Trigonometry · JEE Main previous year questions

If $f(x)=\left|\begin{array}{ccc}2{\cos }^{4}x& 2{\sin }^{4}x& 3+{\sin }^{2}2x\\ 3+2{\cos }^{4}x& 2{\sin }^{4}x& {\sin }^{2}2x\\ 2{\cos }^{4}x& 3+2{\sin }^{4}x& {\sin }^{2}2x\end{array}\right|$, then $\frac{1}{5}{f}^{\prime }(0)$ is equal to :

1. A.2
2. B.1
3. C.0
4. D.6

If $\alpha ,-\frac{\pi }{2}<\alpha <\frac{\pi }{2}$ is the solution of $4\cos \theta +5\sin \theta =1$, then the value of $\tan \alpha$ is

1. A.$\frac{10-\sqrt{10}}{12}$
2. B.$\frac{\sqrt{10}-10}{6}$
3. C.$\frac{\sqrt{10}-10}{12}$
4. D.$\frac{10-\sqrt{10}}{6}$

For $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, if $y(x)=\int \frac{\text{cosec }x+\sin x}{\text{cosec }x\sec x+\tan x{\sin }^{2}x}\mathrm{d}x$, and ${\lim }_{x\to {\left(\frac{\pi }{2}\right)}^{-}}y(x)=0$ then $y\left(\frac{\pi }{4}\right)$ is equal to

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

The sum of the solutions $x\in \mathbb{R}$ of the equation $\frac{3\cos 2x+{\cos }^{3}2x}{{\cos }^{6}x-{\sin }^{6}x}=x^3-x^2+6$ is

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

Let $x=\frac{m}{n}$ ($m,n$ are co-prime natural numbers) be a solution of the equation $\cos \left(2{\sin }^{-1}x\right)=\frac{1}{9}$ and let $\alpha ,\beta (\alpha >\beta )$ be the roots of the equation $m{x}^2-nx-m+n=0$. Then the point $(\alpha ,\beta )$ lies on the line

1. A.$3x-2y=-2$
2. B.$3x+2y=2$
3. C.$5x+8y=9$
4. D.$5x-8y=-9$

Let the set of all $a\in \mathbf{R}$ such that the equation $\cos 2x+a\sin x=2a-7$ has a solution be $[p,q]$ and $r=\tan 9^{\circ }-\tan 27^{\circ }-\frac{1}{\cot 63^{\circ }}+\tan 81^{\circ }$, then $\mathrm{pqr}$ is equal to ____.

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying ${\tan }^{-1}(x)+{\tan }^{-1}(2x)=\frac{\pi }{4}$ is :

1. A.more than 2
2. B.$2$
3. C.$0$
4. D.$1$

If $2{\tan }^2\theta -5\sec \theta =1$ has exactly 7 solutions in the interval $[0,\frac{n\pi }{2}]$, for the least value of $n\in \mathbf{N}$, then ${\sum }_{k=1}^n\frac{k}{2^k}$ is equal to:

1. A.$\frac{1}{2^{14}}(2^{15}-15)$
2. B.$1-\frac{15}{2^{13}}$
3. C.$\frac{1}{2^{15}}(2^{14}-14)$
4. D.$\frac{1}{2^{13}}(2^{14}-15)$

The area of the region enclosed by the curve $f(x)=\max \{\sin x,\cos x\}$, $-\pi \le x\le \pi$ and the $x$-axis is

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

$\underset{0\le x\le \pi }{\max }\left\{x-2\sin x\cos x+\frac{1}{3}\sin 3x\right\}=$

1. A.$\frac{5\pi +2+3\sqrt{3}}{6}$
2. B.$0$
3. C.$\frac{\pi +2-3\sqrt{3}}{6}$
4. D.$\pi$

If $S=\left\{x\in \mathbb{R}:{\sin }^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right)-{\sin }^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi }{4}\right\}$, then $\sum _{x\in S}\left(\sin \left((x^2+x+5)\frac{\pi }{2}\right)-\cos \left((x^2+x+5)\pi \right)\right)$ is equal to ____________.

