Trigonometry · JEE Main previous year questions

2022 34 questions

1. Q192022 · 29 Jul 2022 · Shift 1

   The angle of elevation of the top of a tower from a point A due north of it is $\alpha$ and from a point B at a distance of 9 units due west of A is ${\cos }^{-1}\left(\frac{3}{\sqrt{13}}\right)$. If the distance of the point B from the tower is 15 units, then $\cot \alpha$ is equal to :

   1. A.$\frac{6}{5}$
   2. B.$\frac{9}{5}$
   3. C.$\frac{4}{3}$
   4. D.$\frac{7}{3}$
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   Answer: A

2. Q282022 · 29 Jul 2022 · Shift 1· Numerical answer

   Let $S=\left\{\theta \in (0,2\pi ):7{\cos }^2\theta -3{\sin }^2\theta -2{\cos }^{2}2\theta =2\right\}$. Then, the sum of roots of all the equations $x^2-2\left({\tan }^2\theta +{\cot }^2\theta \right)x+6{\sin }^2\theta =0,\theta \in S$, is ____.

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   Answer: 16

3. Q92022 · 29 Jul 2022 · Shift 2

   Let $m_1,m_2$ be the slopes of two adjacent sides of a square of side a such that $a^2+11a+3\left(m_1^2+m_2^2\right)=220$. If one vertex of the square is $(10(\cos \alpha -\sin \alpha ),10(\sin \alpha +\cos \alpha ))$, where $\alpha \in \left(0,\frac{\pi }{2}\right)$ and the equation of one diagonal is $(\cos \alpha -\sin \alpha )x+(\sin \alpha +\cos \alpha )y=10$, then $72\left({\sin }^4\alpha +{\cos }^4\alpha \right)+a^2-3a+13$ is equal to :

   1. A.119
   2. B.128
   3. C.145
   4. D.155
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   Answer: B

4. Q42022 · 28 Jul 2022 · Shift 1

   Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation ${\cos }^{-1}(x)-2{\sin }^{-1}(x)={\cos }^{-1}(2x)$ is equal to :

   1. A.0
   2. B.1
   3. C.$\frac{1}{2}$
   4. D.$-\frac{1}{2}$
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   Answer: A

5. Q72022 · 28 Jul 2022 · Shift 2

   The sum of the absolute maximum and absolute minimum values of the function $f(x)={\tan }^{-1}(\sin x-\cos x)$ in the interval $[0,\pi ]$ is :

   1. A.0
   2. B.${\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)-\frac{\pi }{4}$
   3. C.${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi }{4}$
   4. D.$\frac{-\pi }{12}$
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   Answer: C

6. Q82022 · 28 Jul 2022 · Shift 2

   Let $x(t)=2\sqrt{2}\cos t\sqrt{\sin 2t}$ and $$y(t)=2\sqrt{2}\sin t\sqrt{\sin 2t},t\in \left(0,\frac{\pi }{2}\right).$$ Then $\frac{1+{\left(\frac{dy}{dx}\right)}^2}{\frac{d^{2}y}{d{x}^2}}$ at $t=\frac{\pi }{4}$ is equal to :

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

7. Q182022 · 27 Jul 2022 · Shift 1

   Let a vertical tower $AB$ of height $2h$ stand on a horizontal ground. Let from a point $P$ on the ground a man can see upto height $h$ of the tower with an angle of elevation $2\alpha$. When from $P$, he moves a distance $d$ in the direction of $\overrightarrow{AP}$, he can see the top $B$ of the tower with an angle of elevation $\alpha$. If $d=\sqrt{7}h$, then $\tan \alpha$ is equal to

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

8. Q212022 · 27 Jul 2022 · Shift 1· Numerical answer

   For $k\in \mathbb{R}$, let the solutions of the equation $\cos \left({\sin }^{-1}\left(x\cot \left({\tan }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)\right)\right)\right)=k,0<|x|<\frac{1}{\sqrt{2}}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x^2-bx-5=0$ are $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ and $\frac{\alpha }{\beta }$, then $\frac{b}{k^2}$ is equal to _________.

