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Evaluate: $\tan^{-1}\left[2\sin\left(2\cos^{-1}\frac{\sqrt{3}}{2}\right)\right]$.

If $\tan^{-1}(x^{2}+y^{2})=a^{2}$, then find $\frac{dy}{dx}$.

Differentiate $2^{\cos^{2}x}$ w.r.t $\cos^{2}x$.

Find the area of the region bounded by the curve $4x^{2}+y^{2}=36$ using integration.

The random variable X has the following probability distribution where a and b are some constants: $P(X)$ for X=1 is 0.2, X=2 is a, X=3 is a, X=4 is 0.2, X=5 is b. If the mean $E(X)=3$, then find values of a and b and hence determine $P(X\ge3)$

Solve the following differential equation $x^{2}dy+y(x+y)dx=0$

Find the particular solution of the differential equation $\frac{dy}{dx}-2xy=3x^{2}e^{x^{2}};y(0)=5.$

Find: $\int\frac{2x+1}{(x+1)^{2}(x-1)}dx$

Evaluate : $\int_{0}^{\pi}\frac{e^{cos~x}}{e^{cos~x}+e^{-cos~x}}d~x$

If $x=a~sin^{3}\theta$, $y=b~cos^{3}\theta$ then find $\frac{d^{2}y}{dx^{2}}$ at $\theta=\frac{\pi}{4}$

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$

Find $\frac{dy}{dx}$ , if $(cos~x)^{y}=(cos~y)^{x}.$

Find the vector equation of the line passing through the point (2, 3, -5) and making equal angles with the co-ordinate axes.

Find: $\int\frac{1}{5+4x-x^{2}}dx$

Find: $\int cos^{3}x~e^{log~sin~x}dx$

Check for differentiability of the function f defined by $f(x)=|x-5|$, at the point $x=5$.

Verify whether the function f defined by $f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}$ is continuous at $x=0$ or not.

Check whether the relation S in the set of real numbers R defined by $S=\{(a,b)$: where $a-b+\sqrt{2}$ is an irrational number is reflexive, symmetric or transitive.

Let $A=R-\{5\}$ and $B=R-\{1\}$. Consider the function $f:A\rightarrow B$, defined by $f(x)=\frac{x-3}{x-5}$ Show that f is one-one and onto.

Using integration, find the area bounded by the ellipse $9x^{2}+25y^{2}=225$, the lines $x=-2,$ $x=2$, and the X-axis.