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ISI B.Stat & B.Math 2014 Objective Paper| Problems & Solutions

Here, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

The system of inequalities
$$
a-b^{2} \geq \frac{1}{4}, b-c^{2} \geq \frac{1}{4}, c-d^{2} \geq \frac{1}{4}, d-a^{2} \geq \frac{1}{4} \quad \text { has }
$$
(A) no solutions
(B) exactly one solution
(C) exactly two solutions
(D) infinitely many solutions.

Problem 2:

Let $\log _{12} 18=a$. Then $\log {24} 16$ is equal to
(A) $\frac{8-4 a}{5-a}$
(B) $\frac{1}{3+a}$
(C) $\frac{4 a-1}{2+3 a}$
(D) $\frac{8-4 a}{5+a}$

Problem 3:

The number of solutions of the equation $\tan x+\sec x=2 \cos x,$ where $0 \leq x \leq \pi$ is
(A) $0$;
(B) $1$;
(C) $2$;
(D) $3$.

Problem 4:

Using only the digits $2,3$ and $9,$ how many six digit numbers can be formed which are divisible by $6$ ?
(A) $41$;
(B) $80$;
(C) $81$;
(D) $161$.

Problem 5:

What is the value of the following integral?

$$
\int_{\frac{1}{2014}}^{2014} \frac{\tan ^{-1} x}{x} d x
$$

(A) $\frac{\pi}{4} \log 2014$;
(B) $\frac{\pi}{2} \log 2014$;
(C) $\pi \log 2014$;
(D) $\frac{1}{2} \log 2014$.

Problem 6:

A light ray travelling along the line $y=1,$ is refiected by a mirror placed along the line $x=2 y .$ The reflected ray travels along the line
(A) $4 x-3 y=5$;
(B) $3 x-4 y=2$;
(C) $x-y=1$;
(D) $2 x-3 y=1$.

Problem 7:

For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Then the number of real solutions of $|2 x-[x]|=4$ is
(A) $1$;
(B) $2$;
(C) $3$;
(D) $4$ .

Problem 8:

What is the ratio of the areas of the regular pentagons inscribed inside and circumscribed around a given circle?
(A) $\cos 36^{\circ}$;
(B) $\cos ^{2} 36^{\circ}$;
(C) $\cos ^{2} 54^{\circ}$;
(D) $\cos ^{2} 72^{\circ}$.

Problem 9:

Let $z_{1}, z_{2}$ be nonzero complex numbers satisfying $\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right| .$ The circumcentre of the triangle with the points $z_{1}, z_{2},$ and the origin as its vertices is given by
(A) $\frac{1}{2}\left(z_{1}-z_{2}\right)$;
(B) $\frac{1}{3}\left(z_{1}+z_{2}\right)$;
(C) $\frac{1}{2}\left(z_{1}+z_{2}\right)$;
(D) $\frac{1}{3}\left(z_{1}-z_{2}\right)$.

Problem 10:

In how many ways can 20 identical chocolates be distributed among 8 students so that each student gets at least one chocolate and exactly two students get at least two chocolates each?
(A) $308$;
(B) $364$;
(C) $616$;
(D) $\left(\begin{array}{c} 8\\2 \end{array} \right) \left(\begin{array}{c} 17 \\ 7 \end{array}\right)$.

Problem 11:

Two vertices of a square lie on a circle of radius $r,$ and the other two vertices lie on a tangent to this circle. Then, each side of the square is

(A) $\frac{3 r}{2}$;
(B) $\frac{4 r}{3}$;
(C) $\frac{6 r}{5}$;
(D) $\frac{8 r}{5}$.

Problem 12:

Let $P$ be the set of all numbers obtained by multiplying five distinct integers between 1 and $100 .$ What is the largest integer $n$ such that $2^{n}$ divides at least one element of $P ?$
(A) $8$;
(B) $20$;
(C) $24$;
(D) $25$.

