ISI M.MATH 2021 Subjective Question Paper with Solutions

This post contains the ISI M.Math 2021 Subjective Questions. It is a valuable resource for Practice if you are preparing for ISI M.Math. You can find some solutions here and try out others while discussing them in the comments below.

ISI M.Math 2021 Problem 1:

Let $M$ be a real $n \times n$ matrix with all diagonal entries equal to $r$ and all non-diagonal entries equal to $s$. Compute the determinant of $M$.

Solution

ISI M.Math 2021 Problem 2:

Let $F[X]$ be the polynomial ring over a field $F$. Prove that the rings $F[X] /\left\langle X^{2}\right)$ and $F[X] /\left\langle X^{2}-1\right\rangle$ are isomorphic if and only if the characteristic of $F$ is $2$

Solution

ISI M.Math 2021 Problem 3:

Let $C$ be a subset of $R$ endowed with the subspace topology. If every continuous real-valued function on $C$ is bounded, then prove that $C$ is compact.

ISI M.Math 2021 Problem 4:

Let $A=\left(a_{i j}\right)$ be a nonzero real $n \times n$ matrix such that $a_{i j}=0$ for $i \geq j$.
If $\sum_{i=0}^{k} c_{i} A^{i}=0$ for some $c_{i} \in \mathbb{R}$, then prove that $c_{0}=c_{1}=0$. Here
$A^{\prime}$ is the i-th power of $A$.

ISI M.Math 2021 Problem 5:

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be the function given by

$g(x)= \begin{cases}x \sin \left(\frac{1}{z}\right),& x \neq 0 \\ 0, & x=0\end{cases}$

Prove that $g(x)$ has a local maximum and a local minimum in the interval $\left(-\frac{1}{m}, \frac{1}{m}\right)$ for any positive integer $m$.

ISI M.Math 2021 Problem 6:

Fix an integer $n \geq 1$, Suppose that $n$ is divisible by distinct natural numbers $k_{1}, k_{2}, k_{3}$ such that

${gcd}\left(k_{1}, k_{2}\right)={gcd}\left(k_{2}, k_{3}\right)={gcd}\left(k_{3}, k_{1}\right)=1$

Pick a random natural number $j$ uniformly from the set $\{1,2,3, \ldots, n\}$. Let $A_{d}$ be the event that $j$ is divisible by $d$. Prove that the events $A_{k_{1}}, A_{k_{2}}, A_{k _{3}}$ are mutually independent.

ISI M.Math 2021 Problem 7:

Let $f:(0,1] \rightarrow[0, \infty)$ be a function. Assume that there exists $M \geq 0$ such that $\sum_{i=1}^{k} f\left(x_{i}\right) \leq M$ for all $k \geq 1$ and for all $x_{1}, \ldots, x_{k} \in[0,1]$. Show that the set $\{x \mid f(x) \neq 0\}$ is countable.

ISI M.Math 2021 Problem 8:

Let $G$ be a group having exactly three subgroups. Prove that $G$ is
cyclic of order $p^{2}$ for some prime $p$.

Solution

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