Let $f_n=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\sum _{k=1}^n{\sin }^{k-1}x\right)\left(\sum _{k=1}^n(2k-1){\sin }^{k-1}x\right)\cos xdx$, $n\in \mathbb{N}$. Then $f_{21}-f_{20}$ is equal to __________

For $x\in (-1,1]$, the number of solutions of the equation ${\sin }^{-1}x=2{\tan }^{-1}x$ is equal to ___________.

If the point $\left(\alpha ,\frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x\cos \theta +y\sin \theta =7$, $\theta \in \left(0,\frac{\pi }{2}\right)$ between the co-ordinates axes, then $\alpha$ is equal to :

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

In a triangle $\mathrm{ABC}$, if $\cos A+2\cos B+\cos C=2$ and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then $\cos A-\cos C$ is equal to

1. A.$\frac{3}{7}$
2. B.$\frac{9}{7}$
3. C.$\frac{10}{7}$
4. D.$\frac{5}{7}$

The number of elements in the set $S=\{\theta \in [0,2\pi ]:3{\cos }^4\theta -5{\cos }^2\theta -2{\sin }^6\theta +2=0\}$ is :

1. A.9
2. B.8
3. C.12
4. D.10

The angle of elevation of the top $\mathrm{P}$ of a tower from the feet of one person standing due South of the tower is $45^{\circ }$ and from the feet of another person standing due west of the tower is $30^{\circ }$. If the height of the tower is $5\text{ meters}$, then the distance (in meters) between the two persons is equal to

1. A.$10$
2. B.$\frac{5}{2}\sqrt{5}$
3. C.$5\sqrt{5}$
4. D.$5$

If $f(x)=\frac{(\tan 1^{\circ })x+{\log }_e(123)}{x{\log }_e(1234)-(\tan 1^{\circ })}$, $x>0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is :

1. A.2
2. B.4
3. C.0
4. D.8

$96\cos \frac{\pi }{33}\cos \frac{2\pi }{33}\cos \frac{4\pi }{33}\cos \frac{8\pi }{33}\cos \frac{16\pi }{33}$ is equal to :

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

Let $S=\left\{x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right):9^{1-{\tan }^{2}x}+9^{{\tan }^{2}x}=10\right\}$ and $\beta ={\sum }_{x\in S}{\tan }^2\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta -14{)}^2$ is equal to :

1. A.16
2. B.32
3. C.8
4. D.64

Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x},x\in [0,\pi ]-\left\{\frac{\pi }{4}\right\}$. Then $f\left(\frac{7\pi }{12}\right)f^{\prime \prime }\left(\frac{7\pi }{12}\right)$ is equal to

1. A.$\frac{2}{3\sqrt{3}}$
2. B.$\frac{2}{9}$
3. C.$\frac{-1}{3\sqrt{3}}$
4. D.$\frac{-2}{3}$

Let $A=\left\{\theta \in (0,2\pi ):\frac{1+2i\sin \theta }{1-i\sin \theta }\text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is :

1. A.$3\pi$
2. B.$\pi$
3. C.$2\pi$
4. D.$4\pi$

The value of $36\left(4{\cos }^2{9}^{\circ }-1\right)\left(4{\cos }^{2}27^{\circ }-1\right)\left(4{\cos }^{2}81^{\circ }-1\right)\left(4{\cos }^{2}243^{\circ }-1\right)$ is :

1. A.$18$
2. B.$36$
3. C.$54$
4. D.$27$

Let $I(x)=\int \frac{x^2(x{\sec }^{2}x+\tan x)}{(x\tan x+1{)}^2}dx$. If $I(0)=0$, then $I\left(\frac{\pi }{4}\right)$ is equal to :

1. A.${\log }_e\frac{(\pi +4{)}^2}{32}-\frac{{\pi }^2}{4(\pi +4)}$
2. B.${\log }_e\frac{(\pi +4{)}^2}{16}-\frac{{\pi }^2}{4(\pi +4)}$
3. C.${\log }_e\frac{(\pi +4{)}^2}{16}+\frac{{\pi }^2}{4(\pi +4)}$
4. D.${\log }_e\frac{(\pi +4{)}^2}{32}+\frac{{\pi }^2}{4(\pi +4)}$

The value of $\tan 9^{\circ }-\tan 27^{\circ }-\tan 63^{\circ }+\tan 81^{\circ }$ is ________.