In this post, you will find ISI B.Stat B.Math 2021 Objective 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
The number of ways one can express $2^{2} 3^{3} 5^{5} 7^{7}$ as a product of two numbers $a$ and $b$, where $\text{gcd}(a, b)=1$, and $1<a<b$, is
Discussion
Problem 2
The sum of all the solutions of $ 2 + \log_2 (x-2) = \log_{(x-2)} 8$ in the interval $(2, \infty)$ is
Problem 3
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that
$$
f(x+1)=\frac{1}{2} f(x) \text { for all } x \in \mathbb{R}
$$
and let $a_{n}=\int_{0}^{n} f(x) d x$ for all integers $n \geq 1$. Then:
(A) $\lim {n \rightarrow \infty} a_{n}$ exists and equals $\int_{0}^{1} f(x) d x$.
(B) $\lim {n \rightarrow \infty} a_{n}$ does not exist.
(C) $\lim {n \rightarrow \infty} a_{n}$ exists if and only if $|\int_{0}^{1} f(x) d x|<1$.
(D) $\lim {n \rightarrow \infty} a_{n}$ exists and equals $2 \int_{0}^{1} f(x) d x$.
Problem 4
Consider the curves $x^{2}+y^{2}-4 x-6 y-12=0,9 x^{2}+4 y^{2}-900=0$ and $y^{2}-6 y-6 x+51=0 .$ The maximum number of disjoint regions into which these curves divide the $X Y$ -plane (excluding the curves themselves), is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .
Problem 5
A box has $13$ distinct pairs of socks. Let $p_{r}$ denote the probability of having at least one matching pair among $a$ bunch of $r$ socks drawn at random from the box. If $r_{0}$ is the maximum possible value of $r$ such that $p_{r}<1$, then the value of $p_{r_{0}}$ is
(A) $1-\frac{12}{ 26C_{12} }$.
(B) $1-\frac{13}{ 26C_{13} }$.
(C) $1-\frac{2^{13}}{ 26C_{13} } .$
(D) $1-\frac{2^{12}}{26C_{12}}$.
Problem 6
Let $a, b, c, d>0$, be any real numbers. Then the maximum prossible value of $c x+d y$, over all points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, must the
(A) $\sqrt{a^{2} c^{2}+b^{2} d^{2}}$.
(B) $\sqrt{a^{2} b^{2}+c^{2} d^{2}}$.
(C) $\sqrt{\frac{a^{2} c^{2}+b^{2} d^{2}}{a^{2}+b^{2}}}$.
(D) $\sqrt{\frac{a^{2} b^{2}+c^{2} d^{2}}{c^{2}+d^{2}}}$.
Problem 7
Let $f(x)=\sin x+\alpha x, x \in \mathbb{R}$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha>1$ or $\alpha<-1$.
(B) $\alpha \geq 1$ or $\alpha \leq-1$.
(C) $a \geq 1$ or $\alpha<-1$.
(D) $\alpha>1$ or $\alpha \leq-1$.
Problem 8
The Value of
$$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots+\frac{1}{1+2+3+\cdots 2021}$$ is
(A) $\frac{2021}{1010}$.
(B) $\frac{2021}{1011}$.
(C) $\frac{2021}{1012}$.
(D) $\frac{2021}{1013}$.
Problem 9
The volume of the region $S=\{(x, y, z):|x|+2|y|+3|z| \leq 6\}$ is
(A) 36 .
(B) 48 .
(C) 72
(D) 6 .
Problem 10:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $\frac{d^{2} f(x)}{d x^{2}}$ is positive for all $x \in \mathbb{R}$, and suppose $f(0)=1, f(1)=4$. Which of the following is not a possible value of $f(2)$ ?
(A) 7 .
(B) 8 .
(C) 9 .
(D) $10$
Problem 11:
Let, $f(x)=e^{-|x|}, x \in \mathbb{R}$,
and $g(\theta)=\int_{-1}^{1} f\left(\frac{x}{\theta}\right) d x, \theta \neq 0$
Then , $\lim _{\theta \rightarrow 0} \frac{g(\theta)}{\theta}$
(A) equals 0 .
(B) equals $+\infty$.
(C) equals 2 .
(D) does not exist.
Problem 12:
The number of different ways to colour the vertices of a square $P Q R S$ using one or more colours from the set \{Red, Blue, Green, Yellow \}$, such that no two adjacent vertices have the same colour is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 84 .
