ISI B.Stat 2011 Objective Paper| Problems & Solutions

Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Group A

Problem 1:

The limit

$$
\lim _{x \rightarrow 0} \frac{1-\cos \left(\sin ^{2} \alpha x\right)}{x}
$$

(A) equals $1$;
(B) equals $\alpha$;
(C) equals $0$ ;
(D) does not exist.

Problem 2:

The set of all $x$ for which the function $f(x)=\log _{\frac{1}{2}}\left(x^{2}-2 x-3\right)$ is defined and monotone increasing is

(A) $(-\infty, 1)$;
(B) $(-\infty,-1) ;$
(C) $(1, \infty)$;
(D) $(3, \infty)$.

Problem 3:

Let a line with slope of $60^{\circ}$ be drawn through the focus $F$ of the parabola $y^{2}=8(x+2)$. If the two points of intersection of the line with the parabola are $A$ and $B$ and the perpendicular bisector of the chord $A B$ intersects the $x$ -axis at the point $P$, then the length of the segment $P F$ is

(A) $\frac{16}{3}$;
(B) $\frac{8}{3}$;
(C) $\frac{16 \sqrt{3}}{3}$;
(D) $8 \sqrt{3}$.

Problem 4:

Suppose $z$ is a complex number with $|z|<1$. Let $w=\frac{1+z}{1-z}$. Which of the following is always true? $[\textrm{Re}(w)$ is the real part of $w$ and $\textrm{Im}(w)$ is the imaginary part of $w .]$

(A) $\textrm{Re}(w)>0$;
(B) $\textrm{Im}(w) \geq 0$;
(C) $|w| \leq 1 ;$
(D) $|w| \geq 1$.

Problem 5:

Among all the factors of $4^{6} 6^{7} 21^{8}$, the number of factors which are perfect squares is

(A) $240$ ;
(B) $360$ ;
(C) $400$ ;
(D) $640$ .

Problem 6:

Let $A$ be the set ${1,2, \ldots, 20}$. Fix two disjoint subsets $S_{1}$ and $S_{2}$ of $A$, each with exactly three elements. How many 3-element subsets of $A$ are there, which have exactly one element common with $S_{1}$ and at least one element common with $S_{2} ?$

(A) $51$ ;
(B) $102$ ;
(C) $135$ ;
(D) $153$ .

Problem 7:

In how many ways can 3 couples sit around a round table such that men and women alternate and none of the couples sit together?

(A) $1$;
(B) $2$ ;
(C) $\frac{5 !}{3}$;
(D) none of these.

Problem 8:

The equation $x^{3}+y^{3}=x y(1+x y)$ represents

(A) two parabolas intersecting at two points;
(B) two parabolas touching at one point;
(C) two non-intersecting hyperbolas;
(D) one parabola passing through the origin.

Problem 9:

Consider the diagram below where $A B Z P$ is a rectangle and $A B C D$ and $C X Y Z$ are squares whose areas add up to $1 .$

The maximum possible area of the rectangle $A B Z P$ is

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

Problem 10:

Let $A$ be the set ${1,2, \ldots, 6} .$ How many functions $f$ from $A$ to $A$ are there such that the range of $f$ has exactly 5 elements?
(A) $240$;
(B) $720$;
(C) $1800$ ;
(D) $10800$ .

Problem 11:

Let $C_{1}, C_{2}$ and $C_{3}$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C_{1}$ touches $C_{2}$ and $C_{2}$ touches $C_{3 .}$ If the area of the smallest circle is 1 unit, then the area of the largest circle is

(A) $\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)^{4}$;
(B) $(1+\sqrt{2})^{2}$
(C) $(2+\sqrt{2})^{2}$
(D) $2^{4}$.

