Trigonometry · JEE Main previous year questions

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

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.$\left[-1,-\frac{1}{2}\right]$

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.$\left(0,\frac{\pi }{2}\right)$

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.$2y+x=4$

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 _______.

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.$\frac{\pi +2}{4}$

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.$\frac{1}{2\sqrt{2}}$

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 ______.

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.$-\frac{3}{4}$

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.$x(1-y)=1$

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.$\frac{2\pi }{3}$

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 $\left(0,\frac{\pi }{2}\right)$

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.$\frac{5}{3\sqrt{3}}$

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 _______.

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.$-\frac{1}{4}$

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 _____.

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.$\frac{\pi }{3}+\frac{1}{6}$

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.$\pi -{\tan }^{-1}\left(\frac{4}{3}\right)$

For $x\in (0,3/2)$, let $f(x)=\sqrt{x}$ , $g(x)=\tan x$ and $h(x)=\frac{1-x^2}{1+x^2}$. If $\varphi (x)=((\mathrm{hof})\mathrm{og})(x)$, then $\varphi \left(\frac{\pi }{3}\right)$ is equal to :

1. A.$\tan \frac{7\pi }{12}$
2. B.$\tan \frac{11\pi }{12}$
3. C.$\tan \frac{\pi }{12}$
4. D.$\tan \frac{5\pi }{12}$

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is :

1. A.$\frac{25}{3}$
2. B.25
3. C.$25\sqrt{3}$
4. D.$\frac{25}{\sqrt{3}}$

The value of ${\sin }^{-1}\left(\frac{12}{13}\right)-{\sin }^{-1}\left(\frac{3}{5}\right)$ is equal to :

1. A.$\pi -{\sin }^{-1}\left(\frac{63}{65}\right)$
2. B.$\frac{\pi }{2}-{\sin }^{-1}\left(\frac{56}{65}\right)$
3. C.$\frac{\pi }{2}-{\cos }^{-1}\left(\frac{9}{65}\right)$
4. D.$\pi -{\cos }^{-1}\left(\frac{33}{65}\right)$

The number of solutions of the equation $1+{\sin }^{4}x={\cos }^{2}3x$, $x\in \left[-\frac{5\pi }{2},\frac{5\pi }{2}\right]$ is :

1. A.5
2. B.3
3. C.7
4. D.4

The derivative of ${\tan }^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$, with respect to $\frac{x}{2}$, where $\left(x\in \left(0,\frac{\pi }{2}\right)\right)$ is :

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

A value of $\theta \in \left(0,\frac{\pi }{3}\right)$, for which $\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & 1+{\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & {\sin }^2\theta & 1+4\cos 6\theta \end{array}\right|=0$, is :

1. A.$\frac{\pi }{18}$
2. B.$\frac{\pi }{9}$
3. C.$\frac{7\pi }{24}$
4. D.$\frac{7\pi }{36}$

If [x] denotes the greatest integer $\le x$, then the system of linear equations $[\sin \theta ]x+[-\cos \theta ]y=0$, $[\cot \theta ]x+y=0$

1. A.has a unique solution if $\theta \in \left(\frac{\pi }{2},\frac{2\pi }{3}\right)$ and have infinitely many solutions if $\theta \in \left(\pi ,\frac{7\pi }{6}\right)$
2. B.have infinitely many solutions if $\theta \in \left(\frac{\pi }{2},\frac{2\pi }{3}\right)$ and has a unique solution if $\theta \in \left(\pi ,\frac{7\pi }{6}\right)$
3. C.have infinitely many solutions if $\theta \in \left(\frac{\pi }{2},\frac{2\pi }{3}\right)\cup \left(\pi ,\frac{7\pi }{6}\right)$
4. D.has a unique solution if $\theta \in \left(\frac{\pi }{2},\frac{2\pi }{3}\right)\cup \left(\pi ,\frac{7\pi }{6}\right)$