ISI Admission Test 2023 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2023
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2023 Question Paper B.Stat B.Math UGB

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Q 1.
Determine all integers n > 1 such that every power of n has an odd
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number of digits.

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Q 2.
c
. and a be de ned inductively by s e
Let a =
e m 1
l a
s ag
0 2 n

l a
ag
r
1+a n−1
a = n ,n ≥ 1.
2
(a) Show that for n = 0, 1, 2, . . .,
π
an = cos θn for some 0 < θn < ,
2
and determine θn .
(b) Using (a) or otherwise, calculate
o m
c
lim 4 (1 −. a ) .
em
n
n

s
n→∞

g la
Q 3.
a
In a triangle ABC, consider points D and E on AC and AB,
respectively, and assume that they do not coincide with any of the
vertices A, B, C. If the segments BD and CE intersect at F ,
consider the areas w, x, y, z of the quadrilateral AEF D and the
triangles BEF, BF C, CDF , respectively.
(a) Prove that y 2 > xz.
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(b) Determine w in terms of x, y, z.
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Q 4.
Let n1 , n2 , · · · , n51 be distinct natural numbers each of which has
exactly 2023 positive integer factors. For instance, 22022 has exactly
2023 positive integer factors 1, 2, 22 , · · · , 22021 , 22022 . Assume that no
prime larger than 11 divides any of the ni ’s. Show that there must
be some perfect cube among the ni ’s. You may use the fact that
2023 = 7 × 17 × 17.
Q 5.
There is a rectangular plot of size 1 × n. This has to be covered by
three types of tiles - red, blue and black. The red tiles are of size 1 × 1,
the blue tiles are of size 1 × 1 and the black tiles are of size 1 × 2. Let
tn denote the number of ways this can be done. For example, clearly
t1 = 2 because we can have either a red or a blue tile. Also, t2 = 5
since we could have tiled the plot as: two red tiles, two blue tiles, a
red tile on the left and a blue tile on the right, a blue tile on the left
and a red tile on the right, or a single black tile.
(a) Prove that t2n+1 = tn (tn−1 + tn+1 ) for all n > 1.
(b) Prove that tn = d≥0 n−d
P
n−2d
d
2 for all n > 0.
Here, 
   m!
m , if 0 ≤ r ≤ m ,
= r!(m−r)!
r 0 , otherwise ,
for integers m, r.
Q 6.
Let {un }n≥1 be a sequence of real numbers de ned as u1 = 1 and
1
un+1 = un + for all n ≥ 1 .
un
√
Prove that un ≤ 3 2 n for all n.

2
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Q 7.
(a) Let n ≥ 1 be an integer. Prove that X n +Y n +Z n can be written as
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a polynomial with integer coe cients in the variables α = X + Y + Z,

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(b) Let .G
c
β = XY + Y Z + ZX and γ = XY Z.
s e
m a
n n n
= x sin(nA) + y sin(nB) + z sin(nC), where
x, y,sz,eA, B, C are real numbers such that A + B + C is an integral l
n

g la of π. Using (a) or otherwise, show that if G = G = 0, then ag
a G = 0 for all positive integers n.
multiple
n
1 2

Q 8.
Let f : [0, 1] → R be a continuous function which is di erentiable on
(0, 1). Prove that either f is a linear function f (x) = ax + b or there
exists t ∈ (0, 1) such that |f (1) − f (0)| < |f ′ (t)|.

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s e
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ag

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