ISI MS(QMBA) 2024 Question Paper with Solutions: 46 Questions, Answer Key & Section-wise Analysis

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 $n\to \infty$.

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

The limit

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

Find the total number of ways of forming two 5-digit numbers $A$ and $B$ such that $B$ can be subtracted from $A$ without using the method of borrowing at any stage.

If $n$ is a positive integer, find the coefficient of $x^{-1}$ in the expansion of $(1+x{)}^n{\left(1+\frac{1}{x}\right)}^n$.

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