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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2015 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let $\mathbb{C}$ denote the set of complex numbers and $S=\{z \in \mathbb{C} \mid \bar{z}=z^{2}\},$ where $\bar{z}$ denotes the complex conjugate of $z .$ Then $S$ has:

(A) two elements

(B) three elements

(C) four elements

(D) six elements.

**Problem 2:**

The number of one-to-one functions from a set with $3$ elements to a set with $6$ elements is

(A) $20$

(B) $120$

(C) $216$

(D) $720$ .

**Problem 3:**

Two sides of a triangle are of length $2 \mathrm{~cm}$ and $3 \mathrm{~cm} .$ Then, the maximum possible area (in $\mathrm{cm}^{2}$ ) of the triangle is:

(A) $2$

(B) $3$

(C) $4$

(D) $6$ .

**Problem 4:**

The number of factors of $2^{15} \times 3^{10} \times 5^{6}$ which are either perfect squares or perfect cubes (or both) is:

(A) $252$

(B) $256$

(C) $260$

(D) $264$ .

**Problem 5:**

The minimum value of the function $f(x)=x^{2}+4 x+\frac{4}{x}+\frac{1}{x^{2}}$ where $x>0$, is

(A) $9.5$

(B) $10$

(C) $15$

(D) $20$ .

**Problem 6:**

The minimum area of the triangle formed by any tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=$ 1 and the coordinate axes is

(A) $a b$

(B) $\frac{a^{2}+b^{2}}{2}$

(C) $\frac{(a+b)^{2}}{4}$

(D) $\frac{a^{2}+a b+b^{2}}{3}$.

**Problem 7:**

The angle between the hyperbolas $x y=1$ and $x^{2}-y^{2}=1$ (at their point of intersection) is

(A) $\frac{\pi}{2}$

(B) $\frac{\pi}{3}$

(C) $\frac{\pi}{4}$

(D) $\frac{\pi}{6}$.

**Problem 8:**

The population of a city doubles in $50$ years. In how many years will it triple, under the assumption that the rate of increase is proportional to the number of inhabitants?

(A) $75$ years

(B) $100$ years

(C) $50 \log_{2}(3)$ years

(D) $50 \log_{e}\left(\frac{3}{2}\right)$ years.

**Problem 9:**

We define a set $\{f_{1}, f_{2}, \ldots, f_{n}\}$ of polynomials to be a linearly dependent set if there exist real numbers $c_{1}, c_{2}, \ldots, c_{n},$ not all zero, such that $c_{1} f_{1}(x)+\cdots+$ $c_{n} f_{n}(x)=0$ for all real numbers $x$. Which of the following is a linearly dependent set?

(A) $\{x, x^{2}, x^{3}\}$

(B) $\{x^{2}-x, 2 x, x^{2}+3 x\}$

(C) $\{x, 2 x^{3}, 5 x^{2}\}$

(D) $\{x^{2}-1,2 x+5, x^{2}+1\}$.

**Problem 10:**

A set of numbers $S$ is said to be multiplicatively closed if $a b \in S$ whenever both $a \in S$ and $b \in S .$ Let $i=\sqrt{-1}$ and $\omega$ be a non-real cube root of unity. Let

$$

S_{1}={a+b i \mid a, b \text { are integers }} \text { and } S_{2}={a+b \omega \mid a, b \text { are integers }}

$$

Which one of the following statements is true?

(A) Both $S_{1}$ and $S_{2}$ are multiplicatively closed.

(B) $S_{1}$ is multiplicatively closed but $S_{2}$ is not.

(C) $S_{2}$ is multiplicatively closed but $S_{1}$ is not.

(D) Neither $S_{1}$ nor $S_{2}$ is multiplicatively closed.

**Problem 11:**

When the product of four consecutive odd positive integers is divided by $5,$ the set of remainder(s) is

(A) $\{0\}$

(B) $\{0,4\}$

(C) $\{0,2,4\}$

(D) $\{0,2,3,4\}$ .

