Trigonometry · JEE Main previous year questions

2021 23 questions

1. Q22021 · 1 Sept 2021 · Shift 2

   ${\cos }^{-1}(\cos (-5))+{\sin }^{-1}(\sin (6))-{\tan }^{-1}(\tan (12))$ is equal to : (The inverse trigonometric functions take the principal values)

   1. A.$3\pi -11$
   2. B.$4\pi -9$
   3. C.$4\pi -11$
   4. D.$3\pi +1$
   Show answer

   Answer: C

2. Q82021 · 1 Sept 2021 · Shift 2

   The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0$, $x=\frac{\pi }{2}$, is :

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

   Answer: A

3. Q172021 · 1 Sept 2021 · Shift 2

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

   1. A.$(3,13\pi /9)$
   2. B.$(2,2\pi /3)$
   3. C.$(2,8\pi /9)$
   4. D.$(3,5\pi /3)$
   Show answer

   Answer: A

4. Q82021 · 26 Aug 2021 · Shift 2

   If $\sum _{r=1}^{50}{\tan }^{-1}\frac{1}{2r^2}=p$, then the value of $\tan p$ is :

   1. A.$\frac{101}{102}$
   2. B.$\frac{50}{51}$
   3. C.100
   4. D.$\frac{51}{50}$
   Show answer

   Answer: B

5. Q112021 · 26 Aug 2021 · Shift 2

   The value of $$2\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{2\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\sin \left(\frac{5\pi }{8}\right)\sin \left(\frac{6\pi }{8}\right)\sin \left(\frac{7\pi }{8}\right)\text{ is :}$$

   1. A.$\frac{1}{4\sqrt{2}}$
   2. B.$\frac{1}{4}$
   3. C.$\frac{1}{8}$
   4. D.$\frac{1}{8\sqrt{2}}$
   Show answer

   Answer: C

6. Q32021 · 27 Jul 2021 · Shift 2

   Let f : R $\to$ R be defined as $f(x+y)+f(x-y)=2f(x)f(y),f\left(\frac{1}{2}\right)=-1$. Then, the value of $\sum _{k=1}^{20}\frac{1}{\sin (k)\sin (k+f(k))}$ is equal to :

   1. A.${\mathrm{cosec}}^2(21)\cos (20)\cos (2)$
   2. B.${\sec }^2(1)\sec (21)\cos (20)$
   3. C.${\mathrm{cosec}}^2(1)\mathrm{cosec}(21)\sin (20)$
   4. D.${\sec }^2(21)\sin (20)\sin (2)$
   Show answer

   Answer: C

7. Q182021 · 25 Jul 2021 · Shift 1

   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 :

   1. A.$8\pi$
   2. B.$11\pi$
   3. C.$12\pi$
   4. D.$9\pi$
   Show answer

   Answer: D

8. Q52021 · 25 Jul 2021 · Shift 2

   The value of $\cot \frac{\pi }{24}$ is :

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

   Answer: B

9. Q102021 · 25 Jul 2021 · Shift 2

   The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is :

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

   Answer: B

10. Q192021 · 22 Jul 2021 · Shift 2

    The number of solutions of ${\sin }^{7}x+{\cos }^{7}x=1$, $x\in [0,4\pi ]$ is equal to

    1. A.$11$
    2. B.$7$
    3. C.$5$
    4. D.$9$
    Show answer

    Answer: C

11. Q102021 · 20 Jul 2021 · Shift 1

    The number of real roots of the equation ${\tan }^{-1}\sqrt{x(x+1)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{4}$ is :

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

    Answer: D

12. Q22021 · 18 Mar 2021 · Shift 1

    The solutions of the equation $\left|\begin{array}{ccc}1+{\sin }^{2}x& {\sin }^{2}x& {\sin }^{2}x\\ {\cos }^{2}x& 1+{\cos }^{2}x& {\cos }^{2}x\\ 4\sin 2x& 4\sin 2x& 1+4\sin 2x\end{array}\right|=0$, $(0<x<\pi )$, are

    1. A.$\frac{\pi }{12},\frac{\pi }{6}$
    2. B.$\frac{\pi }{6},\frac{5\pi }{6}$
    3. C.$\frac{5\pi }{12},\frac{7\pi }{12}$
    4. D.$\frac{7\pi }{12},\frac{11\pi }{12}$
    Show answer

