ISI Entrance Paper 2011 – from Indian Statistical Institute’s B.Stat Entrance

Also see: ISI and CMI Entrance Course at Cheenta

- Let be positive reals with . Then show that
- Consider three positive real numbers a,b and c. Show that there cannot exist two distinct positive integers m and n such that both and hold.
- Let denote the set of real numbers. Suppose a function f:R-> R satisfies f(f(f(x)))=x for all . Show that

(i) f is one-one,

(ii) f cannot be strictly decreasing, and

(iii) if f is strictly increasing, then f(x)=x for all . - Let f be a twice differentiable function on the open interval (-1,1) such that f(0)=1. Suppose f also satisfies and , for all x\ge 0. Show that .
- ABCD is a trapezium such that and >1. Suppose P and Q are points on AC and BD respectively, such that

Prove that PQCD is a parallelogram. - Let be a complex number such that both and have modulus 1. If for a positive integer n, is an n-th root of unity, then show that is also an n-th root of unity and n is a multiple of 6.
- (i) Show that there cannot exists three prime numbers, each greater than 3, which are in arithmetic progression with a common difference less than 5.

(ii) Let k > 3 be an integer. Show that it is not possible for k prime numbers, each greater than k, to be in an arithmetic progression with a common difference less than or equal to k+1. - Let . Arrange in increasing order of magnitude. Justify your answer.
- Consider all non-empty subsets of the set . For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as . For example,

(i) Show that .

(ii) Hence or otherwise, deduce that . - Show that the triangle whose angles satisfy the equality is right angled.

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problem 8 is based on basic inequalities

Let f be a twice differentiable function on the open interval (-1,1) such that f(0)=1. Suppose f also satisfies f(x)≥0,f′(x)≤0 and f”(x)≤f(x), for all x\ge 0. Show that f′(0)≥−2–√.

How to proceed with no. 4?

Can you help me solving the 19 number question ??…

Consider the L-shaped diagram below ..that’s one