ISI MS(QMBA) 2025 Question Paper with Solutions: 47 Questions, Answer Key & Section-wise Analysis

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

The number of factors of 2025 (including 1 and itself) equals

Let $\alpha =(x_1,x_2)$ and $\beta =(y_1,y_2)$ be two vectors in ${\mathbb{R}}^2$ such that $x_1{y}_1+x_2{y}_2=0$ and $x_1^2+x_2^2=y_1^2+y_2^2=1$. Prove that $B=\{\alpha ,\beta \}$ is a basis for ${\mathbb{R}}^2$.

Find ${\lim }_{x\to 0}\left(\frac{\sqrt{(1+x)}-\sqrt{(1+x^2)}}{\sqrt{(1-x^2)}-\sqrt{(1-x)}}\right)$.

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

If the matrix

$$A=\left[\begin{array}{cc}a& 1\\ 2& 3\end{array}\right]$$

has 1 as an eigenvalue, then $\det (A)$ equals

Pick a uniformly chosen random point inside a unit square (a square of side length 1) and draw a circle of radius $\frac{1}{3}$ around the point. Find the probability that the circle lies entirely inside the square.

In the quadratic equation, $x^2+bx+c=0$, both $b$ and $c$ can take any value from 1, 2, 3, 4, 5 and 6. Find the probability that the equation will have real roots.

Find the maximum area of a triangle when the perimeter of the triangle is $s$.

Common Description:

• The symbol $\mathbb{R}$ denotes the set of all real numbers.
• The symbol $\mathbb{Q}$ denotes the set of all rationals numbers.
• The symbol $\mathbb{Z}$ denotes the set of all integers.
• The symbol $\mathbb{N}$ denotes the set of all natural numbers.

Suppose $\sec x+\tan x=3$ for some $x\in (0,\pi /2)$. Then $\sin 2x$ equals