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

Also see: ISI and CMI Entrance Course at Cheenta

  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.