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Show that the area of a parallelogram whose diagonals are represented by $\vec{a}$ and $\vec{b}$ is given by $\frac{1}{2}|\vec{a}\times\vec{b}|$. Also find the area of a parallelogram whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k}$.

Evaluate: $\int_{0}^{\pi}\frac{dx}{a^{2}\cos^{2}x+b^{2}\sin^{2}x}$.

Find: $\int\frac{\cos x}{(4+\sin^{2}x)(5-4\cos^{2}x)}dx$.

A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.

Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.

Evaluate: $\int_{\pi/2}^{\pi}e^{x}\left(\frac{1-\sin x}{1-\cos x}\right)dx$.

Check the differentiability of function $f(x)=x|x|$ at $x=0$.

Find k so that $f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}$ is continuous at $x=-1$.

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.)

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.

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 $\det(-6AB)$.

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

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 $\frac{dy}{dx}=\frac{-1}{(1+x)^{2}}$.