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Explore the Back-StoryThis 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.

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 $M$.

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 $2$

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.

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 $A$.

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 $m$.

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.

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.

Let $G$ be a group having exactly three subgroups. Prove that $G$ is

cyclic of order $p^{2}$ for some prime $p$.

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