ISI Admission Test 2022 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2022
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2022 Question Paper B.Stat B.Math UGB

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Note. In this question-paper, R denotes the set of real numbers.

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1. Consider a board having 2 rows and n columns. Thus there are
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2n cells in the board. Each cell is to be filled in by 0 or 1.
e m
m In how many ways can this be done such that each row sum
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la and each column sum is even? ag
ag (b) In how many ways can this be done such that each row sum
and each column sum is odd?

2. Consider the function
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f (x) = (x − k)4 , x ∈ R,
k=1

o m
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where m > 1 is an integer. Show that f has a unique minimum
and find the point where the m
s e minimum is attained.
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3. Consider the parabolal C : y = 4x and the straight line
agP be a variable point on L. Draw the
2

L : y = x + 2. Let
two tangents from P to C and let Q1 and Q2 denote the two
points of contact on C. Let Q be the mid-point of the line
segment joining Q1 and Q2 . Find the locus of Q as P moves
along L.

4. Let P (x) be an odd degree polynomial in x with real coefficients.
m
m Show that the equation P (P (x)) = 0 has at least as many
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5. For any positive integer n, and i = 1, 2, let f (n) i

a number of divisors of n of the form 3k + i (including 1 and n).
Define, for any positive integer n,

f (n) = f1 (n) − f2 (n).

Find the values of f (52022 ) and f (212022 ).

1
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6. Consider a sequence P1 , P2 , . . . of points in the plane such that
P1 , P2 , P3 are non-collinear and for every n ≥ 4, Pn is the mid-
point of the line segment joining Pn−2 and Pn−3 . Let L denote
the line segment joining P1 and P5 . Prove the following:

(a) The area of the triangle formed by the points Pn , Pn−1 , Pn−2
converges to zero as n goes to infinity.
(b) The point P9 lies on L.

7. Let
P (x) = 1 + 2x + 7x2 + 13x3 , x ∈ R.

Calculate for all x ∈ R,
� � x ��n
lim P .
n→∞ n

8. Find the minimum value of

| sin x + cos x + tan x + cot x + sec x + cosec x|

for real numbers x not multiple of π/2.

9. Find the smallest positive real number k such that the following
inequality holds
1
|z1 + . . . + zn | ≥ (|z1 | + . . . + |zn |) .
k
for every positive integer n ≥ 2 and every choice z1 , . . . , zn of
complex numbers with non-negative real and imaginary parts.
[Hint: First find k that works for n = 2. Then show that the
same k works for any n ≥ 2.]

2

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