ISI Admission Test 2018 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2018
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2018 Question Paper B.Stat B.Math UGB

Notes · Sample Papers · Previous Year Papers · Mock Tests

m
m .co

m .co s e m
se g l a
a

Notations: In the following, N = {1, 2, 3, · · ·} denotes the set of natural
numbers, R denotes the set of real numbers.
m
m .co
m .co
1. Find all pairs (x, y) with x, y real, satisfying the equations:
s e m
s e l a
l a ag
� �
x+y
g
sin = 0, |x| + |y| = 1.
a
2

2. Suppose that P Q and RS are two chords of a circle intersecting at a
point O. It is given that P O = 3 cm and SO = 4 cm. Moreover, the
area of the triangle P OR is 7 cm2 . Find the area of the triangle QOS.

m
3. Let f : R → R be a continuous function such that for all x ∈ R and

.co
for all t ≥ 0,
f (x) = f (et x).

s em
la
Show that f is a constant function.

ag
4. Let f : (0, ∞) → R be a continuous function such that for all x ∈
(0, ∞),
f (2x) = f (x).
Show that the function g defined by the equation
� 2x
dt
g(x) = f (t) for x > 0
x t

is a constant function.
om
m . c
.co s emP.T.O.
s em l a
g la ag
a

1
m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 2

5. Let f : R → R be a differentiable function such that its derivative f �
is a continuous function. Moreover, assume that for all x ∈ R,
� 1
0 ≤ �f � (x)� ≤ .
�
2
Define a sequence of real numbers {an }n∈N by:

a1 = 1,

an+1 = f (an ) for all n ∈ N.

Prove that there exists a positive real number M such that for all
n ∈ N,
|an | ≤ M.

6. Let a ≥ b ≥ c > 0 be real numbers such that for all n ∈ N, there
exist triangles of side lengths an , bn , cn . Prove that the triangles are
isosceles.

7. Let a, b, c ∈ N be such that

a2 + b2 = c2 and c − b = 1.

Prove that

( i ) a is odd,
( ii ) b is divisible by 4,
( iii ) ab + ba is divisible by c.

8. Let n ≥ 3. Let A = ((aij ))1≤i,j≤n be an n × n matrix such that
aij ∈ {1, −1} for all 1 ≤ i, j ≤ n. Suppose that

ak1 = 1 for all 1 ≤ k ≤ n and
n
�
aki akj = 0 for all i �= j.
k=1

Show that n is a multiple of 4.

2

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 2