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ISI B.Stat B.Math 2021 Subjective Paper | Problems & Solutions

In this post, you will find ISI B.Stat B.Math 2021 Subjective Paper with Problems and Solutions. This is a work in progress, so the solutions and discussions will be uploaded soon. You may share your solutions in the comments below.

[Work in Progress]

Problem 1:

There are three cities each of which has exactly the same number of citizens, say $n$. Every citizen in each city has exactly a total of $n+1$ friends in the other two cities. Show that there exist, three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).

Problem 2:

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function satisfying $f(0) \neq 0=f(1)$, Assume also that $f$ satisfies equations $(\mathrm{A})$ and $(\mathrm{B})$ below.

$f(x y)=f(x)+f(y)-f(x) f(y)$ ..... (A)

$f(x-y) f(x) f(y)=f(0) f(x) f(y)$ .... (B)

(i) Determine explicitly the set ${f(a): a \in \mathbb{Z}}$.
(ii) Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set ${b: f(b) \neq 0}$ is infinite.

Problem 3:

Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a_{1}+\frac{a_{2}}{2 !}+\frac{a_{3}}{3 !}+\cdots+\frac{a_{n}}{n !}$$
where $a_{n}$ are integers such that $0 \leq a_{n} \leq n-1$ for all $n>1$.

Problem 4:

Let $g:(0, \infty) \rightarrow(0, \infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x))=x$ for all $x>0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

Problem 5:

Let $a_{0}, a_{1}, \cdots, a_{19} \in \mathbb{R}$ and
$$P(x)=x^{20}+\sum_{i=0}^{19} a_{i} x^{3}, \quad x \in \mathbb{R} .$$
If $P(x)=P(-x)$ for all $x \in \mathbb{R}$, and
$$P(k)=k^{2}, \text { for } k=0,1,2 \cdots, 9$$

then find
$$\lim _{x \rightarrow 0} \frac{P(x)}{\sin ^{2} x}$$

Problem 6:

If a given equilateral triangle $\Delta$ of side length a lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.

Problem 7:

Let $a, b, c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a+b+c=6$ and $a b+b c+a c=9 .$ Suppose $a<b<c$, Show that $0<a<1<b<3<c<4$

Problem 8:

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is $6 \mathrm{~m}$. The square at the bottom has side length $2 \mathrm{~m}$ and the top square has a side length $8 \mathrm{~m}$. Water is filled in at a rate of $\frac{19}{3}$ cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?

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