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
.
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
.