Let $A_1$ and $A_2$ be two arbitrarily chosen subsets of $\{1,2,\dots ,n\}$. The number of ways such that $A_1\cap A_2=\varnothing$ is

Given a circle of radius $R$, a square is inscribed in it. A circle is then inscribed in the square. Again a square in the circle and so on. Let $A_i$ be the area of the $i$th square created in this process. Find the limit of ${\sum }_{i=1}^n{A}_i$ as $n\to \infty$.

The limit

$$\underset{n\to \infty }{\lim }\frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots +\sqrt[n]{n}}{n}$$

Evaluate ${\lim }_{x\to 0}\left(\frac{1}{x^5}{\int }_0^x{e}^{-t^2}dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$.

Let $f,g:[0,1]\to (0,\infty )$ be two continuously differentiable functions such that $f^{\prime }(x)=g^{\prime }(x)$ for all $x\in [0,1]$. Suppose that

$$\frac{f(\frac{1}{2})}{g(\frac{1}{2})}=1.$$

Then $\frac{f(\frac{1}{4})}{g(\frac{1}{4})}$ is equal to

A chemical is used to remove the impurities from the pulp by a bleaching process generating toxic and non-toxic wastes. The toxic waste generated is 0.05 times the square of the amount of chemical used, while the non-toxic waste generated is equal to 0.1 times the amount of chemical used. The pulp yield is 19.1 times the amount of chemical used. Determine the amount of chemical to be used to maximize the pulp yield and minimize the wastes.

Suppose $U$ and $W$ are distinct five-dimensional subspaces of a vector space $V$ with $dim(V)=7$. Find the possible dimensions of $U\cap W$.

Let

$$S=\left\{\left(\begin{array}{cc}a& b\sqrt{5}\\ -b\sqrt{5}& a\end{array}\right):a,b\in \mathbb{Z}\right\}.$$

The number of $A\in S$ such that $A^{-1}$ is also in $S$ is

For $x,y\ge 0$, consider the regions $A=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2\le 1\}$ and $B=\{(x,y)\in {\mathbb{R}}^2:x^4+y^4>1\}$. Then

It is given that the equation $(\cos p-1)x^2+(\cos p)x+\sin p=0$ has a real root. Then $p$ must lie in the interval

For fixed $a_1,a_2,\dots ,a_5\in \mathbb{R}$, the graph of

$$a_1{x}^2+a_{2}x+a_3{y}^2+a_{4}y+a_5=0$$

on the $xy$-plane can never be

A bank has implemented a fraud detection software. The probability that the software detect a fraudulent transaction as fraudulent is 0.99. The probability that the software detect a genuine transaction as genuine is 0.95. From the past experience, it is known that 0.01% of transactions are fraudulent. If the software detect a transaction as fraudulent, what is the probability that the transaction is actually fraudulent?

There are two coins in a box. One of the coins is a fair coin and for the other, the probability of head is $\frac{2}{3}$. A coin is chosen at random from the box and tossed 5 times. Given that at least 4 heads were obtained, what is the conditional probability that the fair coin was chosen?

The 26 letters of the English alphabet are randomly permuted. What is the probability that at least two of the three letters I, N, D come next to each other?