ISI Admission Test 2024 Question Paper M.Math PMB

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ISI Admission Test
2024
Question Paper ·
M.Math PMB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2024 Question Paper M.Math PMB

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• N denotes the set of all positive integers.

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• R denotes the set of all real numbers.

o m m .
c
. : x ∈ X}.
• For a function f : X → Y , Ran(f ) = f (X) denotes the set
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se l
{f (x)

la a g
g
a T : V → V , the symbol T S : V → V denotes the linear map
• For vector spaces V1 , V2 , V3 and linear maps S : V1 → V2 and
2 3 1 3

defined as T S(v) = T (S(v)) for all v ∈ V1 .

1. Let n, m and p be positive integers and suppose g : Rm → Rp be
a linear map which is onto.
m
(a) Prove that there exists a linear map h : Rp → Rm such that

e m = x for all x ∈ R .
s
p

(b) Suppose f : R →la
h is one to one and g(h(x))

and Ran(f ) = a
g R is a linear map which is one to one
n m

Ker(g). Prove that

Ran(f ) ∩ Ran(h) = {0}.

(c) Prove that

Rm = {f (v) + h(x) : v ∈ Rn , x ∈ Rp }.

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m c
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2. Let V be a finite dimensional real vector space and T : V →

em s
la map.
s
a linear map which is diagonalizable. Let λ be an eigenvalue

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T and W = Ker(T − λI), where I : V → V is the identity

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

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

(b) Suppose S : V → V is a linear map such that ST = T S.
Prove that S(U ) ⊆ U and S(W ) ⊆ W.

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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.

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

5. Let f : [0, 1) → R be a bounded continuous function. Define
� x
F (x) = f (t)dt, x ∈ [0, 1).
0

(a) Prove that lim F (x) exists.
x→1−

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

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

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

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

2

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7. Suppose g : [0, 1] → R is a continuous function and f : [0, 1] → R

o m m .
c
is a function satisfying the following conditions:

(i) . f (0) > 0, s e
m
e(ii) f (1) < 0, l a
l a s a g
g
a (iii) the function h : [0, 1] → R defined by h(x) = f (x) + g(x) is
monotonically increasing.

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

(a) Prove that f (x0 ) ≥ 0.

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(b) Prove that there exists t ∈ [x0 , 1] such that g(t) = h(x0 ).

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(c) Prove that f (x0 ) = 0.

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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.
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(b) If f |K is continuous for all compact subsets K of X, then
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c. o prove that f is continuous.
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