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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 be a real matrix with all diagonal entries equal to and all non-diagonal entries equal to . Compute the determinant of .

Let be the polynomial ring over a field . Prove that the rings and are isomorphic if and only if the characteristic of is

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

Let be a nonzero real matrix such that for .

If for some , then prove that . Here

is the i-th power of .

Let be the function given by

Prove that has a local maximum and a local minimum in the interval for any positive integer .

Fix an integer , Suppose that is divisible by distinct natural numbers such that

Pick a random natural number uniformly from the set . Let be the event that is divisible by . Prove that the events are mutually independent.

Let be a function. Assume that there exists such that for all and for all . Show that the set is countable.

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

cyclic of order for some prime .

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.

Let be a real matrix with all diagonal entries equal to and all non-diagonal entries equal to . Compute the determinant of .

Let be the polynomial ring over a field . Prove that the rings and are isomorphic if and only if the characteristic of is

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

Let be a nonzero real matrix such that for .

If for some , then prove that . Here

is the i-th power of .

Let be the function given by

Prove that has a local maximum and a local minimum in the interval for any positive integer .

Fix an integer , Suppose that is divisible by distinct natural numbers such that

Pick a random natural number uniformly from the set . Let be the event that is divisible by . Prove that the events are mutually independent.

Let be a function. Assume that there exists such that for all and for all . Show that the set is countable.

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

cyclic of order for some prime .

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