ISI Admission Test 2020 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2020
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2020 Question Paper B.Stat B.Math UGB

Notes · Sample Papers · Previous Year Papers · Mock Tests

.
.co s e m

s em l a
a ag

1. Let i be a root of the equation x2 + 1 = 0 and let ω be a root of the
equation x2 + x + 1 = 0. Construct a polynomial
m
om
f (x) = a0 + a1 x + . . . + an xn
. co
. c e m
em
where a0 , a1 , . . . , an are all integers such that f (i + ω) = 0.
l as
l as
2. Let a be a fixed real number. Consider the equation
ag
ag (x + 2)2 (x + 7)2 + a = 0, x ∈ R,

where R is the set of real numbers. For what values of a, will the equation
have exactly one double-root?

3. Let A and B be variable points on x-axis and y-axis respectively such
that the line segment AB is in the first quadrant and of a fixed length
2d. Let C be the mid-point of AB and P be a point such that

m
(a) P and the origin are on the opposite sides of AB and,

.co
(b) P C is a line segment of length d which is perpendicular to AB.
m
Find the locus of P .
s e
l a
ag } be such that
4. Let a real-valued sequence {x n n≥1

lim nxn = 0.
n→∞

Find all possible real values of t such that limn→∞ xn (log n)t = 0.

5. Prove that the largest pentagon (in terms of area) that can be inscribed
in a circle of radius 1 is regular (i.e., has equal sides).

6. Prove that the family of curves
m
m x2 y2
.co
.co m
+ = 1

m
a 2 + λ b2 + λ
s e
s e satisfies
− y).g
la
la dx a
dy 2 2 dy dy
(a − b ) = (x + y ) (x

ag dx dx

1
m . c
c. o s e m
s em
For more Question Papers, Sample Papers, Notes & Syllabus visit
gl a
Page 1 of 2

7. Consider a right-angled triangle with integer-valued sides a < b < c
where a, b, c are pairwise co-prime. Let d = c − b. Suppose d divides a.
Then

(a) Prove that d ≤ 2.
(b) Find all such triangles (i.e. all possible triplets a, b, c) with perime-
ter less than 100.

8. A finite sequence of numbers (a1 , . . . , an ) is said to be alternating if

a 1 > a2 , a 2 < a3 , a 3 > a4 , a 4 < a5 , . . . .

or a1 < a2 , a2 > a3 , a3 < a4 , a4 > a5 , . . . .

How many alternating sequences of length 5, with distinct numbers
a1 , . . . , a5 can be formed such that ai ∈ {1, 2, . . . , 20} for i = 1, . . . , 5?

2

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