Trigonometry · JEE Main previous year questions

The least value of $\left({\cos }^2\theta -6\sin \theta \cos \theta +3{\sin }^2\theta +2\right)$ is

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

Let $\frac{\pi }{2}<\theta <\pi$ and $\cot \theta =-\frac{1}{2\sqrt{2}}$. Then the value of $\sin \left(\frac{15\theta }{2}\right)(\cos 8\theta +\sin 8\theta )+\cos \left(\frac{15\theta }{2}\right)(\cos 8\theta -\sin 8\theta )$ is equal to :

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

The number of solutions of ${\tan }^{-1}4x+{\tan }^{-1}6x=\frac{\pi }{6}$, where $-\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}$, is equal to :

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

If $\frac{{\cos }^{2}48^{\circ }-{\sin }^{2}12^{\circ }}{{\sin }^{2}24^{\circ }-{\sin }^2{6}^{\circ }}=\frac{\alpha +\beta \sqrt{5}}{2}$, where $\alpha ,\beta \in \mathbb{N}$, then $\alpha +\beta$ is equal to ____

Let ABC be a triangle. Consider four points ${\mathrm{p}}_1,{\mathrm{p}}_2,{\mathrm{p}}_3,{\mathrm{p}}_4$ on the side AB , five points $p_5,p_6,p_7,p_8,p_9$ on the side $BC$, and four points $p_{10},p_{11},p_{12},p_{13}$ on the side AC . None of these points is a vertex of the triangle ABC . Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1,p_2,\dots ,p_{13}$, is ____

Let a point A lie between the parallel lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ such that its distances from ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, respectively, is :

1. A.$21\sqrt{3}$
2. B.$12\sqrt{2}$
3. C.$15\sqrt{6}$
4. D.27

The value of $\mathrm{cosec}10^{\circ }-\sqrt{3}\sec 10^{\circ }$ is equal to :

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

For a triangle $ABC$, let $\vec{p}=\overrightarrow{BC},\vec{q}=\overrightarrow{CA}$ and $\vec{r}=\overrightarrow{BA}$. If $|\vec{p}|=2\sqrt{3},|\vec{q}|=2$ and $\cos \theta =\frac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$, then $|\vec{p}\times (\vec{q}-3\vec{r}){|}^2+3|\vec{r}{|}^2$ is equal to :

1. A.200
2. B.220
3. C.410
4. D.340

Let the maximum value of ${\left({\sin }^{-1}x\right)}^2+{\left({\cos }^{-1}x\right)}^2$ for $x\in \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$ be $\frac{\mathrm{m}}{\mathrm{n}}{\pi }^2$, where $\gcd (\mathrm{m},\mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to ____.

The value of ${\cot }^{-1}\left(\frac{\sqrt{1+{\tan }^2(2)}-1}{\tan (2)}\right)-{\cot }^{-1}\left(\frac{\sqrt{1+{\tan }^2\left(\frac{1}{2}\right)}+1}{\tan \left(\frac{1}{2}\right)}\right)$ is equal to

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

Let $A=\left\{\theta \in [0,2\pi ]:1+10\mathrm{Re}\left(\frac{2\cos \theta +i\sin \theta }{\cos \theta -3i\sin \theta }\right)=0\right\}$. Then ${\sum }_{\theta \in A}{\theta }^2$ is equal to

1. A.$\frac{21}{4}{\pi }^2$
2. B.$6{\pi }^2$
3. C.$\frac{27}{4}{\pi }^2$
4. D.$8{\pi }^2$

If $10{\sin }^4\theta +15{\cos }^4\theta =6$, then the value of $\frac{27{\mathrm{cosec}}^6\theta +8{\sec }^6\theta }{16{\sec }^8\theta }$ is

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

Considering the principal values of the inverse trigonometric functions, ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}x+\frac{1}{2}\sqrt{1-x^2}\right),-\frac{1}{2}<x<\frac{1}{\sqrt{2}}$, is equal to

1. A.$-\frac{5\pi }{6}-{\sin }^{-1}x$
2. B.$\frac{5\pi }{6}-{\sin }^{-1}x$
3. C.$\frac{\pi }{6}+{\sin }^{-1}x$
4. D.$\frac{\pi }{4}+{\sin }^{-1}x$

Let A be the point of intersection of the lines ${\mathrm{L}}_1:\frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1}$ and ${\mathrm{L}}_2:\frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}$. Let B and C be the points on the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ respectively such that $AB=AC=\sqrt{15}$. Then the square of the area of the triangle $ABC$ is :

1. A.63
2. B.57
3. C.60
4. D.54

The number of solutions of the equation $2x+3\tan x=\pi$, $x\in [-2\pi ,2\pi ]-\left\{\pm \frac{\pi }{2},\pm \frac{3\pi }{2}\right\}$ is:

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

If $y(x)=\left|\begin{array}{ccc}\sin x& \cos x& \sin x+\cos x+1\\ 27& 28& 27\\ 1& 1& 1\end{array}\right|$, $x\in \mathbb{R}$, then $\frac{d^{2}y}{d{x}^2}+y$ is equal to

1. A.28
2. B.27
3. C.-1
4. D.1

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

1. A.$4$
2. B.$3$
3. C.$6$
4. D.$5$

If $\theta \in [-2\pi ,2\pi ]$, then the number of solutions of $2\sqrt{2}{\cos }^2\theta +(2-\sqrt{6})\cos \theta -\sqrt{3}=0$, is equal to:

1. A.8
2. B.6
3. C.10
4. D.12

If $\theta \in [-\frac{7\pi }{6},\frac{4\pi }{3}]$, then the number of solutions of $\sqrt{3}{\mathrm{cosec}}^2\theta -2(\sqrt{3}-1)\mathrm{cosec}\theta -4=0$, is equal to :

1. A.7
2. B.10
3. C.6
4. D.8

If $y=\cos \left(\frac{\pi }{3}+{\cos }^{-1}\frac{x}{2}\right)$, then $(x-y{)}^2+3y^2$ is equal to ____.

If $\sin x+{\sin }^{2}x=1$, $x\in \left(0,\frac{\pi }{2}\right)$, then $({\cos }^{12}x+{\tan }^{12}x)+3({\cos }^{10}x+{\tan }^{10}x+{\cos }^{8}x+{\tan }^{8}x)+({\cos }^{6}x+{\tan }^{6}x)$ is equal to:

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

$\cos \left({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\frac{33}{65}\right)$ is equal to:

1. A.$\frac{33}{65}$
2. B.1
3. C.$\frac{32}{65}$
4. D.0

Let ${}^n{C}_{r-1}=28,{}^n{C}_r=56$ and ${}^n{C}_{r+1}=70$. Let $A(4\cos t,4\sin t),B(2\sin t,-2\cos t)$ and $C\left(3r-n,r^2-n-1\right)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x-1{)}^2+(3y{)}^2=\alpha$, is the locus of the centroid of triangle ABC , then $\alpha$ equals

1. A.18
2. B.8
3. C.20
4. D.6

If for some $\alpha ,\beta ;\alpha \le \beta ,\alpha +\beta =8$ and ${\sec }^2\left({\tan }^{-1}\alpha \right)+{\mathrm{cosec}}^2\left({\cot }^{-1}\beta \right)=36$, then ${\alpha }^2+\beta$ is ____

Let $\mathrm{A}=\left\{x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}:{\log }_{(2/\pi )}|\sin x|+{\log }_{(2/\pi )}|\cos x|=2\right\}$ and $\mathrm{B}=\{x\ge 0:\sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $\mathrm{n}(\mathrm{A}\cup \mathrm{B})$ is equal to :

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