Trigonometry · JEE Main previous year questions

Let $S=\left\{\theta \in [0,2\pi ]:8^{2{\sin }^2\theta }+8^{2{\cos }^2\theta }=16\right\}$. Then $n(S)+{\sum }_{\theta \in S}\left(\sec \left(\frac{\pi }{4}+2\theta \right)\mathrm{cosec}\left(\frac{\pi }{4}+2\theta \right)\right)$ is equal to:

1. A.0
2. B.$-2$
3. C.$-4$
4. D.12

$2\sin \left(\frac{\pi }{22}\right)\sin \left(\frac{3\pi }{22}\right)\sin \left(\frac{5\pi }{22}\right)\sin \left(\frac{7\pi }{22}\right)\sin \left(\frac{9\pi }{22}\right)$ is equal to :

1. A.$\frac{3}{16}$
2. B.$\frac{1}{16}$
3. C.$\frac{1}{32}$
4. D.$\frac{9}{32}$

Let $x=\sin (2{\tan }^{-1}\alpha )$ and $y=\sin \left(\frac{1}{2}{\tan }^{-1}\frac{4}{3}\right)$. If $S=\{\alpha \in \mathbb{R}:y^2=1-x\}$, then ${\sum }_{\alpha \in S}16{\alpha }^3$ is equal to ___________.

Let $\alpha =\tan \left(\frac{5\pi }{16}\sin \left(2{\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\right)$ and $\beta =\cos \left({\sin }^{-1}\left(\frac{4}{5}\right)+{\sec }^{-1}\left(\frac{5}{3}\right)\right)$ where the inverse trigonometric functions take principal values. Then, the equation whose roots are $\alpha$ and $\beta$ is :

1. A.$15x^2-8x-7=0$
2. B.$5x^2-12x+7=0$
3. C.$25x^2-18x-7=0$
4. D.$25x^2-32x+7=0$

Let $S_1=\{x\in [0,12\pi ]:{\sin }^{5}x+{\cos }^{5}x=1\}$ and $S_2=\{x\in [0,8\pi ]:{\sin }^{7}x+{\cos }^{7}x=1\}$ Then $n(S_1)-n(S_2)$ is equal to ____ .

Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\tan x(\cos x-y)$. If the curve passes through the point $\left(\frac{\pi }{4},0\right)$, then the value of ${\int }_0^{\pi /2}ydx$ is equal to :

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

If $\cot \alpha =1$ and $\sec \beta =-\frac{5}{3}$, where $\pi <\alpha <\frac{3\pi }{2}$ and $\frac{\pi }{2}<\beta <\pi$, then the value of $\tan (\alpha +\beta )$ and the quadrant in which $\alpha +\beta$ lies, respectively are :

1. A.$-\frac{1}{7}$ and ${\text{IV}}^{\text{th}}$ quadrant
2. B.$7$ and ${\text{I}}^{\text{st}}$ quadrant
3. C.$-7$ and ${\text{IV}}^{\text{th}}$ quadrant
4. D.$\frac{1}{7}$ and ${\text{I}}^{\text{st}}$ quadrant

If ${\cos }^{-1}\left(\frac{y}{2}\right)={\log }_e{\left(\frac{x}{5}\right)}^5$, $|y|<2$, then :

1. A.$x^2{y}^{\prime \prime }+x{y}^{\prime }-25y=0$
2. B.$x^2{y}^{\prime \prime }-x{y}^{\prime }-25y=0$
3. C.$x^2{y}^{\prime \prime }-x{y}^{\prime }+25y=0$
4. D.$x^2{y}^{\prime \prime }+x{y}^{\prime }+25y=0$

The value of $\cos \left(\frac{2\pi }{7}\right)+\cos \left(\frac{4\pi }{7}\right)+\cos \left(\frac{6\pi }{7}\right)$ is equal to :

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

${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)+{\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)+{\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$ is equal to :

1. A.$\frac{11\pi }{12}$
2. B.$\frac{17\pi }{12}$
3. C.$\frac{31\pi }{12}$
4. D.$-\frac{3\pi }{4}$

The value of $\cot \left({\sum }_{n=1}^{50}{\tan }^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is :

1. A.$\frac{26}{25}$
2. B.$\frac{25}{26}$
3. C.$\frac{50}{51}$
4. D.$\frac{52}{51}$

$\alpha =\sin 36^{\circ }$ is a root of which of the following equation?

