Cheenta

Academy for Gifted Students

How Cheenta works to ensure student success?

Explore the Back-StoryIn 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 $P(x) = x^{20} + \sum_{i=0}^{19} a_{i}x^{i}, x \in \mathbb{R}$

If $P(x)=P(-x)$ for all $x \in \mathbb{R}$ and $P(k)=k^{2}$ for $k=0,1,2,...,9$

then find

**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 $0<a<1<b<3<c<4$

Solution:

**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?

** **

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 $P(x) = x^{20} + \sum_{i=0}^{19} a_{i}x^{i}, x \in \mathbb{R}$

If $P(x)=P(-x)$ for all $x \in \mathbb{R}$ and $P(k)=k^{2}$ for $k=0,1,2,...,9$

then find

**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 $0<a<1<b<3<c<4$

Solution:

**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?

** **

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIALAcademic Programs

Free Resources

Why Cheenta?

Online Live Classroom Programs

Online Self Paced Programs [*New]

Past Papers

More

How to solve Q 3.?