Ramanujan School of Mathematics
Mock Test I Subjective
ISI and CMI Entrance 2019
Short Answer Type Questions
Total time is 2 hours. Each question carries 10 marks. Attempt as many
as you can. Answers without proper explanations will fetch no mark.
1. Find all integers m and n such that m5 − n5 = 16mn.
2. Show that any monic even degree polynomial with coefficients odd natural num-
bers cannot have a rational root.
3. Let I be the incentre of 4ABC. B 0 be the reflection of B in the line AI. Show
that circumcentre of 4BCB 0 lies on circumcircle of 4ABC.
4. Let f be a twice differentiable real valued function satisfying
f (x) + f 00 (x) = −xg(x)f 0 (x)
for every x ∈ R, where g(x) ≥ 0 for all x ∈ R. Show that f (x) is bounded.
5. Let a1 , a2 , · · · , an be the complex-coordinates of the vertices of regular polygon.
Show that,
a1 a2 + a2 a3 + · · · + an a1 = a1 a3 + a2 a4 + · · · + an a2 .
6. The arithmetic derivative D(n) of a positive integer n is defined via the following
rules:
(a) D(1) = 0,
(b) D(p) = 1 for all primes p,
(c) D(ab) = bD(a) + aD(b) for all positive integers a and b.
Find the sum of all positive integers n below 2019 satisfying D(n) = n.
7. Let f : (−1, 1) → R be a twice differentiable function. Suppose that f (1/n) = 1
holds for every n ∈ N. Prove that f 0 (0) = 0. Furthermore, show that f 00 (0) = 0.
8. Let S be a set of n ≥ 1 elements, and let A1 , A2 , · · · , Ak be a family of distinct
subsets of S, with the property that any two of these subsets have non-empty
intersection. And suppose that for any other subset B of S, there is an Ai such
that B ∩ Ai is empty. Prove that k = 2n−1 . (Hint: First show that k ≤ 2n−1 .)
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