Trigonometry · JEE Main previous year questions

2020 23 questions

1. Q222020 · 6 Sept 2020 · Shift 1· Numerical answer

   Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $$f(x)=\{\begin{array}{ll}{x}^5\sin \left(\frac{1}{x}\right)+5x^2,& x<0\\ 0,& x=0\\ x^5\cos \left(\frac{1}{x}\right)+\lambda x^2,& x>0\end{array}$$ The value of $\lambda$ for which $f^{\prime \prime }(0)$ exists, is ______ .

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   Answer: 5

2. Q32020 · 6 Sept 2020 · Shift 2

   Let $\theta =\frac{\pi }{5}$ and $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$. If $B=A+A^4$, then $\det (B)$ :

   1. A.lies in $(1,2)$
   2. B.lies in $(2,3)$
   3. C.is zero.
   4. D.is one.
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   Answer: A

3. Q102020 · 3 Sept 2020 · Shift 1

   If $y^2+{\log }_e({\cos }^{2}x)=y$, $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, then :

   1. A.$|y^{\prime \prime }(0)|=2$
   2. B.$|y^{\prime }(0)|+|y^{\prime \prime }(0)|=3$
   3. C.$y^{\prime \prime }(0)=0$
   4. D.$|y^{\prime }(0)|+|y^{\prime \prime }(0)|=1$
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   Answer: A

4. Q182020 · 3 Sept 2020 · Shift 1

   $2\pi -\left({\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\frac{16}{65}\right)$ is equal to :

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

5. Q132020 · 2 Sept 2020 · Shift 1

   The value of $${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$$ is :

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

6. Q172020 · 2 Sept 2020 · Shift 1

   If the tangent to the curve $y=x+\sin y$ at a point (a, b) is parallel to the line joining $\left(0,\frac{3}{2}\right)$ and $\left(\frac{1}{2},2\right)$, then :

   1. A.b = a
   2. B.|b - a| = 1
   3. C.$b=\frac{\pi }{2}+a$
   4. D.|a + b| = 1
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   Answer: B

7. Q32020 · 2 Sept 2020 · Shift 2

   If the equation ${\cos }^4\theta +{\sin }^4\theta +\lambda =0$ has real solutions for $\theta$, then $\lambda$ lies in the interval :

   1. A.$\left[-\frac{3}{2},-\frac{5}{4}\right]$
   2. B.$\left(-\frac{1}{2},-\frac{1}{4}\right]$
   3. C.$\left(-\frac{5}{4},-1\right]$
   4. D.$\left[-1,-\frac{1}{2}\right]$
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   Answer: D

8. Q152020 · 2 Sept 2020 · Shift 2

   The set of all possible values of $\theta$ in the interval $(0,\pi )$ for which the points $(1,2)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y=1$ is :

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

9. Q172020 · 2 Sept 2020 · Shift 2

   The equation of the normal to the curve $y=(1+x{)}^{2y}+{\cos }^2({\sin }^{-1}x)$ at $x=0$ is :

   1. A.$y=4x+2$
   2. B.$x+4y=8$
   3. C.$y+4x=2$
   4. D.$2y+x=4$
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   Answer: B

10. Q222020 · 2 Sept 2020 · Shift 2· Numerical answer

    If $y={\sum }_{k=1}^{6}k{\cos }^{-1}\left\{\frac{3}{5}\cos kx-\frac{4}{5}\sin kx\right\}$, then $\frac{dy}{dx}$ at $x=0$ is _______.

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    Answer: 91

11. Q62020 · 9 Jan 2020 · Shift 1

    If $f^{\prime }(x)={\tan }^{-1}(\sec x+\tan x)$, $-\frac{\pi }{2}<x<\frac{\pi }{2}$, and $f(0)=0$, then $f(1)$ is equal to :

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

12. Q152020 · 9 Jan 2020 · Shift 1

    The value of $${\cos }^3\left(\frac{\pi }{8}\right)\cos \left(\frac{3\pi }{8}\right)+{\sin }^3\left(\frac{\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)$$ is :

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

13. Q222020 · 9 Jan 2020 · Shift 1· Numerical answer

    The number of distinct solutions of the equation $${\log }_{\frac{1}{2}}|\sin x|=2-{\log }_{\frac{1}{2}}|\cos x|$$ in the interval $[0,2\pi ]$, is ______.

