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May 11, 2014

ISI B.Stat, B.Math Paper 2014 Subjective| Problems & Solution

Here, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

In a class there are $100$ student. We define $\mathbf { A_i} $ as the number of friends of $\mathbf { i^{th} }$ student and $\mathbf { C_i }$ as the number of students who has at least i friends. Prove that $\mathbf { \sum_1^{100} A_i = \sum_0^{99} C_i }$

Problem 2:

$PQR$ be a triangle. Take a point $A$ on or inside the triangle. Let $f(x, y) = ax + by + c$. Show that $\mathbf { f(A) \le max \{ f(P), f(Q) , f(R)\} }$
Solution

Problem 3:

Let $ \mathbf { y = x^4 + ax^3 + bx^2 + cx +d}$ (where $a,b,c,d,\in \mathbb{R})$. it is given that the functions cuts the $x$ axis at least $3$ distinct points. Then show that it either cuts the $x$ axis at $4$ distinct point or $3$ distinct point and at any one of these three points we have a maxima or minima.
Solution

Problem 4:

Let $f(x)$ and $g(x)$ be twice differentiable non decreasing functions defined on an interval $(a,b)$ such that for each $x \in (a,b)$, $f''(x) = g(x)$ and $g''(x) = f(x)$ and also that $f(x) . g(x)$ is a linear in $x$ on $(a,b)$ . Then show that $f(x) = g(x) = 0$.

Problem 5:

Prove that sum of any $12$ consecutive integers cannot be perfect square. Give an example where sum of $11$ consecutive integers is a perfect square.
Solution

Problem 6:

$\mathbf { A = \{(x, y) : x = u + v , y = v , u^2 + v^2 \le 1\} } $ . Then what is the maximum length of a line segment enclosed in this area.

Problem 7:

Let $f(x)$ be a non decreasing function defined on the domain $\mathbf {[0, \infty) }$ . Then show that if $\mathbf { 0\le x < y < z < \infty , (z-x) \int_y^z f(u) du \ge (z-y) \int_x^z f(u) du }$

Problem 8:

$n (> 1)$ lotus leaf's are arranged in a circle. A frog jumps from a particular leaf by the following rule: It always moves clockwise. From staring point it skips $1$ leaf and jumps to the next. Then it skips $2$ leaves and jumps to the following. That is in the $3rd$ jump it skips $3$ leaves and $4th$ jump it skips $4$ leaves and so on. In this manner it keeps moving round and round the circle of leaves. It may go to one leaf more than once. If it reaches each leaf at least once then n (the number of leaves) cannot be odd.
Solution

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22 comments on “ISI B.Stat, B.Math Paper 2014 Subjective| Problems & Solution”

  1. In the 1st problem, the summation of Ai will be over 1 to 100

    In the 2nd one, it is to be shown that f(A) ≤ max {f(P),f(Q),f(R)}

    In the 4th one, it is given that f(x)g(x) is a linear function

  2. Sir, i solved only 4 problems correctly,may i have a chance for ISI??(75% in objective)
    This problems are number 4,5,6,7

  3. I don't think I have understood the first question correctly .
    Consider a situation where each student has only one friend. This is possible if we see 100 students as 50 pairs where in each pair, each student is friend of the other.
    In that case sum_{i=1}^{100}{A}_{i}  = 100 . Also {C}_{0} = 100 because all students have more than 0 friends. Note , {C}_{1} = 100 because all students have at least 1 friend.
    Therefore sum_{i=0}^{99}{C}_{i} = 200 . This counter example hence proves the given equation false . But we are required to prove it true . I think I have made a mistake in understanding the question .

    1. In the first question ci is no. of students having strictly greater than i friends. Or the summation of ci should run from 1 to 99. As such the problem is wrong.

      1. Yes, I agree, I tried to use induction to solve the problem, and it returns me an ambiguity. The solution will be correct, only if, as you have stated, ci summn. runs from 1 to 99

    1. Let u be Kcos(A) and v be Ksin(A), where mod(K) is less than 1. By solving the Q, we obtain an equation of the type x^2+ (2)y^2- 2xy = K^2, this is the equation of an ellipse,,,

  4. Sir,minimum how many questions should i attempt in the objective and the subjective paper so as to get selected for the interview?

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