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# ISI B.Stat & B.Math 2015 Objective Paper| problems & solutions

Here, 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.

Here, 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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