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 $x$ be an irrational number. If $a, b, c$ and $d$ are rational numbers such that $\frac{a x+b}{cx+d}$ is a rational number, which of the following must be true?

(A) $a d=b c$

(B) $a c=b d$

(C) $a b=c d$

(D) $a=d=0$

**Problem 2:**

Let $z=x+i y$ be a complex number which satisfies the equation $(z+\bar{z}) z=2+4 i$. Then

(A) $y=\pm 2$;

(B) $x=\pm 2$;

(C) $x=\pm 3$;

(D) $y=\pm 1$.

**Problem 3:**

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^{n}+b^{n}$ is prime, what can you say about $n ?$

(A) The integer $n$ must be $2$ ;

(B) The integer $n$ need not be $2,$ but must be a power of $2$ ;

(C) The integer $n$ need not be a power of $2,$ but must be even;

(D) None of the above is necessarily true.

**Problem 4:**

For how many real values of $p$ do the equations $x^{2}+p x+1=0$ and $x^{2}+x+p=0$ have exactly one common root?

(A) $0$

(B) $1$ ;

(C) $2$ ;

(D) $3 .$

**Problem 5:**

The limit $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n^{2}}\right)^{\frac{8 n^{3}}{n} \sin \left(\frac{\pi}{2 n}\right)}$$ is

(A) $\infty$;

(B) $1 ;$

(C) $e^{4}$;

(D) $4$ .

**Problem 6:**

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is

(A) $40.5$ minutes;

(B) $81$ minutes;

(C) $60.75$ minutes;

(D) $20.25$ minutes.

**Problem 7:**

Let $A B$ be a fixed line segment. Let $P$ be a moving point such that $\angle A P B$ is equal to a constant acute angle. Then, which of the following curves does the point $P$ move along?

(A) a circle;

(B) an ellipse;

(C) the boundary of the common region of $2$ identical intersecting circles with centres outside the common region;

(D) the boundary of the union of $2$ identical intersecting circles with centres outside the common region.

**Problem 8:**

A circular pit of radius $r$ metres and depth $2$ metres is dug and the removed soil is piled up as a cone with the bottom of the pit as its base. What proportion of the volume of the cone is above the ground level?

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

(B) $\frac{8}{27}$;

(C) $\frac{4 r^{2}}{9}$;

(D) $\frac{r^{3}}{27}$.

**Problem 9:**

The algebraic sum of the perpendicular distances from $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ $C\left(x_{3}, y_{3}\right),$ to a line is zero. Then the line must pass through the

(A) orthocentre of $\triangle A B C$;

(B) centroid of $\triangle A B C$ :

(C) incentre of $\triangle A B C$;

(D) circumcentre of $\triangle A B C$.

**Problem 10:**

The value of the integral

$$

\int_{\pi / 2}^{5 \pi / 2} \frac{e^{\tan ^{-1}(\sin x)}}{e^{\tan ^{-1}(\sin x)}+e^{\tan ^{-1}(\cos x)}} d x

$$

equals

(A) $1 ;$

(B) $\pi$;

(C) $e$;

(D) none of these.

**Problem 11:**

If the function $f(x)=\frac{(x-1)(x-2)}{x-a},$ for $x \neq a,$ takes all values in $(-\infty, \infty),$ then we must have

(A) $a \leq 1$;

(B) $a \geq 2$;

(C) $a \leq 1$ or $a \geq 2$ ;

(D) $1 \leq a \leq 2$.

**Problem 12:**

In how many ways can you choose three distinct numbers from the set $\{1,2,3, \ldots, 19,20\}$ such that their product is divisible by $4 ?$

(A) $795$

(B) $810$

(C) $855$

(D) $1665$

**Problem 13:**

Consider the function $f(x)=e^{2 x}-x^{2} .$ Then

(A) $f(x)=0$ for some $x<0$ but $f(x) \neq 0$ for every $x>0$;

(B) $f(x)=0$ for some $x>0$ but $f(x) \neq 0$ for every $x<0$;

(C) $f(x_{1})=0$ for some $x_{1}<0$ and $f(x_{2})=0$ for some $x_{2}>0$;

(D) $f(x) \neq 0$ for every $x$.

