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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let $C$ be the circle $x^{2}+y^{2}+4 x+6 y+9=0$. The point $(-1,-2)$ is

(A) inside $C$ but not the centre of $C$;

(B) outside $C$;

(C) on $C$ :

(D) the centre of $C$.

**Problem 2:**

The number of distinct real roots of the equation

$$

\left(x+\frac{1}{x}\right)^{2}-5\left(x+\frac{1}{x}\right)+6=0

$$ is

(A) $1$;

(B) $2$;

(C) $3$;

(D) $4$.

**Problem 3:**

The set of complex numbers $z$ satisfying the equation

$$

(3+7 i) z+(10-2 i) \bar{z}+100=0

$$

represents, in the Argand plane,

(A) a straight line;

(B) a pair of intersecting straight lines;

(C) a pair of distinct parallel straight lines;

(D) a point.

**Problem 4:**

Let $X$ be the set ${1,2,3, \ldots, 10}$ and $P$ the subset ${1,2,3,4,5} .$ The number of subsets $Q$ of $X$ such that $P \cap Q={3}$ is

(A) $1$ ;

(B) $2^{4}$

(C) $2^{5}$

(D) $2^{9}$.

**Problem 5:**

The number of triplets $(a, b, c)$ of integers such that $a<b<c$ and $a, b, c$ are sides of a triangle with perimeter $21$ is

(A) $7$ ;

(B) $8$;

(C) $11$;

(D) $12$.

**Problem 6:**

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $a x^{2}+2 b x+c=0$ and $d x^{2}+2 e x+f=0$ have a common root, then $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in

(A) A.P.

(B) G.P.;

(C) H.P.;

(D) none of the above.

**Problem 7:**

The number of solutions of the equation $\sin ^{-1} x=2 \tan ^{-1} x$ is

(A) $1$ ;

(B) $2$ ;

(C) $3$;

(D) $5$.

**Problem 8:**

Suppose $x^{2}+p x+q=0$ has two real roots $\alpha$ and $\beta$ with $|\alpha| \neq|\beta| .$ If $\alpha^{4}$ and $\beta^{4}$ are the roots of $x^{2}+r x+s=0,$ then the equation $x^{2}-4 q x+2 q^{2}+r=0$ has

(A) one positive and one negative root;

(B) two distinct positive roots;

(C) two distinct negative roots:

(D) no real roots.

**Problem 9:**

Suppose $A B C D$ is a quadrilateral such that $\angle B A C=50^{\circ}, \angle C A D=60^{\circ}$ $\angle C B D=30^{\circ}$ and $\angle B D C=25^{\circ} .$ If $E$ is the point of intersection of $A C$ and $B D,$ then the value of $\angle A E B$ is

(A) $75^{\circ}$;

(B) $85^{\circ}$;

(C) $95^{\circ}$;

(D) $110^{\circ}$.

**Problem 10:**

Let $\mathbb{R}$ be the set of all real numbers. The function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=x^{3}-3 x^{2}+6 x-5$

(A) one-to-one, but not onto:

(B) one-to-one and onto;

(C) onto, but not one-to-one;

(D) neither one-to-one nor onto.

**Problem 11:**

The angles of a convex pentagon are in A.P. Then, the minimum possible value of the smallest angle is

(A) $30^{\circ}$

(B) $36^{\circ}$

(C) $45^{\circ}$;

(D) $54^{\circ}$.

**Problem 12:**

The number of points $(b, c)$ lying on the circle $x^{2}+(y-3)^{2}=8,$ such that the quadratic equation $t^{2}+b t+c=0$ has real roots, is

(A) infinite;

(B) $2$ ;

(C) $4$ ;

(D) $0 .$

**Problem 13:**

Let $L$ be the point $(t, 2)$ and $M$ be a point on the $y$ -axis such that $L M$ has slope $-t .$ Then the locus of the midpoint of $L M,$ as $t$ varies over all real values, is

(A) $y=2+2 x^{2}$

(B) $y=1+x^{2}$

(C) $y=2-2 x^{2}$

(D) $y=1-x^{2}$

**Problem 14**:

Suppose $x, y \in(0, \pi / 2)$ and $x \neq y .$ Which of the following statements is true?

(A) $2 \sin (x+y)<\sin 2 x+\sin 2 y$ for all $x, y$

(B) $2 \sin (x+y)>\sin 2 x+\sin 2 y$ for all $x, y$

(C) There exist $x, y$ such that $2 \sin (x+y)=\sin 2 x+\sin 2 y$;

(D) None of the above.

**Problem 15:**

A triangle $A B C$ has a fixed base $B C$. If $A B: A C=1: 2,$ then the locus of the vertex $A$ is

(A) a circle whose centre is the midpoint of $B C$;

(B) a circle whose centre is on the line $B C$ but not the midpoint of $B C$;

(C) a straight line;

(D) none of the above.

**Problem 16:**

Suppose $e^{-x} \sin x-e^{-x} \cos x+\cos x=0$ for some $x>0$. Then

(A) $\sin x>0$;

(B) $\sin x \cos x>0$;

(C) $\cos x>0$;

(D) $\sin x \cos x<0$.

**Problem 17:**

Let $f(x)=x^{6}-3 x^{2}-10 .$ The set of all values taken by $f(x)$ as $x$ varies over the interval $[-2,2]$ is

(A) $[-12,-10]$ ;

(B) $[-10,42]$;

(C) $[-12,42]$;

(D) $[-10,12]$ .