Problem 13:
Define $a=p^{3}+p^{2}+p+11$ and $b=p^{2}+1$, where $p$ is any prime number. Let $d=g c d(a, b)$. Then the set of possible values of $d$ is
(A) ${1,2,5}$.
(B) ${2,5,10}$.
(C) ${1,5,10}$.
(D) ${1,2,10}$.
Problem 14:
Consider all $2 \times 2$ matrices whose entries are distinct and taken from the set $\{1,2,3,4\}$. The sum of determinants of all such matrices is
(A) 24 .
(B) 10 .
(C) 12 .
(D) 0 .
Problem 15:
Let $a, b, c$ and $d$ be four non-negative real numbers where $a+b+c+d= 1$. The number of different ways one can choose these numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=\max \{a, b, c, d\}$ is
(A) 1 .
(B) 5 .
(C) 11 .
(D) 15 .
Problem 16:
The polynomial $x^{4}+4 x+c=0$ has at least one real root if and only if
(A) $c<2$.
(B) $c \leq 2$.
(C) $c<3$.
(D) $c \leq 3$.
problem 17:
The number of all integer solutions of the equation $x^{2}+y^{2}+x-y=$ 2021 is
(A) 5 .
(B) 7 .
(C) 1 .
(D) $0 .$
Problem 18:
The number of different values of $a$ for which the equation $x^{3}-x+a=$ 0 has two identical real roots is
(A) 0 .
(B) 1 .
(C) $2 .$
(D) 3 .
Problem 19:
Suppose $f(x)$ is a twice differentiable function on $[a, b]$ such that $f(a)=0=f(b)$
and $x^{2} \frac{d^{2} f(x)}{d x^{2}}+4 x \frac{d f(x)}{d x}+2 f(x)>0$ for all $x \in(a, b)$
Then,
(A) $f$ is negative for all $x \in(a, b)$.
(B) $f$ is positive for all $x \in(a, b)$.
(C) $f(x)=0$ for exactly one $x \in(a, b)$.
(D) $f(x)=0$ for at least two $x \in(a, b)$.
Problem 20:
Consider the following two subsets of $\mathbb{C}$ :
$A=\{\frac{1}{z}:|z|=2\}$ and $B=\{\frac{1}{z}:|z-1|=2\} .$
Then ,
(A) $A$ is a circle, but $B$ is not a circle.
(B) $B$ is a circle, but $A$ is not a circle.
(C) $A$ and $B$ are both circles.
(D) Neither $A$ nor $B$ is a circle.
Problem 21:
For a positive integer $n$, the equation
$$x^{2}=n+y^{2}, \quad x, y$$ integers
does not have a solution if and only if
(A) $n=2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .
problem 22:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be any twice differentiable function such that its second
derivative is continuous and $\frac{d f(x)}{d x} \neq 0$ for all $x \neq 0$.
If $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=\pi$, then ,
(A) for all $x \neq 0, \quad f(x)>f(0)$.
(B) for all $x \neq 0, \quad f(x)0$
(C) for all $x, \quad \frac{d^{2} f(x)}{d x^{2}}>0$
(D) for all $x, \quad \frac{d^{2} f(x)}{d x^{2}}<0$.
Problem 23:
Let us denote the fractional part of a real number $x$ by ${x}$ (note:
${x}=x-[x]$ where $[x]$ is the integer part of $x$ ). Then,
$$\lim _{n \rightarrow \infty}\{(3+2 \sqrt{2})^{n}\}$$
(A) equals 0.
(D) equals 1 .
(C) equals $\frac{1}{2}$.
(D) does not exist.
Problem 24:
Let,
$$p(x)=x^{3}-3 x^{2}+2 x, x \in \mathbb{R}$$
$f_{0}(x)= \begin{cases}\int_{0}^{x} p(t) d t, & x \geq 0 \ -\int_{x}^{0} p(t) d t, & x<0\end{cases}$,
$f_{1}(x)=e^{f_{0}(x)}, \quad f_{2}(x)=e^{f_{1}(x)}, \quad \ldots \quad, f_{n}(x)=e^{f_{n-1}(x)}$
How many roots does the equation $\frac{d f_{n}(x)}{d x}=0$ have in the interval $(-\infty, \infty) ?$
(A) 1 .
(B) 3 .
(C) $n+3$.
(D) $3 n$.
