Twelve points are located on the circumference of a circle. Lines are drawn to connect all possible pairs of points. How many lines are drawn?

Let $X$ be a non-empty set and let $\mathcal{P}(X)$ be its power set. Define two operations ‘+’ and ‘$\ast$’ on $\mathcal{P}(X)$ as follows. For $A,B\in \mathcal{P}(X)$, let

$$A+B=A\cup B,A\ast B=A\cap B.$$

If $G_1=(\mathcal{P}(X),+)$ and $G_2=(\mathcal{P}(X),\ast )$, then,

Show that the sequence $\{S_n\}$, where $S_n={\left(1+\frac{1}{n}\right)}^2$, is convergent and that ${\lim }_{n\to \infty }{\left(1+\frac{1}{n}\right)}^n$ lies between 2 and 3.

Find the length of the arc of the parabola $x^2=4y$ from the vertex to the point $x=2$, where $x\in R$ and $y\in R$.

The set of positive real numbers $x$ for which the series

$$\sum _{n=1}^{\infty }\frac{1}{x^n+x^{-n}}$$

diverges

One of the subjects of a postgraduate course offered by a reputed institute is handled online by a faculty from another city. Suppose the video conferencing software used by the faculty is 99% reliable, his internet connection is 98% reliable and the internet connectivity at the institute is also 98% reliable, what is the chance of having failure-free online classes? The online classes will work only if the video conferencing software and internet connectivity at the faculty premise and the institute works. Suppose another video conferencing software is available in the market which is also 99% reliable and an alternative internet service provider is available at faculty's city with 98% reliability. Can purchasing and keeping one more video conferencing software or internet facility or both and keeping them as standby improve the chance of failure-free operation of the online class to 98%?

A 1-inch-diameter coin is thrown on a table covered with a grid of lines two inches apart. What is the probability that the coin lands in a square without touching any of the lines of the grid?

Calculate the mean deviation for the arithmetic progression $a,a+d,a+2d,a+3d,\dots ,a+2nd$.

The system of equations

$$\begin{array}{rl}x+2y+3z& =c\\ 2x-y+z& =2\\ ax+y+4z& =b\end{array}$$

has no solution if

Let $C$ be the circle in the $xy$-plane which passes through the origin and touches the line $y=-2$. If the centre of $C$ lies on the straight line $x+y=1$, then the radius of $C$ is

Find the length of the arc of the parabola $x^2=4y$ from the vertex to the point $x=2$, where $x\in R$ and $y\in R$.

What is the sum of the maximum and the minimum values of

$$\frac{1}{1+(2\cos x-4\sin x{)}^2}$$

over $x\in \mathbb{R}$?