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   Answer: 12

9. Q182022 · 27 Jul 2022 · Shift 2

   The angle of elevation of the top $\mathrm{P}$ of a vertical tower $\mathrm{PQ}$ of height $10$ from a point $\mathrm{A}$ on the horizontal ground is $45^{\circ }$. Let $\mathrm{R}$ be a point on $\mathrm{AQ}$ and from a point $\mathrm{B}$, vertically above $\mathrm{R}$, the angle of elevation of $\mathrm{P}$ is $60^{\circ }$. If $\angle \mathrm{BAQ}=30^{\circ }$, $\mathrm{AB}=\mathrm{d}$ and the area of the trapezium $\mathrm{PQRB}$ is $\alpha$, then the ordered pair $(d,\alpha )$ is :

   1. A.$\left(10(\sqrt{3}-1),25\right)$
   2. B.$\left(10(\sqrt{3}-1),\frac{25}{2}\right)$
   3. C.$\left(10(\sqrt{3}+1),25\right)$
   4. D.$\left(10(\sqrt{3}+1),\frac{25}{2}\right)$
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   Answer: A

10. Q192022 · 27 Jul 2022 · Shift 2

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

    1. A.$S=\left\{\frac{\pi }{12}\right\}$
    2. B.$S=\left\{\frac{2\pi }{3}\right\}$
    3. C.${\sum }_{\theta \in S}\theta =\frac{\pi }{2}$
    4. D.${\sum }_{\theta \in S}\theta =\frac{3\pi }{4}$
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    Answer: C

11. Q142022 · 26 Jul 2022 · Shift 1

    $\tan \left(2{\tan }^{-1}\frac{1}{5}+{\sec }^{-1}\frac{\sqrt{5}}{2}+2{\tan }^{-1}\frac{1}{8}\right)$ is equal to :

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

12. Q162022 · 26 Jul 2022 · Shift 1

    If $a={\lim }_{n\to \infty }{\sum }_{k=1}^n\frac{2n}{n^2+k^2}$ and $f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}$, $x\in (0,1)$, then :

    1. A.$2\sqrt{2}f\left(\frac{a}{2}\right)=f^{\prime }\left(\frac{a}{2}\right)$
    2. B.$f\left(\frac{a}{2}\right)f^{\prime }\left(\frac{a}{2}\right)=\sqrt{2}$
    3. C.$\sqrt{2}f\left(\frac{a}{2}\right)=f^{\prime }\left(\frac{a}{2}\right)$
    4. D.$f\left(\frac{a}{2}\right)=\sqrt{2}{f}^{\prime }\left(\frac{a}{2}\right)$
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    Answer: C

13. Q192022 · 26 Jul 2022 · Shift 1

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

    1. A.0
    2. B.$-2$
    3. C.$-4$
    4. D.12
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    Answer: C

14. Q142022 · 25 Jul 2022 · Shift 2

    $2\sin \left(\frac{\pi }{22}\right)\sin \left(\frac{3\pi }{22}\right)\sin \left(\frac{5\pi }{22}\right)\sin \left(\frac{7\pi }{22}\right)\sin \left(\frac{9\pi }{22}\right)$ is equal to :

    1. A.$\frac{3}{16}$
    2. B.$\frac{1}{16}$
    3. C.$\frac{1}{32}$
    4. D.$\frac{9}{32}$
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    Answer: B

15. Q292022 · 25 Jul 2022 · Shift 2· Numerical answer

    Let $x=\sin (2{\tan }^{-1}\alpha )$ and $y=\sin \left(\frac{1}{2}{\tan }^{-1}\frac{4}{3}\right)$. If $S=\{\alpha \in \mathbb{R}:y^2=1-x\}$, then ${\sum }_{\alpha \in S}16{\alpha }^3$ is equal to ___________.