**Problem 12:**

Consider the equation $x^{2}+y^{2}=2015$ where $x \geq 0$ and $y \geq 0$. How many solutions $(x, y)$ exist such that both $x$ and $y$ are non-negative integers?

(A) None

(B) Exactly one

(C) Exactly two

(D) Greater than two.

**Problem 13:**

Let $P$ be a point on the circle $x^{2}+y^{2}-9=0$ above the $x$ -axis, and $Q$ be a point on the circle $x^{2}+y^{2}-20 x+96=0$ below the $x$ -axis such that the line joining $P$ and $Q$ is tangent to both these circles. Then the length of $P Q$ is

(A) $5 \sqrt{2}$ units

(B) $5 \sqrt{3}$ units

(C) $5 \sqrt{6}$ units

(D) $6 \sqrt{5}$ units.

**Problem 14:**

Let $S=\{(x, y)| x, y \text{ are positive integers }\}$ viewed as a subset of the plane. For every point $P$ in $S,$ let $d_{P}$ denote the sum of the distances from $P$ to the point $(8,0)$ and the point $(0,12)$. The number of points $P$ in $S$ such that $d_{P}$ is the least among all elements in the set $S$, is

(A) $0$

(B) $3$

(C) $8$

(D) $1 .$

**Problem 15:**

Let $A, B$ and $C$ be the angles of a triangle. Suppose that $\tan A$ and $\tan B$ are the roots of the equation $x^{2}-8 x+5=0$. Then $\cos ^{2} C-8 \cos C \sin C+5 \sin ^{2} C$ equals

(A) $-1$

(B) $0$

(C) $1$

(D) $2$.

**Problem 16:**

Let $A=\{a_{1}, a_{2}, \ldots, a_{10}\}$ and $B={1,2}$. The number of functions $f: A \rightarrow B$ for which the sum $f\left(a_{1}\right)+\cdots+f\left(a_{10}\right)$ is an even number, is

(A) $128$

(B) $256$

(C) $512$

(D) $768$ .

**Problem 17:**

Define $sgn(x)=\begin{cases} 1 & \text { if } x>0 \\-1 & \text { if } x<0 \\ 0 & \text { if } x=0 \end{cases}.$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function given by $f(x)=(x+1) sgn\left(x^{2}-1\right)$ where $\mathbb{R}$ is the set of real numbers. Then the number of discontinuities of $f$ is:

(A) $0$

(B) $1$

(C) $2$

(D) $3$ .

**Problem 18:**

Suppose $X$ is a subset of real numbers and $f: X \rightarrow X$ is a bijection (that is, one-to-one and onto) satisfying $f(x)>x$ for all $x \in X .$ Then $X$ cannot be:

(A) the set of integers

(B) the set of positive integers

(C) the set of positive real numbers

(D) the set of real numbers.

**Problem 19:**

The set of real numbers $x$ satisfying the inequality

$$

\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9

$$

is:

(A) $\left[-\frac{1}{2}, 0\right) \cup\left(0, \frac{45}{8}\right)$

(B) $\left[-\frac{1}{2}, 0\right) \cup\left(\frac{45}{8}, \infty\right)$

(C) $\left[-\frac{1}{2}, 0\right) \cup(0, \infty)$

(D) $\left(0, \frac{45}{8}\right) \cup\left(\frac{45}{8}, \infty\right)$.

**Problem 20:**

Let $A B C D E F G H I J$ be a 10 -digit number, where all the digits are distinct. Further, $A>B>C, \quad A+B+C=9, \quad D>E>F>G$ are consecutive odd numbers and $H>I>J$ are consecutive even numbers. Then $A$ is

(A) $8$

(B) $7$

(C) $6$

(D) $5$.

**Problem 21:**

Let $A=\{(a, b, c): a, b, c$ are prime numbers, $a<b<c, a+b+c=30\} .$ The number of elements in $A$ is

(A) $0$

(B) $1$

(C) $2$

(D) $3$.