Problem 12:

Let $[x]$ denote the largest integer less than or equal to $x$. Then $\int_{0}^{n^{\frac{1}{k}}}\left[x^{k}+n\right] d x$ equals

(A) $n^{2}+\sum_{i=1}^{n} i^{\frac{1}{k}}$;
(B) $2 n^{\frac{1+k}{k}}-\sum_{i=1}^{n} i^{\frac{1}{k}} ;$

(C) $2 n^{\frac{1+k}{k}}-\sum_{i=1}^{n-1} i^{\frac{1}{k}}$;
(D) None of these.

Problem 13:

Consider the function

$$
f(x)=\begin{cases}
x(x-1) e^{2 x}, & \text { if } x \leq 0; \\
x(1-x) e^{-2 x}, & \text { if } x>0
\end{cases}.
$$

Then $f(x)$ attains its maximum value at

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

Problem 14:

Consider the function $f(x)=\frac{x^{n}(1-x)^{n}}{n !},$ where $n \geq 1$ is a fixed integer. Let $f^{(k)}$ denote the $k$ -th derivative of $f$. Which of the following is true for all $k \geq 1 ?$

(A) $f^{(k)}(0)$ and $f^{(k)}(1)$ are integers;
(B) $f^{(k)}(0)$ is an integer, but not $f^{(k)}(1)$;
(C) $f^{(k)}(1)$ is an integer, but not $f^{(k)}(0)$;
(D) Neither $f^{(k)}(1)$ nor $f^{(k)}(0)$ is an integer.

Problem 15:

The number of solutions of the equation $\sin (\cos \theta)=\theta, \quad-1 \leq \theta \leq 1,$ is

(A) $0 ;$
(B) $1$;
(C) $2$;
(D) $3$.

Problem 16:

Suppose $A B C D$ is a parallelogram and $P, Q$ are points on the sides $B C$ and $C D$ respectively, such that $P B=\alpha B C$ and $D Q=\beta D C .$ If the area of the triangles $\triangle A B P, \Delta A D Q$ and $\Delta P C Q$ are 15,15 and 4 respectively, then the area of $\Delta A P Q$ is

(A) $14$ ;
(B) $15$ ;
(C) $16$ ;
(D) $18$ .

Problem 17:

Consider an equilateral triangle $A B C$ with side $2.1 \mathrm{~cm} .$ You want to place a number of smaller equilateral triangles, each with side $1 \mathrm{~cm} .$, over the triangle $A B C,$ so that the triangle $A B C$ is fully covered. What is the minimum number of smaller triangles that you need?

(A) $4$;
(B) $5$;
(C) $6$ ;
(D) $7$ .

Problem 18:

A regular tetrahedron has all its vertices on a sphere of radius $R$. Then the length of each edge of the tetrahedron is

(A) $\frac{\sqrt{2}}{\sqrt{3}} R$;
(B) $\frac{\sqrt{3}}{2} R$;
(C) $\frac{4}{3} R$;
(D) $\frac{2 \sqrt{2}}{\sqrt{3}} R$.

Problem 19:

Consider the L-shaped brick in the diagram below.

If an ant starts from $A$, find the minimum distance it has to travel along the surface to reach $B$.

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

Problem 20:

Let $f(x)=(\tan x)^{\frac{3}{2}}-3 \tan x+\sqrt{\tan x}$. Consider the three integrals

$$
I_{1}=\int_{0}^{1} f(x) d x, \quad I_{2}=\int_{0.3}^{1.3} f(x) d x, \quad I_{3}=\int_{0.5}^{1.5} f(x) d x
$$

Then,

(A) $I_{1}>I_{2}>I_{3}$;
(B) $I_{2}>I_{1}>I_{3}$;
(C) $I_{3}>I_{1}>I_{2}$;
(D) $I_{1}>I_{3}>I_{2}$.

Group B

Problem 21:

Let $a<b<c$ be three real numbers and $w$ denote a complex cube root of unity. If $\left(a+b w+c w^{2}\right)^{3}+\left(a+b w^{2}+c w\right)^{3}=0,$ then which of the following must be true?

(A) $a+b+c=0$
(B) $a b c=0$;
(C) $a b+b c+c a=0$;
(D) $b=\frac{c+a}{2}$.

