INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

Here, 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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