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Let $A=\{1,2,3\}$ and $B=\{4,5,6\}$. A relation R from A to B is defined as $R=\{(x,y):x+y=6, x\in A, y\in B\}$. (i) Write all elements of R. (ii) Is R a function? Justify. (iii) Determine domain and range of R.

If $f:R^{+}\rightarrow R$ is defined as $f(x) = \log_{a} x$ ($a > 0$ and $a\ne1$), prove that f is a bijection. ($R^{+}$ is a set of all positive real numbers.)

For the curve $y=5x-2x^{3}$ if x increases at the rate of $2\text{ units/s}$, then how fast is the slope of the curve changing when $x=2$?

Calculate the area of the region bounded by the curve $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the x-axis using integration.

Find domain of $\sin^{-1}\sqrt{x-1}$.

Simplify $\sin^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)$.

If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that f is an increasing function.

Find the least value of 'a' so that $f(x)=2x^{2}-ax+3$ is an increasing function on $[2, 4]$.

Let A and B be two square matrices of order 3 such that $\det(A) = 3$ and $\det(B) = -4$. Find the value of $\det(-6AB)$.

If $A=\begin{bmatrix}1&2&0\\ -2&-1&-2\\ 0&-1&1\end{bmatrix}$, then find $A^{-1}$. Hence, solve the system of linear equations: $x-2y=10$, $2x-y-z=8$, $-2y+z=7$.

Given $A=\begin{bmatrix}-4&4&4\\ -7&1&3\\ 5&-3&-1\end{bmatrix}$ and $B=\begin{bmatrix}1&-1&1\\ 1&-2&-2\\ 2&1&3\end{bmatrix}$, find AB. Hence, solve the system of linear equations: $x-y+z=4$, $x-2y-2z=9$, $2x+y+3z=1$.

Find the image A' of the point A(2, 1, 2) in the line $l:\vec{r}=4\hat{i}+2\hat{j}+2\hat{k}+\lambda(\hat{i}-\hat{j}-\hat{k})$. Also, find the equation of line joining AA'. Find the foot of perpendicular from point A on the line l.

Find the shortest distance between the lines: $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ and $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$.

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

Using integration, find the area of the region bounded by the line $y=5x+2$, the x-axis and the ordinates $x=-2$ and $x=2$.

Find: $\int\frac{1}{x}\sqrt{\frac{x+a}{x-a}}dx$.

Two dice are thrown. Defined are the following two events A and B: $A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne3\}$ where (x, y) denote a point in the sample space. Check if events A and B are independent or mutually exclusive.

A die with number 1 to 6 is biased such that probability of $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.

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

If $y=\log(\sqrt{x}+\frac{1}{\sqrt{x}})^{2}$, then show that $x(x+1)^{2}y_{2}+(x+1)^{2}y_{1}=2$.