Problem 22:

Suppose $f$ is continuously differentiable up to 3 rd order and satisfies

$$
\int_{0}^{1}\{6 f(x)+x^{3} f^{\prime \prime \prime}(x)\} d x=f^{\prime \prime}(1)
$$

Which of the following must be true?

(A) $f(1)=0$;
(B) $f^{\prime}(1)=2 f(1) ;$
(C) $f^{\prime}(1)=f(1) ;$
(D) $f^{\prime}(1)=0$

Problem 23:

Let $f(x)=a x^{2}+b x+c$ for some real numbers $a, b$ and $c .$ If $f(-5) \geq 10$ $f(-3)<6$ and $f(2) \geq 7,$ then which of the following cannot be true?

(A) $f(3)=6$
(B) $f(3) \geq 16$;
(C) $f(4)=5$
(D) $f(4) \geq 6.2$.

Problem 24:

Consider the sequence $x_{n}, n \geq 1,$ defined as:

$$
x_{n}=\left(1+\frac{2}{n^{a}}\right)^{-n^{b}} n^{c}
$$

where $a, b$ and $c$ are real numbers. Which of the following are true?

(A) if $b<a, x_n \to \infty$ as $n \to\ infty$;

(B) if $a<b, x_n \to \infty$ as $n \to\ infty$;

(C) if $a =b$ and $c>0, x_n \to \infty$ as $n\to \infty$;

(D) if $a =b$ and $c>0, x_n \to \infty$ as $n\to \infty$;

Problem 25:

The value of $n^{\frac{1}{n}}-1$

(A) tends to 0 as $n \rightarrow \infty$;
(B) is greater than $\frac{\log n}{n}$ for all $n \geq 3$
(C) is greater than $\log n$ for all $n \geq 3$
(D) is greater than $\frac{1}{\sqrt{n}}$ for all $n \geq 3$.

Problem 26:

If the complex numbers $1+i$ and $5-3 i$ represent two diagonally opposite vertices of a square, which of the following complex numbers can represent another vertex of the square?

(A) $5+2 i$
(B) $3+2 \sqrt{2}-i$
(C) $1-3 i$
(D) $4+2 \sqrt{2}+2 \sqrt{2} i$.

Problem 27:

Suppose $x$ and $y$ are two positive numbers satisfying the equation $x^{y}=y^{x}$ Which of the following are true?

(A) For all $x>1,$ there always exist a $y>x$ such that the above equation holds;
(B) For all $x>e,$ there is always a $y>x$ such that the above equation holds:
(C) For all $1x$ such that the above equation holds;
(D) If $x<1,$ then $y$ must be equal to $x$.

Problem 28:

Consider 6 points on the plane no three of which are collinear. An edge is a straight line joining one point to another. Two points are called connected if one can go from one point to another through edges. Suppose you are only told how many edges are there in total, but not where they are. Which of the following are true?

(A) If you are told that there are $7$ edges, you cannot be sure that all pairs of points are connected;
(B) If you are told that there are $9$ edges, you can always ensure that all pairs of points are connected;
(C) If you are told that there are $12$ edges, you cannot be sure that all pairs of points are connected;
(D) If you are told that there are $13$ edges, you can always ensure that all pairs of points are connected.

Problem 29:

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable everywhere. Which of the following conditions imply that $|f(x)|$ is also differentiable?

(A) $f(x)=0$ whenever $f^{\prime}(x)=0$;
(B) $f^{\prime}(x)=0$ whenever $f(x)=0$;
(C) $f^{\prime}(x)$ never takes the value $0$ ;
(D) $f(x)$ never takes the value $0$ .

Problem 30:

Let the coordinates of the centre of a circle be $\left(-\frac{7}{10}, 2 \sqrt{2}\right)$. Then the number of points $(x, y)$ on the circle such that both $x$ and $y$ are rational

(A) cannot be $3$ or more;
(B) at least $1,$ but at most $2$ ;
(C) at least $2,$ but infinitely many;
(D) infinitely many.

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