Trigonometry · JEE Main previous year questions

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.$0$

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.$0$

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.$2y^{\prime }+3{\pi }^{2}y=0$

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.$\frac{\pi }{12}$

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

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

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.$\frac{3}{\sqrt{5}}-\frac{3}{\sqrt{2}}+1$

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 ____.

If the sum of all the solutions of ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)+{\cot }^{-1}\left(\frac{1-x^2}{2x}\right)=\frac{\pi }{3}$, $-1<x<1,x\ne 0$, is $\alpha -\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to ____.

If m and n respectively are the numbers of positive and negative values of $\theta$ in the interval $[-\pi ,\pi ]$ that satisfy the equation $\cos 2\theta \cos \frac{\theta }{2}=\cos 3\theta \cos \frac{9\theta }{2}$, then mn is equal to ____.

${\tan }^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right)+{\sec }^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right)$ is equal to :

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

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

Let $S=\{\theta \in [0,2\pi ):\tan (\pi \cos \theta )+\tan (\pi \sin \theta )=0\}$. Then ${\sum }_{\theta \in S}{\sin }^2\left(\theta +\frac{\pi }{4}\right)$ is equal to _______.

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.$\frac{7}{3}$

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 ____.

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

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

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.$\frac{-\pi }{12}$

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

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.$\sqrt{7}-\sqrt{3}$

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 _________.

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.$\left(10(\sqrt{3}+1),\frac{25}{2}\right)$

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.${\sum }_{\theta \in S}\theta =\frac{3\pi }{4}$

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

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

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.$f\left(\frac{a}{2}\right)=\sqrt{2}{f}^{\prime }\left(\frac{a}{2}\right)$