    Answer: D

13. Q302021 · 18 Mar 2021 · Shift 1· Numerical answer

    The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2\pi ]$ is

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    Answer: 1

14. Q42021 · 18 Mar 2021 · Shift 2

    If $15{\sin }^4\alpha +10{\cos }^4\alpha =6$, for some $\alpha \in \mathrm{R}$, then the value of $27{\sec }^6\alpha +8{\mathrm{cosec}}^6\alpha$ is equal to :

    1. A.500
    2. B.400
    3. C.250
    4. D.350
    Show answer

    Answer: C

15. Q122021 · 16 Mar 2021 · Shift 1

    If for $x\in \left(0,\frac{\pi }{2}\right)$, ${\log }_{10}\sin x+{\log }_{10}\cos x=-1$ and ${\log }_{10}(\sin x+\cos x)=\frac{1}{2}({\log }_{10}n-1)$, $n>0$, then the value of n is equal to :

    1. A.16
    2. B.9
    3. C.12
    4. D.20
    Show answer

    Answer: C

16. Q182021 · 16 Mar 2021 · Shift 1

    The number of roots of the equation, $(81{)}^{{\sin }^{2}x}+(81{)}^{{\cos }^{2}x}=30$ in the interval $[0,\pi ]$ is equal to :

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

    Answer: C

17. Q112021 · 26 Feb 2021 · Shift 1

    If $\frac{{\sin }^{-1}x}{a}=\frac{{\cos }^{-1}x}{b}=\frac{{\tan }^{-1}y}{c}$; $0<x<1$, then the value of $\cos \left(\frac{\pi c}{a+b}\right)$ is :

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

    Answer: D

18. Q212021 · 26 Feb 2021 · Shift 1· Numerical answer

    The value of the integral ${\int }_0^{\pi }|\sin 2x|dx$ is _________.

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    Answer: 2

19. Q292021 · 26 Feb 2021 · Shift 1· Numerical answer

    If $\sqrt{3}({\cos }^{2}x)=(\sqrt{3}-1)\cos x+1$, the number of solutions of the given equation when $x\in \left[0,\frac{\pi }{2}\right]$ is _________.

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    Answer: 1

20. Q52021 · 26 Feb 2021 · Shift 2

    If $0<a,b<1$, and ${\tan }^{-1}a+{\tan }^{-1}b=\frac{\pi }{4}$, then the value of $(a+b)-\left(\frac{a^2+b^2}{2}\right)+\left(\frac{a^3+b^3}{3}\right)-\left(\frac{a^4+b^4}{4}\right)+\dots \dots$ is :

    1. A.${\log }_{e}2$
    2. B.e
    3. C.${\log }_e\left(\frac{e}{2}\right)$
    4. D.$e^2-1$
    Show answer

    Answer: A

21. Q92021 · 26 Feb 2021 · Shift 2

    Let $A_1$ be the area of the region bounded by the curves $y=\sin x$, $y=\cos x$ and y-axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by the curves $y=\sin x$, $y=\cos x$, x-axis and $x=\frac{\pi }{2}$ in the first quadrant. Then,

    1. A.$A_1:A_2=1:\sqrt{2}$ and $A_1+A_2=1$
    2. B.$A_1=A_2$ and $A_1+A_2=\sqrt{2}$
    3. C.$2A_1=A_2$ and $A_1+A_2=1+\sqrt{2}$
    4. D.$A_1:A_2=1:2$ and $A_1+A_2=1$
    Show answer

    Answer: A

22. Q72021 · 25 Feb 2021 · Shift 2

    If $0<x,y<\pi$ and $\cos x+\cos y-\cos (x+y)=\frac{3}{2}$, then $\sin x+\cos y$ is equal to :

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

    Answer: A

23. Q82021 · 25 Feb 2021 · Shift 2

    $\mathrm{cosec}\left[2{\cot }^{-1}(5)+{\cos }^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :

    1. A.$\frac{75}{56}$
    2. B.$\frac{65}{56}$
    3. C.$\frac{56}{33}$
    4. D.$\frac{65}{33}$
    Show answer

    Answer: B