $(a,b)\subset (0,2\pi )$ be the largest interval for which ${\sin }^{-1}(\sin \theta )-{\cos }^{-1}(\sin \theta )>0,\theta \in (0,2\pi )$, holds. If $\alpha x^2+\beta x+{\sin }^{-1}\left(x^2-6x+10\right)+{\cos }^{-1}\left(x^2-6x+10\right)=0$ and $\alpha -\beta =b-a$, then $\alpha$ is equal to :

    1. A.$\frac{\pi }{16}$
    2. B.$\frac{\pi }{48}$
    3. C.$\frac{\pi }{8}$
    4. D.$\frac{\pi }{12}$
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    Answer: D

22. Q72023 · 30 Jan 2023 · Shift 1

    If the solution of the equation ${\log }_{\cos x}\cot x+4{\log }_{\sin x}\tan x=1,x\in \left(0,\frac{\pi }{2}\right)$, is ${\sin }^{-1}\left(\frac{\alpha +\sqrt{\beta }}{2}\right)$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to :

    1. A.3
    2. B.6
    3. C.4
    4. D.5
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    Answer: C

23. Q82023 · 30 Jan 2023 · Shift 1

    If $\tan 15^{\circ }+\frac{1}{\tan 75^{\circ }}+\frac{1}{\tan 105^{\circ }}+\tan 195^{\circ }=2a$, then the value of $\left(a+\frac{1}{a}\right)$ is :

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

24. Q92023 · 29 Jan 2023 · Shift 2

    The area of the region $A=\left\{(x,y):|\cos x-\sin x|\le y\le \sin x,0\le x\le \frac{\pi }{2}\right\}$ is

    1. A.$\sqrt{5}+2\sqrt{2}-4.5$
    2. B.$1-\frac{3}{\sqrt{2}}+\frac{4}{\sqrt{5}}$
    3. C.$\sqrt{5}-2\sqrt{2}+1$
    4. D.$\frac{3}{\sqrt{5}}-\frac{3}{\sqrt{2}}+1$
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    Answer: C

25. Q232023 · 29 Jan 2023 · Shift 2· Numerical answer

    Let $\alpha =8-14i,A=\left\{z\in \mathbb{C}:\frac{\alpha z-\bar{\alpha }\bar{z}}{z^2-(\bar{z}{)}^2-112i}=1\right\}$ and $B=\{z\in \mathbb{C}:|z+3i|=4\}$. Then $\sum _{z\in A\cap B}(\mathrm{Re}z-\mathrm{Im}z)$ is equal to ____.

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    Answer: 14