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A relation R is defined on $N\times N$ (where N is the set of natural numbers) as: $(a, b)~R~(c,d)\Leftrightarrow a-c=b-d$ Show that R is an equivalence relation.

Show that a function $f:R\rightarrow R$ defined by $f(x)=\frac{2x}{1+x^{2}}$ is neither one-one nor onto. Further, find set A so that the given function $f:R\rightarrow A$ becomes an onto function.

Find: $\int x^{2}\cdot sin^{-1}(x^{3/2})dx$

A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.

The position vectors of vertices of $\Delta$ ABC are $A(2\hat{i}-\hat{j}+\hat{k}),$ $B(\hat{i}-3\hat{j}-5\hat{k})$ and $C(3\hat{i}-4\hat{j}-4\hat{k})$ Find all the angles of $\Delta$ AВС.

Find the general solution of the differential equation : $y~dx=(x+2y^{2})~dy$

Find the particular solution of the differential equation given by $2xy+y^{2}-2x^{2}\frac{dy}{dx}=0$ $y=2$, when $x=1.$

Find: $\int\frac{1}{x[(log~x)^{2}-3~log~x-4]}dx$

Evaluate : $\int_{-2}^{2}\sqrt{\frac{2-x}{2+x}}dx$

Show that: $\frac{d}{dx}(|x|)=\frac{x}{|x|},x\ne0$

If $x=e^{cos~3t}$ and $y=e^{sin~3t}$ , prove that $\frac{dy}{dx}=-\frac{y~log~x}{x~log~y}$

Show that $f(x)=e^{x}-e^{-x}+x-tan^{-1}x$ is strictly increasing in its domain.

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

If M and m denote the local maximum and local minimum values of the function $f(x)=x+\frac{1}{x}(x\ne0)$ respectively, find the value of $(M-m)$

If $y=cosec(cot^{-1}x)$, then prove that $\sqrt{1+x^{2}}\frac{dy}{dx}-x=0$ .

If $f(x)=|tan~2x|$, then find the value of $f^{\prime}(x)$ at $x=\frac{\pi}{3}$

Two vertices of the parallelogram ABCD are given as $A(-1,2,1)$ and $B(1,-2,5)$. If the equation of the line passing through C and D is $\frac{x-4}{1}=\frac{y+7}{-2}=\frac{z-8}{2}$ then find the distance between sides AB and CD. Hence, find the area of parallelogram ABCD.

Find the equation of the line passing through the point of intersection of the lines $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and $\frac{x-1}{0}=\frac{y}{-3}=\frac{z-7}{2}$ and perpendicular to these given lines.

Using integration, find the area of the region enclosed between the circle $x^{2}+y^{2}=16$ and the lines $x=-2$ and $x=2.$

The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.