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Explore the Back-StoryHere, 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 inqualities

$a-b^{2} \geq \frac{1}{4}$, $b-c^{2} \geq \frac{1}{4}$, $c-d^{2} \geq \frac{1}{4}$, $d-a^{2} \geq \frac{1}{4}$

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 reflected 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)$\left(\begin{array}{l}8 \ 2\end{array}\right)\left(\begin{array}{c}17 \ 7\end{array}\right)$

**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| 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) $55^{\circ}$.

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 inqualities

$a-b^{2} \geq \frac{1}{4}$, $b-c^{2} \geq \frac{1}{4}$, $c-d^{2} \geq \frac{1}{4}$, $d-a^{2} \geq \frac{1}{4}$

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 reflected 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)$\left(\begin{array}{l}8 \ 2\end{array}\right)\left(\begin{array}{c}17 \ 7\end{array}\right)$

**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| 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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