ISI Admission Test 2024 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2024
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2024 Question Paper B.Stat B.Math UGB

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Q1. Find, with proof, all possible values of t such that

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1 + 21/3 + 31/3 + · · · + n1/3
� �

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lim =c
m
n→∞ nt

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for some real number c > 0. Also find the corresponding values of c.
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a Q2. Suppose n ≥ 2. Consider the polynomial

Qn (x) = 1 − xn − (1 − x)n .

Show that the equation Qn (x) = 0 has only two real roots, namely 0
and 1.

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Q3. Let ABCD be a quadrilateral with all internal angles < π.
. in the picture below. Let
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e of the shaded triangles shown.
Squares are drawn on each side as shown
Δ , Δ , Δ and Δ denote the sareas
1 2 3

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4

Prove that
a
Δ − Δ + Δ − Δ = 0.
1 2 3 4

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F

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A

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 3

Q4. Let f : R → R be a function which is differentiable at 0. Define
another function g : R → R as follows:

f (x) sin( 1 ) if x �= 0,
x
g(x) =
0 if x = 0.

Suppose that g is also differentiable at 0. Prove that

g � (0) = f � (0) = f (0) = g(0) = 0.

Q5. Let P (x) be a polynomial with real coefficients. Let α1 , . . . , αk
be the distinct real roots of P (x) = 0. If P � is the derivative of P ,
show that for each i = 1, 2, . . . , k,

(x − αi )P � (x)
lim = ri ,
x→αi P (x)

for some positive integer ri .

�2024
Q6. Let x1 , . . . , x2024 be non-negative real numbers with i=1 xi = 1.
Find, with proof, the minimum and maximum possible values of the
expression
1012
� 2024
�
xi + x2i .
i=1 i=1013

2

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Q7. Consider a container of the shape obtained by revolving a segment
of the parabola x = 1 + y 2 around the y-axis as shown below. The
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container is initially empty. Water is poured at a constant rate of

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1 cm /s into the container. Let h(t) be the height of water inside the
container .at time t. Find the time t when the rate of change of h(t) s e
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is maximum.

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Q8. In a sports tournament involving N teams, each team plays every
other team exactly once. At the end of every match, the winning
team gets 1 point and the losing team gets 0 points. At the end of
the tournament, the total points received by the individual teams are
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arranged in decreasing order as follows:
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a
a N −k
≤ xk ≤ N −
k+1
.
2 2

3
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 3