Problem 25:
For $0 \leq x<2 \pi$, the number of solutions of the equation
$$\sin ^{2} x+2 \cos ^{2} x+3 \sin x \cos x=0$$
is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .
Problem 26:
Let $f: \mathbb{R} \rightarrow[0, \infty)$ be a continuous function such that
$f(x+y)=f(x) f(y)$
for all $x, y \in \mathbb{R}$. Suppose that $f$ is differentiable at $x=1$ and
$\left.\frac{d f(x)}{d x}\right|_{x=1}=2 .$
Then, the value of $f(1) \log _{e} f(1)$ is
(A) $e$.
(B) 2 .
$(\mathrm{C}) \log _{e} 2$
(D) 1.
Problem 27:
The expression $\sum_{k=0}^{10} 2^{k} \tan \left(2^{k}\right)$ equals
(A) $\cot 1+2^{11} \cot \left(2^{11}\right)$
(B) $\cot 1-2^{10} \cot \left(2^{10}\right)$.
(C) $\cot 1+2^{10} \cot \left(2^{10}\right)$.
(D) $\cot 1-2^{11} \cot \left(2^{11}\right)$.
Problem 28:
If the maximum and minimum values of $\sin ^{6} x+\cos ^{6} x$, as $x$ takes all
real values, are $a$ and $b$, respectively, then $a-b$ equals
(A) $\frac{1}{2}$.
(B) $\frac{2}{3}$.
(C) $\frac{3}{4}$.
(D) 1 .
Problem 29:
If two real numbers $x$ and $y$ satisfy $(x+5)^{2}+(y-10)^{2}=196$, then the minimum possible value of
$x^{2}+2 x+y^{2}-4 y$ is
(A) $271-112 \sqrt{5}$.
(B) $14-4 \sqrt{5}$.
(C) $276-112 \sqrt{5}$.
(D) $9-4 \sqrt{5}$.
Problem 30:
Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by
$f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0, \ 0, & x=0 .\end{cases}$,
Then,
(A) $f$ is discontinuous.
(B) $f$ is continuous but not differentiable.
(C) $f$ is differentiable and its derivative is discontinuous.
(D) $f$ is differentiable and its derivative is continuous.
In this post, you will find ISI B.Stat B.Math 2021 Objective 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
The number of ways one can express $2^{2} 3^{3} 5^{5} 7^{7}$ as a product of two numbers $a$ and $b$, where $\text{gcd}(a, b)=1$, and $1<a<b$, is
Discussion
Problem 2
The sum of all the solutions of $ 2 + \log_2 (x-2) = \log_{(x-2)} 8$ in the interval $(2, \infty)$ is
Problem 3
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that
$$
f(x+1)=\frac{1}{2} f(x) \text { for all } x \in \mathbb{R}
$$
and let $a_{n}=\int_{0}^{n} f(x) d x$ for all integers $n \geq 1$. Then:
(A) $\lim {n \rightarrow \infty} a_{n}$ exists and equals $\int_{0}^{1} f(x) d x$.
(B) $\lim {n \rightarrow \infty} a_{n}$ does not exist.
(C) $\lim {n \rightarrow \infty} a_{n}$ exists if and only if $|\int_{0}^{1} f(x) d x|<1$.
(D) $\lim {n \rightarrow \infty} a_{n}$ exists and equals $2 \int_{0}^{1} f(x) d x$.
Problem 4
Consider the curves $x^{2}+y^{2}-4 x-6 y-12=0,9 x^{2}+4 y^{2}-900=0$ and $y^{2}-6 y-6 x+51=0 .$ The maximum number of disjoint regions into which these curves divide the $X Y$ -plane (excluding the curves themselves), is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .
Problem 5
A box has $13$ distinct pairs of socks. Let $p_{r}$ denote the probability of having at least one matching pair among $a$ bunch of $r$ socks drawn at random from the box. If $r_{0}$ is the maximum possible value of $r$ such that $p_{r}<1$, then the value of $p_{r_{0}}$ is
(A) $1-\frac{12}{ 26C_{12} }$.
(B) $1-\frac{13}{ 26C_{13} }$.
(C) $1-\frac{2^{13}}{ 26C_{13} } .$
(D) $1-\frac{2^{12}}{26C_{12}}$.
Problem 6
Let $a, b, c, d>0$, be any real numbers. Then the maximum prossible value of $c x+d y$, over all points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, must the
(A) $\sqrt{a^{2} c^{2}+b^{2} d^{2}}$.
