ISI Admission Test 2019 Question Paper B.Stat B.Math UGB

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ISI 2019
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2019 Question Paper B.Stat B.Math UGB

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2019

Notation.

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R denotes the set of all real numbers.

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C denotes the set of all complex numbers.

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1. Prove that the positive integers n that cannot be written as a sum of

g l r consecutive positive integers, with r > 1, are of the form n = 2l for
a some l ≥ 0.

2. Let f : (0, ∞) → R be defined by
� 1 �
f (x) = lim cosn .
n→∞ nx
(a) Show that f has exactly one point of discontinuity.
(b) Evaluate f at its point of discontinuity.

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c
. If f (z) = z + 2, then draw a
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3. Let Ω = {z = x + iy ∈ C : |y| ≤ 1}. 2

sketch of
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f (Ω) = {f (z) : z ∈ Ω}.

4. Let f : R → R be a twice differentiable function such that
� x+y
1
f (t) dt = f (x), for all x ∈ R, y > 0.
2y x−y

Show that there exist a, b ∈ R such that f (x) = ax + b for all x ∈ R.

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m 5. A subset S of the plane is called convex if given any two points x
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and y in S, the line segment joining x and y is contained in S. A

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quadrilateral is called convex if the region enclosed by the edges of the

las quadrilateral is a convex set.
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a is a rectangle
ag Show that given a convex quadrilateral Q of area 1, there
R of area 2 such that Q can be drawn inside R.

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6. For all natural numbers n, let
� �
√
�
An = 2 − 2 + 2 + · · · + 2 (n many radicals).

(a) Show that for n ≥ 2,
π
An = 2 sin .
2n+1

(b) Hence, or otherwise, evaluate the limit

lim 2n An .
n→∞

7. Let f be a polynomial with integer coefficients. Define
� �
a1 = f (0), a2 = f (a1 ) = f f (0) ,

and
an = f (an−1 ) for n ≥ 3.
If there exists a natural number k ≥ 3 such that ak = 0, then prove
that either a1 = 0 or a2 = 0.

8. Consider the following subsets of the plane:

C1 = {(x, y) : x > 0, y = x1 }

and
C2 = {(x, y) : x < 0, y = −1 + x1 }.
Given any two points P = (x, y) and Q = (u, v) of the plane, their
distance d(P, Q) is defined by
�
d(P, Q) = (x − u)2 + (y − v)2 .

Show that there exists a unique choice of points P0 ∈ C1 and Q0 ∈ C2
such that

d(P0 , Q0 ) ≤ d(P, Q) for all P ∈ C1 and Q ∈ C2 .

2

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