Trigonometry · JEE Main previous year questions

If $\sum _{r=1}^{50}{\tan }^{-1}\frac{1}{2r^2}=p$, then the value of $\tan p$ is :

1. A.$\frac{101}{102}$
2. B.$\frac{50}{51}$
3. C.100
4. D.$\frac{51}{50}$

The value of $$2\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{2\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\sin \left(\frac{5\pi }{8}\right)\sin \left(\frac{6\pi }{8}\right)\sin \left(\frac{7\pi }{8}\right)\text{ is :}$$

1. A.$\frac{1}{4\sqrt{2}}$
2. B.$\frac{1}{4}$
3. C.$\frac{1}{8}$
4. D.$\frac{1}{8\sqrt{2}}$

Let f : R $\to$ R be defined as $f(x+y)+f(x-y)=2f(x)f(y),f\left(\frac{1}{2}\right)=-1$. Then, the value of $\sum _{k=1}^{20}\frac{1}{\sin (k)\sin (k+f(k))}$ is equal to :

1. A.${\mathrm{cosec}}^2(21)\cos (20)\cos (2)$
2. B.${\sec }^2(1)\sec (21)\cos (20)$
3. C.${\mathrm{cosec}}^2(1)\mathrm{cosec}(21)\sin (20)$
4. D.${\sec }^2(21)\sin (20)\sin (2)$

The sum of all values of x in $[0,2\pi ]$, for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to :

1. A.$8\pi$
2. B.$11\pi$
3. C.$12\pi$
4. D.$9\pi$

The value of $\cot \frac{\pi }{24}$ is :

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

The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is :

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

The number of solutions of ${\sin }^{7}x+{\cos }^{7}x=1$, $x\in [0,4\pi ]$ is equal to

1. A.$11$
2. B.$7$
3. C.$5$
4. D.$9$

The number of real roots of the equation ${\tan }^{-1}\sqrt{x(x+1)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{4}$ is :

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

The solutions of the equation $\left|\begin{array}{ccc}1+{\sin }^{2}x& {\sin }^{2}x& {\sin }^{2}x\\ {\cos }^{2}x& 1+{\cos }^{2}x& {\cos }^{2}x\\ 4\sin 2x& 4\sin 2x& 1+4\sin 2x\end{array}\right|=0$, $(0<x<\pi )$, are

1. A.$\frac{\pi }{12},\frac{\pi }{6}$
2. B.$\frac{\pi }{6},\frac{5\pi }{6}$
3. C.$\frac{5\pi }{12},\frac{7\pi }{12}$
4. D.$\frac{7\pi }{12},\frac{11\pi }{12}$

The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2\pi ]$ is

If $15{\sin }^4\alpha +10{\cos }^4\alpha =6$, for some $\alpha \in \mathrm{R}$, then the value of $27{\sec }^6\alpha +8{\mathrm{cosec}}^6\alpha$ is equal to :

1. A.500
2. B.400
3. C.250
4. D.350

If for $x\in \left(0,\frac{\pi }{2}\right)$, ${\log }_{10}\sin x+{\log }_{10}\cos x=-1$ and ${\log }_{10}(\sin x+\cos x)=\frac{1}{2}({\log }_{10}n-1)$, $n>0$, then the value of n is equal to :

1. A.16
2. B.9
3. C.12
4. D.20

The number of roots of the equation, $(81{)}^{{\sin }^{2}x}+(81{)}^{{\cos }^{2}x}=30$ in the interval $[0,\pi ]$ is equal to :

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

If $\frac{{\sin }^{-1}x}{a}=\frac{{\cos }^{-1}x}{b}=\frac{{\tan }^{-1}y}{c}$; $0<x<1$, then the value of $\cos \left(\frac{\pi c}{a+b}\right)$ is :

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

The value of the integral ${\int }_0^{\pi }|\sin 2x|dx$ is _________.

If $\sqrt{3}({\cos }^{2}x)=(\sqrt{3}-1)\cos x+1$, the number of solutions of the given equation when $x\in \left[0,\frac{\pi }{2}\right]$ is _________.

If $0<a,b<1$, and ${\tan }^{-1}a+{\tan }^{-1}b=\frac{\pi }{4}$, then the value of $(a+b)-\left(\frac{a^2+b^2}{2}\right)+\left(\frac{a^3+b^3}{3}\right)-\left(\frac{a^4+b^4}{4}\right)+\dots \dots$ is :

1. A.${\log }_{e}2$
2. B.e
3. C.${\log }_e\left(\frac{e}{2}\right)$
4. D.$e^2-1$

Let $A_1$ be the area of the region bounded by the curves $y=\sin x$, $y=\cos x$ and y-axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by the curves $y=\sin x$, $y=\cos x$, x-axis and $x=\frac{\pi }{2}$ in the first quadrant. Then,

1. A.$A_1:A_2=1:\sqrt{2}$ and $A_1+A_2=1$
2. B.$A_1=A_2$ and $A_1+A_2=\sqrt{2}$
3. C.$2A_1=A_2$ and $A_1+A_2=1+\sqrt{2}$
4. D.$A_1:A_2=1:2$ and $A_1+A_2=1$

If $0<x,y<\pi$ and $\cos x+\cos y-\cos (x+y)=\frac{3}{2}$, then $\sin x+\cos y$ is equal to :

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

$\mathrm{cosec}\left[2{\cot }^{-1}(5)+{\cos }^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :

1. A.$\frac{75}{56}$
2. B.$\frac{65}{56}$
3. C.$\frac{56}{33}$
4. D.$\frac{65}{33}$

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 ______ .

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.

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.$|y^{\prime }(0)|+|y^{\prime \prime }(0)|=1$

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

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

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.$\frac{1}{2}\left(1-i\sqrt{3}\right)$