**Problem 18:**

$N$ is a $50$ digit number. All the digits except the $26$ th from the right are $1 .$ If $N$ is divisible by $13,$ then the unknown digit is

(A) $1$ ;

(B) $3$;

(C) $7$ ;

(D) $9$.

**Problem 19:**

If $f(x)=x^{n-1} \log x$, then the $n$ -th derivative of $f$ equals

(A) $\frac{(n-1) !}{x}$;

(B) $\frac{n}{x}$;

(C) $(-1)^{n-1} \frac{(n-1) !}{x} ;$

(D) $\frac{1}{x}$.

**Problem 20:**

Suppose $a<b$. The maximum value of the integral

$$

\int_{a}^{b}\left(\frac{3}{4}-x-x^{2}\right) d x

$$

over all possible values of $a$ and $b$ is

(A) $\frac{3}{4}$;

(B) $\frac{4}{3}$;

(C) $\frac{3}{2}$ ;

(D)$ \frac{2}{3}$ .

**Problem 21:**

For any $n \geq 5,$ the value of $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2^{n}-1}$ lies between

(A) $0$ and $\frac{n}{2}$;

(B) $\frac{n}{2}$ and $n$

(C) $n$ and $2 n$;

(D) none of the above.

**Problem 22:**

Let $\omega$ denote a cube root of unity which is not equal to $1 .$ Then the number of distinct elements in the set

$$

\{(1+\omega+\omega^{2}+\cdots+\omega^{n})^{m}: m, n=1,2,3, \cdots\}

$$

is

(A) $4$ ;

(B) $5$ ;

(C) $7$ ;

(D) infinite.

**Problem 23:**

The graph of a function $f$ defined on (0,2) with the property $\lim_{x \to 0+} f(x) =\lim_{x \to 2-} f(x) =\infty$ is as follows:

The graph of the derivative of the above function looks like:

**Problem 24:**

The value of the integral

$$

\int_{2}^{3} \frac{d x}{\log _{e} x}

$$

(A) is less than $2$ ;

(B) is equal to $2$;

(C) lies in the interval $(2,3)$ ;

(D) is greater than $3$ .

**Problem 25:**

A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is $4$ metres and the radius of the base is $1$ metre. The volume of the sphere is the same as that of the cone. Then, the distance between the centre of the sphere and the vertex of the cone is

(A) $4$ metres;

(B) $\sqrt{17}$ metres;

(C) $\sqrt{15}$ metres;

(D) $5$ metres.

**Problem 26:**

For each positive integer $n$, define a function $f_{n}$ on $[0,1]$ as follows:

$f_{n}(x)=\begin{cases}0 \quad \text { if } \quad x=0 \\ \sin \frac{\pi}{2 n} \quad \text { if } \quad 0<x \leq \frac{1}{n} \\ \sin \frac{2 \pi}{2 n} \quad \text { if } \quad \frac{1}{n}<x \leq \frac{2}{n} \\ \sin \frac{3 \pi}{2 n} \quad \text { if } \quad \frac{2}{n}<x \leq \frac{3}{n} \\ \vdots \quad \vdots \quad \vdots \\ \sin \frac{n \pi}{2 n} \quad \text { if } \frac{n-1}{n}<x \leq 1 .\end{cases}$

Then, the value of $\lim_{n \to \infty} \int_0^1 f_n(x) dx$ is

(A) $\pi$ :

(B) $1$:

(C) $\frac{1}{\pi}$

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

**Problem 27:**

The limit

$$

\lim _{n \rightarrow \infty}\left(1+\frac{1}{n^{2}+\cos n}\right)^{n^{2}+n}

$$

(A) does not exist;

(B) equals $1$ ;

(C) equals $e$;

(D) equals $e^{2}$.

**Problem 28:**

Let $K$ be the set of all points $(x, y)$ such that $|x|+|y| \leq 1 .$ Given a point $A$ in the plane, let $F_{A}$ be the point in $K$ which is closest to $A$. Then the points $A$ for which $F_{A}=(1,0)$ are

(A) all points $A=(x, y)$ with $x \geq 1$;

(B) all points $A=(x, y)$ with $x \geq y+1$ and $x \geq 1-y$;

(C) all points $A=(x, y)$ with $x \geq 1$ and $y=0$;

(D) all points $A=(x, y)$ with $x \geq 0$ and $y=0$.

**Problem 29:**

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players $A, B, C, D, E$ and $F$ play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, $B$ has 8 points and $C$ has 4 points. It is also known that $\mathrm{E}$ won against $\mathrm{F}$. In the next set of games $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$ win their games against $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ respectively. If $\mathrm{A}, \mathrm{B}$ and $\mathrm{D}$ move to the final round, the final scores of $\mathrm{E}$ and $\mathrm{F}$ are, respectively,

(A) $4$ and $2$ ;

(B) $2$ and $4 ;$

(C) $2$ and $2$ ;

(D) $4$ and $4$ .

**Problem 30:**

The number of ways in which one can select six distinct integers from the set $\{1,2,3, \cdots, 49\},$ such that no two consecutive integers are selected, is

(A) $\left(\begin{array}{c}49 \\ 6\end{array}\right)-5\left(\begin{array}{c}48 \\ 5\end{array}\right)$;

(B) $\left(\begin{array}{c}43 \\ 6\end{array}\right)$;

(C) $\left(\begin{array}{c}25 \\ 6\end{array}\right)$;

(D) $\left(\begin{array}{c}44 \\ 6\end{array}\right)$.

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