ISI Admission Test 2021 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2021
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2021 Question Paper B.Stat B.Math UGB

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m .co

m .co s e m
se g l a
a

Notations: The set of integers {· · · , −2, −1, 0, 1, 2 · · · } is denoted by
Z. The set of real numbers is denoted by R.

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1. There are three cities each of which has exactly the same number of

. c o
citizens, say n. Every citizen in each city has exactly a total of n + 1
e m
e m
friends in the other two cities. Show that there exist three people, one
l as
l as
from each city, such that they are friends. We assume that friendship
ag
ag is mutual (that is, a symmetric relation).
2. Let f : Z → Z be a function satisfying f (0) 6= 0 = f (1). Assume also
that f satisfies equations (A) and (B) below.

f (xy) = f (x) + f (y) − f (x)f (y) (A)
f (x − y)f (x)f (y) = f (0)f (x)f (y) (B)

o m
c
for all integers x, y.

m . : a ∈ Z}.
(i) Determine explicitly the set {f (a)
e
(ii) Assuming that there is asnon-zero
prove that the set {b : f g l a integer a such that f (a) 6= 0,

a (b) 6= 0} is infinite.
3. Prove that every positive rational number can be expressed uniquely
as a finite sum of the form
a2 a3 an
a1 + + + ··· + ,
2! 3! n!
where an are integers such that 0 ≤ an ≤ n − 1 for all n > 1.

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4. Let g : (0, ∞) → (0, ∞) be a differentiable function whose derivative
. the
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is continuous, and such that g(g(x)) = x for all x > 0. If g is not
m s e
e la
identity function, prove that g must be strictly decreasing.

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ag 5. Let a , a , · · · , a ∈ R and
0 1 19

X
19
20
P (x) = x + ai xi , x ∈ R.
i=0

If P (x) = P (−x) for all x ∈ R, and

P (k) = k 2 , for k = 0, 1, 2 · · · , 9,

m .
c. o
1
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 2

then find
P (x)
lim .
x→0 sin2 x

6. If a given equilateral triangle ∆ of side length a lies in the union of
five equilateral triangles of side length b, show that there exist four
equilateral triangles of side length b whose union contains ∆.

7. Let a, b, c be three real numbers which are roots of a cubic polynomial,
and satisfy a + b + c = 6 and ab + bc + ac = 9. Suppose a < b < c.
Show that
0 < a < 1 < b < 3 < c < 4.

8. A pond has been dug at the Indian Statistical Institute as an inverted
truncated pyramid with a square base (see figure below). The depth
of the pond is 6m. The square at the bottom has side length 2m and
the top square has side length 8m. Water is filled in at a rate of 19
3
cubic meters per hour. At what rate is the water level rising exactly
1 hour after the water started to fill the pond?

2

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