(B) $\sqrt{a^{2} b^{2}+c^{2} d^{2}}$.
(C) $\sqrt{\frac{a^{2} c^{2}+b^{2} d^{2}}{a^{2}+b^{2}}}$.
(D) $\sqrt{\frac{a^{2} b^{2}+c^{2} d^{2}}{c^{2}+d^{2}}}$.
Problem 7
Let $f(x)=\sin x+\alpha x, x \in \mathbb{R}$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha>1$ or $\alpha<-1$.
(B) $\alpha \geq 1$ or $\alpha \leq-1$.
(C) $a \geq 1$ or $\alpha<-1$.
(D) $\alpha>1$ or $\alpha \leq-1$.
Problem 8
The Value of
$$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots+\frac{1}{1+2+3+\cdots 2021}$$ is
(A) $\frac{2021}{1010}$.
(B) $\frac{2021}{1011}$.
(C) $\frac{2021}{1012}$.
(D) $\frac{2021}{1013}$.
Problem 9
The volume of the region $S=\{(x, y, z):|x|+2|y|+3|z| \leq 6\}$ is
(A) 36 .
(B) 48 .
(C) 72
(D) 6 .
Problem 10:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $\frac{d^{2} f(x)}{d x^{2}}$ is positive for all $x \in \mathbb{R}$, and suppose $f(0)=1, f(1)=4$. Which of the following is not a possible value of $f(2)$ ?
(A) 7 .
(B) 8 .
(C) 9 .
(D) $10$
Problem 11:
Let, $f(x)=e^{-|x|}, x \in \mathbb{R}$,
and $g(\theta)=\int_{-1}^{1} f\left(\frac{x}{\theta}\right) d x, \theta \neq 0$
Then , $\lim _{\theta \rightarrow 0} \frac{g(\theta)}{\theta}$
(A) equals 0 .
(B) equals $+\infty$.
(C) equals 2 .
(D) does not exist.
Problem 12:
The number of different ways to colour the vertices of a square $P Q R S$ using one or more colours from the set \{Red, Blue, Green, Yellow \}$, such that no two adjacent vertices have the same colour is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 84 .
Problem 13:
Define $a=p^{3}+p^{2}+p+11$ and $b=p^{2}+1$, where $p$ is any prime number. Let $d=g c d(a, b)$. Then the set of possible values of $d$ is
(A) ${1,2,5}$.
(B) ${2,5,10}$.
(C) ${1,5,10}$.
(D) ${1,2,10}$.
Problem 14:
Consider all $2 \times 2$ matrices whose entries are distinct and taken from the set $\{1,2,3,4\}$. The sum of determinants of all such matrices is
(A) 24 .
(B) 10 .
(C) 12 .
(D) 0 .
Problem 15:
Let $a, b, c$ and $d$ be four non-negative real numbers where $a+b+c+d= 1$. The number of different ways one can choose these numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=\max \{a, b, c, d\}$ is
(A) 1 .
(B) 5 .
(C) 11 .
(D) 15 .
Problem 16:
The polynomial $x^{4}+4 x+c=0$ has at least one real root if and only if
(A) $c<2$.
(B) $c \leq 2$.
(C) $c<3$.
(D) $c \leq 3$.
problem 17:
The number of all integer solutions of the equation $x^{2}+y^{2}+x-y=$ 2021 is
(A) 5 .
(B) 7 .
(C) 1 .
(D) $0 .$
Problem 18:
The number of different values of $a$ for which the equation $x^{3}-x+a=$ 0 has two identical real roots is
(A) 0 .
(B) 1 .
(C) $2 .$
(D) 3 .
Problem 19:
Suppose $f(x)$ is a twice differentiable function on $[a, b]$ such that $f(a)=0=f(b)$
and $x^{2} \frac{d^{2} f(x)}{d x^{2}}+4 x \frac{d f(x)}{d x}+2 f(x)>0$ for all $x \in(a, b)$
Then,
(A) $f$ is negative for all $x \in(a, b)$.
(B) $f$ is positive for all $x \in(a, b)$.
(C) $f(x)=0$ for exactly one $x \in(a, b)$.
(D) $f(x)=0$ for at least two $x \in(a, b)$.