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    Answer: 130

16. Q112022 · 30 Jun 2022 · Shift 1

    Let $\alpha =\tan \left(\frac{5\pi }{16}\sin \left(2{\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\right)$ and $\beta =\cos \left({\sin }^{-1}\left(\frac{4}{5}\right)+{\sec }^{-1}\left(\frac{5}{3}\right)\right)$ where the inverse trigonometric functions take principal values. Then, the equation whose roots are $\alpha$ and $\beta$ is :

    1. A.$15x^2-8x-7=0$
    2. B.$5x^2-12x+7=0$
    3. C.$25x^2-18x-7=0$
    4. D.$25x^2-32x+7=0$
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    Answer: C

17. Q302022 · 30 Jun 2022 · Shift 1· Numerical answer

    Let $S_1=\{x\in [0,12\pi ]:{\sin }^{5}x+{\cos }^{5}x=1\}$ and $S_2=\{x\in [0,8\pi ]:{\sin }^{7}x+{\cos }^{7}x=1\}$ Then $n(S_1)-n(S_2)$ is equal to ____ .

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    Answer: 4

18. Q112022 · 28 Jun 2022 · Shift 2

    Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\tan x(\cos x-y)$. If the curve passes through the point $\left(\frac{\pi }{4},0\right)$, then the value of ${\int }_0^{\pi /2}ydx$ is equal to :

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

19. Q192022 · 28 Jun 2022 · Shift 2

    If $\cot \alpha =1$ and $\sec \beta =-\frac{5}{3}$, where $\pi <\alpha <\frac{3\pi }{2}$ and $\frac{\pi }{2}<\beta <\pi$, then the value of $\tan (\alpha +\beta )$ and the quadrant in which $\alpha +\beta$ lies, respectively are :

    1. A.$-\frac{1}{7}$ and ${\text{IV}}^{\text{th}}$ quadrant
    2. B.$7$ and ${\text{I}}^{\text{st}}$ quadrant
    3. C.$-7$ and ${\text{IV}}^{\text{th}}$ quadrant
    4. D.$\frac{1}{7}$ and ${\text{I}}^{\text{st}}$ quadrant
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    Answer: A

20. Q62022 · 27 Jun 2022 · Shift 1

    If ${\cos }^{-1}\left(\frac{y}{2}\right)={\log }_e{\left(\frac{x}{5}\right)}^5$, $|y|<2$, then :

    1. A.$x^2{y}^{\prime \prime }+x{y}^{\prime }-25y=0$
    2. B.$x^2{y}^{\prime \prime }-x{y}^{\prime }-25y=0$
    3. C.$x^2{y}^{\prime \prime }-x{y}^{\prime }+25y=0$
    4. D.$x^2{y}^{\prime \prime }+x{y}^{\prime }+25y=0$
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    Answer: D

21. Q152022 · 27 Jun 2022 · Shift 1

    The value of $\cos \left(\frac{2\pi }{7}\right)+\cos \left(\frac{4\pi }{7}\right)+\cos \left(\frac{6\pi }{7}\right)$ is equal to :

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

22. Q162022 · 27 Jun 2022 · Shift 1

    ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)+{\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)+{\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$ is equal to :

    1. A.$\frac{11\pi }{12}$
    2. B.$\frac{17\pi }{12}$
    3. C.$\frac{31\pi }{12}$
    4. D.$-\frac{3\pi }{4}$
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    Answer: A

23. Q132022 · 27 Jun 2022 · Shift 2

    The value of $\cot \left({\sum }_{n=1}^{50}{\tan }^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is :

    1. A.$\frac{26}{25}$
    2. B.$\frac{25}{26}$
    3. C.$\frac{50}{51}$
    4. D.$\frac{52}{51}$
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    Answer: A

24. Q142022 · 27 Jun 2022 · Shift 2

    $\alpha =\sin 36^{\circ }$ is a root of which of the following equation?

    1. A.$16x^4-10x^2-5=0$
    2. B.$16x^4+20x^2-5=0$
    3. C.$16x^4-20x^2+5=0$
    4. D.$4x^4-10x^2+5=0$
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    Answer: C

25. Q162022 · 27 Jun 2022 · Shift 2

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

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