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If $A_{1}$ denotes the area of region bounded by $y^{2}=4x,$ $x=1$ and x-axis in the first quadrant and $A_{2}$ denotes the area of region bounded by $y^{2}=4x,$ $x=4$, find $A_{1}:A_{2}$.

If $A=\begin{bmatrix}1&cot~x\\ -cot~x&1\end{bmatrix}$ show that $A^{\prime}A^{-1}=\begin{bmatrix}-cos~2x&-sin~2x\\ sin~2x&-cos~2x\end{bmatrix}$

Solve the following system of equations, using matrices: $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$ $\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$ , $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$ where x, y, $z\ne0$

Find the equation of the line which bisects the line segment joining points $A(2,3,4)$ and $B(4,5,8)$ and is perpendicular to the lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$

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 $x=\frac{\pi}{3}$