Chain Rule

Q. Let $m$ and $n$ be two positive integers greater than 1. If
$\underset{\alpha \to 0}{lim}\frac{e^{\cos \left({\alpha }^n\right)}-e}{{\alpha }^m}=-\frac{e}{2}$
Then the value of $\frac{m}{n}$ is

Q.

The value of $\cos \left({36}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({36}^{\mathrm{o}}+\mathrm{A}\right)+\cos \left({54}^{\mathrm{o}}-\mathrm{A}\right)\cos \left({54}^{\mathrm{o}}+\mathrm{A}\right)$ is

Q. $f\left(x\right)$ satisfies the relation $f\left(x\right)-\lambda \underset{0}{\overset{\pi /2}{\int }}\sin x\cos tf\left(t\right)dt=\sin x.$

If $\lambda >2,$ then $f\left(x\right)$ decreases in which of the following interval?

1. $(0,\pi )$
2. $(\frac{\pi }{2},\frac{3\pi }{2})$
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
4. none of these

Q. $\stackrel{1}{\underset{-1}{\int }}\frac{x^3+|x|+3}{x^2+4|x|+3}dx$ is

1. $\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}\ln \left(\sin \alpha \right)d\alpha$
2. $-\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}\ln \left(\cos \alpha \right)d\alpha$
3. $-\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}\ln \left(\sin 2\alpha \right)d\alpha$
4. $-\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}(\ln \left(\sin \alpha \right)+\ln \left(\cos \alpha \right))d\alpha$

Q. $\int \frac{x^3-x^2+x-1}{x-1}dx$

Q. Let $f$ and $g$ be two continuous functions. Then ${\int }_{-\pi /2}^{\pi /2}\{f\left(x\right)+f(-x)\}\{g\left(x\right)-g(-x)\}dx$ is equal to

1. $1$
2. $\pi$
3. $0$
4. $-1$

Q. Evaluate ${\int }_0^{\infty }\sin xdx$

Q. If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}-=\alpha$, then $4x^2-4xy\cos \alpha +y^2$ is equal to

1. 4
2. $2{\sin }^2\alpha$
3. $-4{\sin }^2\alpha$
4. $4{\sin }^2\alpha$

Q. $\stackrel{1}{\underset{-1}{\int }}\frac{x^3+|x|+3}{x^2+4|x|+3}dx$ is

1. $\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}\ln \left(\sin \alpha \right)d\alpha$
2. $-\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}\ln \left(\cos \alpha \right)d\alpha$
3. $-\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}\ln \left(\sin 2\alpha \right)d\alpha$
4. $-\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}(\ln \left(\sin \alpha \right)+\ln \left(\cos \alpha \right))d\alpha$

Q. If $f\left(x\right)=|x^2-5x+4|$, then value of ${\int }_0^{2}f\left(x\right)dx$ is

1. $2$
2. $0$
3. $1$
4. $3$

Q. Represent $\sin \alpha -i\cos \alpha$ in polar form.

Q. If rolle's theorem is applicable on $f\left(x\right)=x^{\alpha }\tan x$ in $[-\frac{\pi }{4},\frac{\pi }{4}],$ then the value of $\alpha +1$ can be

1. $0$
2. $1$
3. $2$
4. $3$

Q. If $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{3x}{10}$ and $f(n\pi +x)=f\left(x\right),$ then the least positive integral value of $n$ is