ISI Admission Test 2026 Question Paper M.Math PMB

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ISI Admission Test
2026
Question Paper ·
M.Math PMB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2026 Question Paper M.Math PMB

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The symbol R will denote the set of all real numbers while C will denote the set of all
complex numbers.

1. Let n, m and p be positive integers and suppose g : Rm → Rp be a linear map which is
onto.

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(a) Prove that there exists a linear map h : Rp → Rm such that h is one-to-one and
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(b) Suppose f : .R → R is a linear map which is one-to-one and Ran(f ) = Ker(g). se
g(h(x)) = x for all x ∈ Rp .
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s em l a
n m

la a g
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Prove that

a
Ran(f ) ∩ Ran(h) = {0}.
(c) Prove that
Rm = {f (v) + h(x) : v ∈ Rn , x ∈ Rp }.

2. Let V be a finite-dimensional real vector space and T : V → V a linear map which is
diagonalizable. Let λ be an eigenvalue of T and W = Ker(T − λI), where I : V → V
is the identity map.

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(a) Show that there exists a subspace U of V such that

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U ∩ W = {0} and V = {u + w : u ∈ U, w ∈ W }.

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(b) Suppose S : V → V is a linear map such that ST = T S. Prove that S(U ) ⊆ U
and S(W ) ⊆ W.
a
3. Let R be a commutative ring with unity. Suppose R has exactly two non-zero prime
ideals P1 and P2 . Assume that P1 ∩ P2 = (0).

(a) Prove that P1 and P2 are maximal ideals.
(b) Prove that R is a product of two fields.
(c) Find all the ideals of the ring R.

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4. Let S3 be the symmetric group on three elements. Compute the number of group

c. 5.o Let f : [0, 1) → R be a bounded continuous function. Define se
m
homomorphisms from S3 to S3 .

s em g l a
gla f (t) dt, x ∈ [0, 1). a
a
Z x
F (x) =
0

(a) Prove that lim− F (x) exists.
x→1

(b) Define F (1) = lim− F (x). Suppose f is also monotonically increasing. Prove that
x→1
lim− F (1+h)−F
h
(1)
exists.
h→0

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6. Suppose g : [0, 1] → R is a continuous function and f : [0, 1] → R is a function
satisfying the following conditions:

(a) f (0) > 0,
(b) f (1) < 0,
(c) The function h : [0, 1] → R defined by h(x) = f (x) + g(x) is increasing.

Let A = {x ∈ [0, 1] : f (x) ≥ 0} and let x0 = sup A.

(a) Prove that f (x0 ) ≥ 0.
(b) Prove that there exists t ∈ [x0 , 1] such that g(t) = h(x0 ).
(c) Prove that f (x0 ) = 0.

7. Suppose f : R → R is a continuous function such that
Z N
lim |f (x)| dx < ∞.
N →∞ 0

(a) Show that
Z N
1
lim √
xf (x) dx = 0.
N →∞ N N

(b) Using (a) or otherwise, show that
Z N
1
lim xf (x) dx = 0.
N →∞ N 0

8. Let X and Y be metric spaces and f : X → Y be a function.

(a) If f is continuous and K is a compact subset of X, prove that the function
f |K : K → Y defined by
f |K (x) = f (x), x ∈ K,
is continuous.
(b) If f |K is continuous for all compact subsets K of X, then prove that f is contin-
uous.

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