Problem 13:

Consider the function $f(x)=a x^{3}+b x^{2}+c x+d,$ where $a, b, c$ and $d$ are real numbers with $a>0$. If $f$ is strictly increasing, then the function $g(x)=$ $f^{\prime}(x)-f^{\prime \prime}(x)+f^{\prime \prime \prime}(x)$ is
(A) zero for some $x \in \mathbb{R}$;
(B) positive for all $x \in \mathbb{R}$;
(C) negative for all $x \in \mathbb{R}$;
(D) strictly increasing.

Problem 14:

Let $A$ be the set of all points $(h, k)$ such that the area of the triangle formed by $(h, k),(5,6)$ and (3,2) is 12 square units. What is the least possible length of a line segment joining (0,0) to a point in $A ?$
(A) $\frac{4}{\sqrt{5}}$;
(B) $\frac{8}{\sqrt{5}}$;
(C) $\frac{12}{\sqrt{5}}$;
(D) $\frac{16}{\sqrt{5}}$.

Problem 15:

Let $P$=$\{a b c: a, b, c \text{ positive integers }, a^{2}+b^{2}=c^{2},\text { and }3 \text { divides } c\}$ . What is the largest integer $n$ such that $3^{n}$ divides every element of $P$?
(A) $1$;
(B) $2$;
(C) $3$;
(D) $4$.

Problem 16:

Let $A_{0}=\emptyset$ (the empty set). For each $i=1,2,3, \ldots,$ define the set $A_{i}=$ $A_{i-1} \cup\{A_{i-1}\} .$ The set $A_{3}$ is
(A) $\emptyset$;
(B) $\{\emptyset\}$;
(C) ${\emptyset,\{\emptyset}\}$;
(D) $\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$.

Problem 17:

Let $f(x)=\frac{1}{x-2} \cdot$ The graphs of the functions $f$ and $f^{-1}$ intersect at

(A) $(1+\sqrt{2}, 1+\sqrt{2})$ and $(1-\sqrt{2}, 1-\sqrt{2})$;
(B) $(1+\sqrt{2}, 1+\sqrt{2})$ and $\left(\sqrt{2},-1-\frac{1}{\sqrt{2}}\right)$;
(C) $(1-\sqrt{2}, 1-\sqrt{2})$ and $\left(-\sqrt{2},-1+\frac{1}{\sqrt{2}}\right)$;
(D) $\left(\sqrt{2},-1-\frac{1}{\sqrt{2}}\right)$ and $\left(-\sqrt{2},-1+\frac{1}{\sqrt{2}}\right)$.

Problem 18:

Let $N$ be a number such that whenever you take $N$ consecutive positive integers, at least one of them is coprime to $374 .$ What is the smallest possible value of $N ?$
(A) $4$;
(B) $5$;
(C) $6$;
(D) $7$.

Problem 19:

Let $A_{1}, A_{2}, \ldots, A_{18}$ be the vertices of a regular polygon with 18 sides. How many of the triangles $\Delta A_{i} A_{j} A_{k}, 1 \leq i<j<k \leq 18,$ are isosceles but not equilateral?
(A) $63$;
(B) $70$;
(C) $126$;
(D) $144$.

Problem 20:

The limit $\lim _{x \rightarrow 0} \frac{\sin ^{\alpha} x}{x}$ exists only when
(A) $\alpha \geq 1$;
(B) $\alpha=1$;
(C) $|\alpha| \leq 1$;
(D) $\alpha$ is a positive integer.

Problem 21:

Consider the region $R=\{(x, y): x^{2}+y^{2} \leq 100, \sin (x+y)>0\} .$ What is the area of $R ?$
(A) $25 \pi$;
(B) $50 \pi$;
(C) $50$;
(D) $100 \pi-50$.

Problem 22:

Considcr a cyclic trapezium whose circumcentre is on one of the sides. If the ratio of the two parallel sides is $1: 4,$ what is the ratio of the sum of the two oblique sides to the longer parallel side?

(A) $\sqrt{3}: \sqrt{2}$;
(B) $3: 2$;
(C) $\sqrt{2}: 1$;
(D) $\sqrt{5}: \sqrt{3}$.

