ISI Entrance Paper 2010 – from Indian Statistical Institute’s B.Stat Entrance

Also see: ISI and CMI Entrance Course at Cheenta

  1. Let \mathbf{a_1,a_2,\cdots, a_n } and \mathbf{ b_1,b_2,\cdots, b_n } be two permutations of the numbers \mathbf{1,2,\cdots, n }. Show that \mathbf{\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2 }
  2. Let a,b,c,d be distinct digits such that the product of the 2-digit numbers \mathbf{\overline{ab}} and \mathbf{\overline{cb}} is of the form \mathbf{\overline{ddd}}. Find all possible values of a+b+c+d.
  3. Let \mathbf{I_1, I_2, I_3} be three open intervals of \mathbf{\mathbb{R}} such that none is contained in another. If \mathbf{I_1\cap I_2 \cap I_3} is non-empty, then show that at least one of these intervals is contained in the union of the other two.
  4. A real valued function f is defined on the interval (-1,2). A point \mathbf{x_0} is said to be a fixed point of f if \mathbf{f(x_0)=x_0}. Suppose that f is a differentiable function such that f(0)>0 and f(1)=1. Show that if f'(1)>1, then f has a fixed point in the interval (0,1).
  5. Let A be the set of all functions \mathbf{f:\mathbb{R} to \mathbb{R}} such that f(xy)=xf(y) for all \mathbf{x,y \in \mathbb{R}}.(a) If \mathbf{f \in A} then show that f(x+y)=f(x)+f(y) for all x,y \mathbf{\in \mathbb{R}}(b) For \mathbf{g,h \in A}, define a function \( \mathbf{g \circ h} \) by \mathbf{(g \circ h)(x)=g(h(x))} for \mathbf{x \in \mathbb{R}}. Prove that \mathbf{g \circ h} is in A and is equal to \mathbf{h \circ g}.
  6. Consider the equation \mathbf{n^2+(n+1)^4=5(n+2)^3}(a) Show that any integer of the form 3m+1 or 3m+2 can not be a solution of this equation.(b) Does the equation have a solution in positive integers?
  7. Consider a rectangular sheet of paper ABCD such that the lengths of AB and AD are respectively 7 and 3 centimetres. Suppose that B’ and D’ are two points on AB and AD respectively such that if the paper is folded along B’D’ then A falls on A’ on the side DC. Determine the maximum possible area of the triangle AB’D’.
  8. Take r such that \mathbf{1\le r\le n}, and consider all subsets of r elements of the set \mathbf{{1,2,\ldots,n}}. Each subset has a smallest element. Let F(n,r) be the arithmetic mean of these smallest elements. Prove that: \mathbf{F(n,r)={n+1\over r+1}}.
  9. Let \mathbf{f: \mathbb{R}^2 to \mathbb{R}^2} be a function having the following property: For any two points A and B in \mathbf{\mathbb{R}^2}, the distance between A and B is the same as the distance between the points f(A) and f(B).Denote the unique straight line passing through A and B by l(A,B)(a) Suppose that C,D are two fixed points in \mathbf{\mathbb{R}^2}. If X is a point on the line l(C,D), then show that f(X) is a point on the line l(f(C),f(D)).(b) Consider two more point E and F in \mathbf{\mathbb{R}^2} and suppose that l(E,F) intersects l(C,D) at an angle \mathbf{\alpha}. Show that l(f(C),f(D)) intersects l(f(E),f(F)) at an angle alpha. What happens if the two lines l(C,D) and l(E,F) do not intersect? Justify your answer.
  10. There are 100 people in a queue waiting to enter a hall. The hall has exactly 100 seats numbered from 1 to 100. The first person in the queue enters the hall, chooses any seat and sits there. The n-th person in the queue, where n can be 2, . . . , 100, enters the hall after (n-1)-th person is seated. He sits in seat number n if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which 100 seats can be filled up, provided the 100-th person occupies seat number 100.