# I.S.I. B.Stat Entrance Subjective 2011

1. Let $$x_1, x_2, \cdots , x_n$$ be positive reals with $$x_1+x_2+\cdots+x_n=1$$. Then show that $$\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}$$
2. Consider three positive real numbers a,b and c. Show that there cannot exist two distinct positive integers m and n such that both $$\mathbf{a^m+b^m=c^m}$$ and $$\mathbf{ a^n+b^n=c^n}$$ hold.
3. Let $$\mathbb{R}$$ denote the set of real numbers. Suppose a function f:R-> R satisfies f(f(f(x)))=x for all $$x\in \mathbb{R}$$. 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 $$x \in \mathbb{R}$$.
4. Let f be a twice differentiable function on the open interval (-1,1) such that f(0)=1. Suppose f also satisfies $$f(x) \ge 0, f'(x) \le 0$$ and $$f”(x)\le f(x)$$, for all x\ge 0. Show that $$f'(0) \ge -\sqrt2$$.
5. ABCD is a trapezium such that $$\mathbf{AB\parallel DC}$$ and $$\mathbf{\frac{AB}{DC}=\alpha}$$ >1. Suppose P and Q are points on AC and BD respectively, such that $$\mathbf{\frac{AP}{AC}=\frac{BQ}{BD}=\frac{\alpha -1}{\alpha+1}}$$
Prove that PQCD is a parallelogram.
6. Let $$\mathbf{\alpha }$$ be a complex number such that both $$\mathbf{ \alpha }$$ and $$\mathbf{\alpha+1 }$$ have modulus 1. If for a positive integer n, $$\mathbf{ 1+\alpha }$$ is an n-th root of unity, then show that $$\mathbf{ \alpha }$$ is also an n-th root of unity and n is a multiple of 6.
7. (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.
8.  Let $$\mathbf{I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} , dx , n=1,2,3,4}$$ . Arrange $$\mathbf{I_1, I_2, I_3, I_4 }$$ in increasing order of magnitude. Justify your answer.
9.  Consider all non-empty subsets of the set $$\mathbf{{1,2\cdots,n}}$$. For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as $$\mathbf{S_n}$$. For example, $$\mathbf{S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3} }$$
(i) Show that $$\mathbf{S_n=\frac1n+\left(1+\frac1n\right)S_{n-1} }$$.
(ii) Hence or otherwise, deduce that $$\mathbf{S_n=n}$$.
10.  Show that the triangle whose angles satisfy the equality $$\mathbf{\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2}$$ is right angled.

## 4 Replies to “I.S.I. B.Stat Entrance Subjective 2011”

1. When I originally commented I appear to have clicked the -Notify me when new comments
are added- checkbox and now each time a comment is added I
receive 4 emails with the same comment. Is there an easy method you can remove me from that service?

Kudos!

2. 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?