1. A.$16x^4-10x^2-5=0$
2. B.$16x^4+20x^2-5=0$
3. C.$16x^4-20x^2+5=0$
4. D.$4x^4-10x^2+5=0$

Let for some real numbers $\alpha$ and $\beta$, $a=\alpha -i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)x+\bar{a}y=0$ has more than one solution, then $\frac{\alpha }{\beta }$ is equal to

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

If ${\sin }^2(10^{\circ })\sin (20^{\circ })\sin (40^{\circ })\sin (50^{\circ })\sin (70^{\circ })=\alpha -\frac{1}{16}\sin (10^{\circ })$, then $16+{\alpha }^{-1}$ is equal to ____.

$16\sin (20^{\circ })\sin (40^{\circ })\sin (80^{\circ })$ is equal to :

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

If the inverse trigonometric functions take principal values then ${\cos }^{-1}\left(\frac{3}{10}\cos \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)+\frac{2}{5}\sin \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)\right)$ is equal to :

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

The number of values of x in the interval $\left(\frac{\pi }{4},\frac{7\pi }{4}\right)$ for which $14{\mathrm{cosec}}^{2}x-2{\sin }^{2}x=21-4{\cos }^{2}x$ holds, is $\underline{}$.

The value of $2\sin (12^{\circ })-\sin (72^{\circ })$ is :

1. A.$\frac{\sqrt{5}(1-\sqrt{3})}{4}$
2. B.$\frac{1-\sqrt{5}}{8}$
3. C.$\frac{\sqrt{3}(1-\sqrt{5})}{2}$
4. D.$\frac{\sqrt{3}(1-\sqrt{5})}{4}$

The value of ${\tan }^{-1}\left(\frac{\cos \left(\frac{15\pi }{4}\right)-1}{\sin \left(\frac{\pi }{4}\right)}\right)$ is equal to :

1. A.$-\frac{\pi }{4}$
2. B.$-\frac{\pi }{8}$
3. C.$-\frac{5\pi }{12}$
4. D.$-\frac{4\pi }{9}$

If the angle made by the tangent at the point $(x_0,y_0)$ on the curve $x=12(t+\sin t\cos t)$, $y=12(1+\sin t{)}^2$, $0<t<\frac{\pi }{2}$, with the positive x-axis is $\frac{\pi }{3}$, then $y_0$ is equal to:

1. A.$6\left(3+2\sqrt{2}\right)$
2. B.$3\left(7+4\sqrt{3}\right)$
3. C.$27$
4. D.$48$

If $y={\tan }^{-1}\left(\sec x^3-\tan x^3\right)$, $\frac{\pi }{2}<x^3<\frac{3\pi }{2}$, then

1. A.$x{y}^{\prime \prime }+2y^{\prime }=0$
2. B.$x^2{y}^{\prime \prime }-6y+\frac{3\pi }{2}=0$
3. C.$x^2{y}^{\prime \prime }-6y+3\pi =0$
4. D.$x{y}^{\prime \prime }-4y^{\prime }=0$

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

1. A.8
2. B.5
3. C.6
4. D.7

${\cos }^{-1}(\cos (-5))+{\sin }^{-1}(\sin (6))-{\tan }^{-1}(\tan (12))$ is equal to : (The inverse trigonometric functions take the principal values)

1. A.$3\pi -11$
2. B.$4\pi -9$
3. C.$4\pi -11$
4. D.$3\pi +1$

The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0$, $x=\frac{\pi }{2}$, is :

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

If n is the number of solutions of the equation $2\cos x\left(4\sin \left(\frac{\pi }{4}+x\right)\sin \left(\frac{\pi }{4}-x\right)-1\right)=1$, $x\in [0,\pi ]$ and S is the sum of all these solutions, then the ordered pair (n, S) is :

1. A.$(3,13\pi /9)$
2. B.$(2,2\pi /3)$
3. C.$(2,8\pi /9)$
4. D.$(3,5\pi /3)$