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# ISI B.Stat B.Math 2021 Objective Paper | Problems & Solutions

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

• (A) 5
• (B) 6
• (C) 7
• (D) 8

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

• (A) $\frac{35}{8}$.
• (B) $5$
• (C) $\frac{49}{8}$
• (D)$\frac{55}{8}$

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.

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