Problem 23:

Consider the function $f(x)=\{\log _{e}\left(\frac{4+\sqrt{2 x}}{x}\right)\}^{2}$ for $x>0 .$ Then

(A) $f$ decreases upto some point and increases after that;
(B) $f$ increases upto some point and decreases after that;
(C) $f$ increases initially, then decreases and then again increases;
(D) $f$ decreases initially, then increases and then again decreases.

Problem 24:

What is the number of ordered triplets $(a, b, c),$ where $a, b, c$ are positive integers (not necessarily distinct), such that $a b c=1000 ?$

(A) $64$;
(B) $100$;
(C) $200$;
(D) $560$.

Problem 25:

Let $f:(0, \infty) \rightarrow(0, \infty)$ be a function differentiable at $3,$ and satisfying $f(3)=$ $3 f^{\prime}(3)>0 .$ Then the limit

$$
\lim _{x \rightarrow \infty}\left(\frac{f\left(3+\frac{3}{x}\right)}{f(3)}\right)^{x}
$$

(A) exists and is equal to $3$;
(B) exists and is equal to $e$;
(C) exists and is always equal to $f(3)$;
(D) need not always exist.

Problem 26:

Let $z$ be a non-zero complex number such that $\left|z-\frac{1}{z}\right|=2$. What is the maximum value of $|z| ?$

(A) $1$;
(B) $\sqrt{2}$;
(C) $2$;
(D) $1+\sqrt{2}$.

Problem 27:

The minimum value of

$$
|\sin x+\cos x+\tan x+cosec x+\sec x+\cot x| \text { is }
$$

(A) $0$;
(B) $2 \sqrt{2}-1$;
(C) $2 \sqrt{2}+1$;
(D) $6$.

Problem 28:

For any function $f: X \rightarrow Y$ and any subset $A$ of $Y$, define

$$
f^{-1}(A)={x \in X: f(x) \in A}
$$

Let $A^{c}$ denote the complement of $A$ in $Y$. For subsets $A_{1}, A_{2}$ of $Y$, consider the following statements:

(i) $f^{-1}\left(A_{1}^{c} \cap A_{2}^{c}\right)=\left(f^{-1}\left(A_{1}\right)\right)^{c} \cup\left(f^{-1}\left(A_{2}\right)\right)^{c}$
(ii) If $f^{-1}\left(A_{1}\right)=f^{-1}\left(A_{2}\right)$ then $A_{1}=A_{2}$.

Then,
(A) both (i) and (ii) are always true;
(B) (i) is always true, but (ii) may not always be true;
(C) (ii) is always true, but (i) may not always be true;
(D) neither (i) nor (ii) is always true.

Problem 29:

Let $f$ be a function such that $f^{\prime \prime}(x)$ exists, and $f^{\prime \prime}(x)>0$ for all $x \in[a, b] .$ For any point $c \in[a, b],$ let $A(c)$ denote the area of the region bounded by $y=f(x)$ the tangent to the graph of $f$ at $x=c$ and the lines $x=a$ and $x=b .$ Then

(A) $A(c)$ attains its minimum at $c=\frac{1}{2}(a+b)$ for any such $f$;
(B) $A(c)$ attains its maximum at $c=\frac{1}{2}(a+b)$ for any such $f$;
(C) $A(c)$ attains its minimum at both $c=a$ and $c=b$ for any such $f$;
(D) the points $c$ where $A(c)$ attains its minimum depend on $f$.

Problem 30:

In $\triangle A B C,$ the lines $B P, B Q$ trisect $\angle A B C$ and the lines $C M, C N$ trisect $\angle A C B .$ Let $B P$ and $C M$ intersect at $X$ and $B Q$ and $C N$ intersect at $Y .$ If $\angle A B C=45^{\circ}$ and $\angle A C B=75^{\circ},$ then $\angle B X Y$ is

(A) $45^{\circ}$;
(B) $47 \frac{1^{\circ}}{2}$;
(C) $50^{\circ}$;
(D) $55^{\circ}$.

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