**Problem 22:**

Let $f(x)=\begin{cases}\frac{|\sin x|}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases}.$

Then $\int_{-1}^{1} f(x) d x$ is equal to

(A) $\frac{2 \pi}{3}$

(B) $\frac{3 \pi}{8}$

(C)$-\frac{\pi}{4}$

(D) $0$ .

**Problem 23:**

Let $f:(0,2) \cup(4,6) \rightarrow \mathbb{R}$ be a differentiable function. Suppose also that $f^{\prime}(x)=1$ for all $x \in(0,2) \cup(4,6) .$ Which of the following is ALWAYS true?

(A) $f$ is increasing

(B) $f$ is one-to-one

(C) $f(x)=x$ for all $x \in(0,2) \cup(4,6)$

(D) $f(5.5)-f(4.5)=f(1.5)-f(0.5)$

**Problem 24:**

Consider 50 evenly placed points on a circle with centre at the origin and radius $R$ such that the arc length between any two consecutive points is the same. The complex numbers represented by these points form

(A) an arithmetic progression with common difference $\left(\cos \left(\frac{2 \pi}{50}\right)+i \sin \left(\frac{2 \pi}{50}\right)\right)$

(B) an arithmetic progression with common difference $\left(R \cos \left(\frac{2 \pi}{50}\right)+i R \sin \left(\frac{2 \pi}{50}\right)\right)$

(C) a geometric progression with common ratio $\left(\cos \left(\frac{2 \pi}{50}\right)+i \sin \left(\frac{2 \pi}{50}\right)\right)$

(D) a geometric progression with common ratio $\left(R \cos \left(\frac{2 \pi}{50}\right)+i R \sin \left(\frac{2 \pi}{50}\right)\right)$

**Problem 25:**

Given two complex numbers $z, w$ with unit modulus (i.e., $|z|=|w|=1$ ), which of the following statements will ALWAYS be correct?

(A) $|z+w|<\sqrt{2}$ and $|z-w|<\sqrt{2}$

(B) $|z+w| \leq \sqrt{2}$ and $|z-w| \geq \sqrt{2}$

(C) $|z+w| \geq \sqrt{2}$ or $|z-w| \geq \sqrt{2}$

(D) $|z+w|<\sqrt{2}$ or $|z-w|<\sqrt{2}$

**Problem 26:**

The number of points in the region $\{(x, y): x^{2}+y^{2} \leq 4\}$ satisfying $\tan ^{4} x+$ $\cot ^{4} x+1=3 \sin ^{2} y$ is

(A) $1$

(B) $2$

(C) $3$

(D) $4$.

**Problem 27:**

If all the roots of the equation $x^{4}-8 x^{3}+a x^{2}+b x+16=0$ are positive, then $a+b$

(A) must be $-8$

(B) can be any number strictly between $-16$ and $-8$

(C) must be $-16$

(D) can be any number strictly between $-8$ and $0$

**Problem 28:**

Let $O$ denote the origin and $A, B$ denote respectively the points (-10,0) and (7,0) on the $x$ -axis. For how many points $P$ on the $y$ -axis will the lengths of all the line segments $P A, P O$ and $P B$ be positive integers?

(A) $0$

(B) $2$

(C) $4$

(D) infinite.

**Problem 29:**

Let $G(x)=\int_{-x^{3}}^{x^{3}} f(t) d t,$ where $x$ is any real number and $f$ is a continuous function such that $f(t)>1$ for all real $t .$ Then,

(A) $G^{\prime}(0)=0$ and $G$ has a local maximum or minimum at $x=0$.

(B) For any real number $c$, the equation $G(x)=c$ has a unique solution.

(C) There exists a real number $c$ such that $G(x)=c$ has no solution.

(D) There exists a real number $c$ such that $G(x)=c$ has more than one solution.

**Problem 30:**

There are $2 n+1$ real numbers having the property that the sum of any $n$ of them is less than the sum of the remaining $n+1 .$ Then,

(A) all the numbers must be positive

(B) all the numbers must be negative

(C) all the numbers must be equal

(D) such a system of real numbers cannot exist.

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