ISI Admission Test 2020 Question Paper M.Math PMB

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ISI Admission Test 2020
Question Paper ·
M.Math PMB
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ISI Admission Test 2020 Question Paper M.Math PMB

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Notation. In the following, R denotes the set of real numbers.

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(1) (a) Let {fn } be a sequence of continuous real-valued functions on [0, 1]

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converging uniformly on [0, 1] to a function f . Suppose for all n ≥ 1 there

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exists xn ∈ [0, 1] such that fn (xn ) = 0. Show that there exists x ∈ [0, 1]
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such that f (x) = 0.

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(b) Give an example of a sequence {fn } of continuous real-valued func-
tions on [0, ∞) converging uniformly on [0, ∞) to a function f , such that
for each n ≥ 1 there exists xn ∈ [0, ∞) satisfying fn (xn ) = 0, but f
satisfies f (x) 6= 0 for all x ∈ [0, ∞).

(2) Let f : [0, 1] → R be a continuous function. Show that
n   
Y 1 k R1
lim 1+ f = e 0 f (x)dx .
n→∞
k=1
n n

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(3) (a) Let f : R → R be a twice continuously differentiable function.

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Show that

a
f (x + h) + f (x − h) − 2f (x)

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= f 00 (x)
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lim

a
h→0 h 2

for all x ∈ R.
(b) Show that if f further satisfies
Z x+y
1
f (t)dt = f (x)
2y x−y
for all x ∈ R, y > 0, then there exist a, b ∈ R such that f (x) = ax + b for
all x ∈ R.

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(4) Let f : R → R be a twice continuously differentiable function. Show

c. o (5) Let J be a 2×2 real matrix such that J = −I, where I is thesidentity
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that if f is bounded and f 00 (x) ≥ 0 for all x ∈ R then f must be constant.

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matrix.

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(a) Show that if v ∈ R and v 6= 0, then the vectors v, Jv ∈ R are
linearly independent.
(b) Show that there exists an invertible 2 × 2 real matrix U such that
 
−1 0 −1
U JU =
1 0

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(6) Suppose V is a 3-dimensional real vector space and T : V → V is a
linear map such that T 3 = 0 and T 2 6= 0.
(a) Show that there exists a vector v ∈ V such that the set {v, T (v), T 2 (v)}
is a basis of V .
(b) Suppose S : V → V is another linear map such that S 3 = 0 and
S 2 6= 0. Show that there exists an invertible linear map U : V → V such
that S = U T U −1 .

(7) Let K be a field, and let R be the ring K[x]. Let I ⊂ R be the ideal
generated by (x − 1)(x − 2). Find all maximal ideals of the ring R/I.

(8) Let G be a finite group, and let H be a normal subgroup of G. Let
P be a Sylow p-subgroup of H.
(a) Show that for all g ∈ G, there exists h ∈ H such that gP g −1 = hP h−1 .
(b) Let N = {g ∈ G|gP g −1 = P }. Let HN be the set HN = {hn|h ∈
H, n ∈ N }. Show that G = HN .

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