Trigonometry · JEE Main previous year questions

The normal to the curve $x=a(1+\cos \theta ),y=a\sin \theta$ at ${}^{\prime }{\theta }^{\prime }$ always passes through the fixed point

1. A.$(a,a)$
2. B.$(0,a)$
3. C.$(0,0)$
4. D.$(a,0)$

The period of ${\sin }^2\theta$ is

1. A.${\pi }^2$
2. B.$\pi$
3. C.$2\pi$
4. D.$\pi /2$

${\cot }^{-1}\left(\sqrt{\cos \alpha }\right)-{\tan }^{-1}\left(\sqrt{\cos \alpha }\right)=x$, then sin x is equal to :

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

${\int }_0^{10\pi }|\sin x|dx$ is

1. A.$20$
2. B.$8$
3. C.$10$
4. D.$18$

${\lim }_{x\to 0}\frac{\sqrt{1-\cos 2x}}{\sqrt{2}x}$ is

1. A.$1$
2. B.$-1$
3. C.zero
4. D.does not exist

Locus of mid point of the portion between the axes of $x\cos \alpha +y\sin \alpha =p$ where $p$ is constant is :

1. A.$x^2+y^2=\frac{4}{p^2}$
2. B.$x^2+y^2=4p^2$
3. C.$\frac{1}{x^2}+\frac{1}{y^2}=\frac{2}{p^2}$
4. D.$\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$

The number of solution of $\tan x+\sec x=2\cos x$ in $[0,2\pi ]$ is

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

In a triangle with sides $a,b,c,r_1>r_2>r_3$ (which are the ex-radii) then :

1. A.$a>b>c$
2. B.$a<b<c$
3. C.$a>b$ and $b<c$
4. D.$a<b$ and $b>c$