- 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 \(\mathbf { \sum_1^{100} A_i = \sum_0^{99} C_i }\)
- Let 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
- Let \(\mathbf { y = x^4 + ax^3 + bx^2 + cx +d , a,b,c,d,e \mathbb{R}}\). 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
- Let f(x) and g(x) be twice differentiable non decreasing functions such that f”(x) = g(x) and g”(x) = f(x) and f(x) . g(x) is a linear function. Then show that f(x) = g(x) = 0.
- 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
- \(\mathbf { A = {(x, y) , x = u + v , y = v , u^2 + v^2 \le 1} } \) . Then what is the maximum length of a line segment enclosed in this area
- Let f(x) be a non decreasing function defined on the domain \(\mathbf {[0, \infty) }\) . Then show that if \(\mathbf { 0\le x < y < z < \infty , (z-x) \int_y^z f(u) du \ge (z-y) \int_x^z f(u) du }\)
- n (> 1) lotus leafs 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

(These problems are collected from student feedback. Courtesy Krishnendu Bhowmick)

*Related*

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

Thanks for the clarification.

Problem 4th is f(x), g(x) are non-decreasing functions. Wouldn’t work without that?

Was it given? ‘Non – decreasing’ part?

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

It is possible.5+ is better.

Yeah, the question specifically stated f and g are non-decreasing.

sir pls give inside out of isi b.stats interview… i will be highly obliged…. thanks .

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 $latex sum_{i=1}^{100}{A}_{i} = 100$ . Also $latex {C}_{0} = 100 $ because all students have more than $latex 0 $ friends. Note , $latex {C}_{1} = 100$ because all students have at least 1 friend.

Therefore $latex 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 .

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.

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

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What is the answer to the 6th question?

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,,,

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

22/30 and 4/6 should suffice

For the undergraduate course?I’m never able to solve more than 12 in the objective paper

Yes for B Stat Bmath entrance

I didn’t expect ISI questions to be this easy

sir the frog and leaf problem. ….can it be solved without using modulo? ?

Possible… But it would be modular arithmetic in disguise

Plz give the solution of the first problem.