ISI Admission Test 2017 Question Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2017
Question Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2017 Question Paper B.Stat B.Math UGB

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(1) Let the sequence {an }n≥1 be defined by
an = tan(nθ),
where tan(θ) = 2. Show that for all n, an is a rational number which
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can be written with an odd denominator.
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(2) Consider a circle of radius 6 as given in the diagram below. Let B,
C, D and E be points on the circle such that BD and CE, when
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ag extended, intersect at A. If AD and AE have length 5 and 4 respec-
tively, and DBC is a right angle, then show that the length of BC is
√
12+9 15
5 .

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(3) Suppose f : R → R is a function given by
(
1 if x = 1,
f (x) = 10

1

e(x −1) + (x − 1)2 sin x−1 if x 6= 1.
(a) Find f 0 (1).
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100 
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(b) Evaluate lim 100 u − u f 1+ .
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u→∞ u
s
k=1

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a (1, −1), (−1, 1),
ag (4) Let S be the square formed by the four vertices (1, 1),
and (−1, −1). Let the region R be the set of points inside S which
are closer to the centre than to any of the four sides. Find the area
of the region R.

P.T.O.

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(5) Let g : N → N with g(n) being the product of the digits of n.
(a) Prove that g(n) ≤ n for all n ∈ N.
(b) Find all n ∈ N, for which n2 − 12n + 36 = g(n).

(6) Let p1 , p2 , p3 be primes with p2 6= p3 , such that 4 + p1 p2 and 4 + p1 p3
are perfect squares. Find all possible values of p1 , p2 , p3 .

(7) Let A = {1, 2, . . . , n}. For a permutation P = (P (1), P (2), · · · , P (n))
of the elements of A, let P (1) denote the first element of P . Find the
number of all such permutations P so that for all i, j ∈ A:
• if i < j < P (1), then j appears before i in P ; and
• if P (1) < i < j, then i appears before j in P .

(8) Let k, n and r be positive integers.
(a) Let Q(x) = xk + a1 xk+1 + · · · + an xk+n be a polynomial with
real coefficients. Show that the function Q(x)
xk
is strictly positive
for all real x satisfying
1
0 < |x| < n .
P
1+ |ai |
i=1
(b) Let P (x) = b0 + b1 x + · · · + br xr be a non-zero polynomial with
real coefficients. Let m be the smallest number such that bm 6=
0. Prove that the graph of y = P (x) cuts the x-axis at the origin
(i.e. P changes sign at x = 0) if and only if m is an odd integer.

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