Problem 20:
Consider the following two subsets of $\mathbb{C}$ :
$A=\{\frac{1}{z}:|z|=2\}$ and $B=\{\frac{1}{z}:|z-1|=2\} .$
Then ,
(A) $A$ is a circle, but $B$ is not a circle.
(B) $B$ is a circle, but $A$ is not a circle.
(C) $A$ and $B$ are both circles.
(D) Neither $A$ nor $B$ is a circle.
Problem 21:
For a positive integer $n$, the equation
$$x^{2}=n+y^{2}, \quad x, y$$ integers
does not have a solution if and only if
(A) $n=2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .
problem 22:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be any twice differentiable function such that its second
derivative is continuous and $\frac{d f(x)}{d x} \neq 0$ for all $x \neq 0$.
If $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=\pi$, then ,
(A) for all $x \neq 0, \quad f(x)>f(0)$.
(B) for all $x \neq 0, \quad f(x)0$
(C) for all $x, \quad \frac{d^{2} f(x)}{d x^{2}}>0$
(D) for all $x, \quad \frac{d^{2} f(x)}{d x^{2}}<0$.
Problem 23:
Let us denote the fractional part of a real number $x$ by ${x}$ (note:
${x}=x-[x]$ where $[x]$ is the integer part of $x$ ). Then,
$$\lim _{n \rightarrow \infty}\{(3+2 \sqrt{2})^{n}\}$$
(A) equals 0.
(D) equals 1 .
(C) equals $\frac{1}{2}$.
(D) does not exist.
Problem 24:
Let,
$$p(x)=x^{3}-3 x^{2}+2 x, x \in \mathbb{R}$$
$f_{0}(x)= \begin{cases}\int_{0}^{x} p(t) d t, & x \geq 0 \ -\int_{x}^{0} p(t) d t, & x<0\end{cases}$,
$f_{1}(x)=e^{f_{0}(x)}, \quad f_{2}(x)=e^{f_{1}(x)}, \quad \ldots \quad, f_{n}(x)=e^{f_{n-1}(x)}$
How many roots does the equation $\frac{d f_{n}(x)}{d x}=0$ have in the interval $(-\infty, \infty) ?$
(A) 1 .
(B) 3 .
(C) $n+3$.
(D) $3 n$.
Problem 25:
For $0 \leq x<2 \pi$, the number of solutions of the equation
$$\sin ^{2} x+2 \cos ^{2} x+3 \sin x \cos x=0$$
is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .
Problem 26:
Let $f: \mathbb{R} \rightarrow[0, \infty)$ be a continuous function such that
$f(x+y)=f(x) f(y)$
for all $x, y \in \mathbb{R}$. Suppose that $f$ is differentiable at $x=1$ and
$\left.\frac{d f(x)}{d x}\right|_{x=1}=2 .$
Then, the value of $f(1) \log _{e} f(1)$ is
(A) $e$.
(B) 2 .
$(\mathrm{C}) \log _{e} 2$
(D) 1.
Problem 27:
The expression $\sum_{k=0}^{10} 2^{k} \tan \left(2^{k}\right)$ equals
(A) $\cot 1+2^{11} \cot \left(2^{11}\right)$
(B) $\cot 1-2^{10} \cot \left(2^{10}\right)$.
(C) $\cot 1+2^{10} \cot \left(2^{10}\right)$.
(D) $\cot 1-2^{11} \cot \left(2^{11}\right)$.
Problem 28:
If the maximum and minimum values of $\sin ^{6} x+\cos ^{6} x$, as $x$ takes all
real values, are $a$ and $b$, respectively, then $a-b$ equals
(A) $\frac{1}{2}$.
(B) $\frac{2}{3}$.
(C) $\frac{3}{4}$.
(D) 1 .
Problem 29:
If two real numbers $x$ and $y$ satisfy $(x+5)^{2}+(y-10)^{2}=196$, then the minimum possible value of
$x^{2}+2 x+y^{2}-4 y$ is
(A) $271-112 \sqrt{5}$.
(B) $14-4 \sqrt{5}$.
(C) $276-112 \sqrt{5}$.
(D) $9-4 \sqrt{5}$.
Problem 30:
Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by
$f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0, \ 0, & x=0 .\end{cases}$,
Then,
(A) $f$ is discontinuous.
(B) $f$ is continuous but not differentiable.
(C) $f$ is differentiable and its derivative is discontinuous.
(D) $f$ is differentiable and its derivative is continuous.