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    Answer: 8

14. Q32020 · 9 Jan 2020 · Shift 2

    If $x=2\sin \theta -\sin 2\theta$ and $y=2\cos \theta -\cos 2\theta$, $\theta \in [0,2\pi ]$, then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\pi$ is :

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

15. Q72020 · 9 Jan 2020 · Shift 2

    If $x={\sum }_{n=0}^{\infty }(-1{)}^n{\tan }^{2n}\theta$ and $y={\sum }_{n=0}^{\infty }{\cos }^{2n}\theta$ for $0<\theta <\frac{\pi }{4}$, then :

    1. A.$x(1+y)=1$
    2. B.$y(1-x)=1$
    3. C.$y(1+x)=1$
    4. D.$x(1-y)=1$
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    Answer: B

16. Q32020 · 8 Jan 2020 · Shift 1

    Let $f(x)=(\sin ({\tan }^{-1}x)+\sin ({\cot }^{-1}x){)}^2-1$, $|x|>1$. If $\frac{dy}{dx}=\frac{1}{2}\frac{d}{dx}\left({\sin }^{-1}(f(x))\right)$ and $y\left(\sqrt{3}\right)=\frac{\pi }{6}$, then $y(-\sqrt{3})$ is equal to :

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

17. Q162020 · 8 Jan 2020 · Shift 1

    Let $f(x)=x{\cos }^{-1}(-\sin |x|)$, $x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$, then which of the following is true?

    1. A.f' is decreasing in $\left(-\frac{\pi }{2},0\right)$ and increasing in $\left(0,\frac{\pi }{2}\right)$
    2. B.$f^{\prime }(0)=-\frac{\pi }{2}$
    3. C.f is not differentiable at x = 0
    4. D.f' is increasing in $\left(-\frac{\pi }{2},0\right)$ and decreasing in $\left(0,\frac{\pi }{2}\right)$
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    Answer: A

18. Q182020 · 8 Jan 2020 · Shift 1

    Let the volume of a parallelopiped whose coterminous edges are given by $\vec{u}=\hat{i}+\hat{j}+\lambda \hat{k}$, $\vec{v}=\hat{i}+\hat{j}+3\hat{k}$ and $\vec{w}=2\hat{i}+\hat{j}+\hat{k}$ be 1 cu. unit. If $\theta$ be the angle between the edges $\vec{u}$ and $\vec{w}$, then $\cos \theta$ can be :

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

19. Q212020 · 8 Jan 2020 · Shift 2· Numerical answer

    If $\frac{\sqrt{2}\sin \alpha }{\sqrt{1+\cos 2\alpha }}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2\beta }{2}}=\frac{1}{\sqrt{10}}$, $\alpha ,\beta \in \left(0,\frac{\pi }{2}\right)$ then $\tan (\alpha +2\beta )$ is equal to _______.

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    Answer: 1

20. Q142020 · 7 Jan 2020 · Shift 1

    If $y(\alpha )=\sqrt{2\left(\frac{\tan \alpha +\cot \alpha }{1+{\tan }^2\alpha }\right)+\frac{1}{{\sin }^2\alpha }}$, $\alpha \in \left(\frac{3\pi }{4},\pi \right)$ $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is :

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

21. Q222020 · 7 Jan 2020 · Shift 1· Numerical answer

    Let A(1, 0), B(6, 2) and $C\left(\frac{3}{2},6\right)$ be the vertices of a triangle ABC. If P is a Point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point $\left(-\frac{7}{6},-\frac{1}{3}\right)$, is _____.

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    Answer: 5

22. Q112020 · 7 Jan 2020 · Shift 2

    If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $(0,2\pi )-\{\pi \}$ which satisfy the equation, $2{\cot }^2\theta -\frac{5}{\sin \theta }+4=0$, then ${\int }_{{\theta }_1}^{{\theta }_2}{\cos }^{2}3\theta d\theta$ is equal to :

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

23. Q122020 · 7 Jan 2020 · Shift 2

    If $\frac{3+i\sin \theta }{4-i\cos \theta }$, $\theta \in [0,2\pi ]$, is a real number, then an argument of $\sin \theta +i\cos \theta$ is :

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