**Problem 14**:

For $k \geq 1$, the value of

$$

\left(\begin{array}{c}

n \\

0

\end{array}\right)+\left(\begin{array}{c}

n+1 \\

1

\end{array}\right)+\left(\begin{array}{c}

n+2 \\

2

\end{array}\right)+\cdots+\left(\begin{array}{c}

n+k \\

k

\end{array}\right)

$$

equals

(A) $\left(\begin{array}{c}n+k+1 \\ n+k\end{array}\right)$;

(B) $(n+k+1)\left(\begin{array}{c}n+k \\ n+1\end{array}\right)$

(C) $\left(\begin{array}{c}n+k+1 \\ n+1\end{array}\right)$

(D) $\left(\begin{array}{c}n+k+1 \\ n\end{array}\right)$.

**Problem 15:**

In a triangle $A B C,$ angle $A$ is twice the angle $B$. Then which of the following has to be true?

(A) $a^{2}=b(b+c)$

(B) $b^{2}=a(a+c)$

(C) $c^{2}=a(a+b)$

(D) $a b=c(a+c)$

**Problem 16:**

The point $(x, y)$ on the line $x+y=10$ for which $\min \{4-x, 5-y\}$ is the largest is

(A) $\left(\frac{9}{2}, \frac{11}{2}\right)$

(B) $(5,5)$

(C) $\left(\frac{11}{2}, \frac{9}{2}\right)$

(D) none of these.

**Problem 17:**

The value of $$\sin ^{-1}\cot [\sin ^{-1}\{\frac{1}{2}(1-\sqrt{\frac{5}{6}})\}+\cos ^{-1} \sqrt{\frac{2}{3}}+\sec ^{-1} \sqrt{\frac{8}{3}}]$$

is

(A) $0$;

(B) $\pi / 6$;

(C) $\pi / 4$;

(D) $\pi / 2$.

**Problem 18:**

Which of the following graphs represents the function

$$

f(x)=\int_{0}^{\sqrt{x}} e^{-u^{2} / x} d u, \quad \text { for } \quad x>0 \quad \text { and } \quad f(0)=0 ?

$$

**Problem 19:**

Consider a triangle $A B C$ with the sides $a, b, c$ in A.P. Then the largest possible value of the angle $B$ is

(A) $60^{\circ}$

(B) $67 \frac{1}{2}^{\circ}$;

(C) $75^{\circ}$

(D) $82 \frac{1}{2}^{\circ}$.

**Problem 20:**

If $a_{n}=\left(1+\frac{1}{n^{2}}\right)\left(1+\frac{2^{2}}{n^{2}}\right)^{2}\left(1+\frac{3^{2}}{n^{2}}\right)^{3} \cdots\left(1+\frac{n^{2}}{n^{2}}\right)^{n},$ then

$\lim_{n\to \infty} a_n^{\frac{-1}{n^2}}$

is

(A) $0$;

(B) $1$;

(C) $e$;

(D) $\sqrt{e} / 2$.

**Problem 21:**

If $f(x)=e^{x} \sin x,$ then $\left.\frac{d^{10}}{d x^{10}} f(x)\right|_{x=0}$ equals

(A) $1$ ;

(B) $-1$ ;

(C) $10 ;$

(D) $32$ .

**Problem 22:**

Consider a circle with centre $O .$ Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C=43^{\circ}$ and $\angle B P D=18^{\circ},$ then the value of $\angle B O D$ is

(A) $36^{\circ}$

(B) $29^{\circ}$

(C) $7^{\circ}$

(D) $25^{\circ}$,

**Problem 23:**

Consider a triangle $A B C$. The median $A D$ meets the side $B C$ at the point $D$. A point $E$ on $A D$ is such that $A E: D E=1: 3 .$ The straight line $B E$ extended meets the side $A C$ at a point $F$. Then $A F: F C$ equals

(A) $1: 6$;

(B) $1: 7$;

(C) $1: 4$ ;

(D) $1: 3$ .

**Problem 24:**

A person standing at a point $A$ finds the angle of elevation of a nearby tower to be $60^{\circ}$. From $A$, the person walks a distance of $100$ feet to a point $B$ and then walks again to another point $C$ such that $\angle A B C=120^{\circ} .$ If the angles of elevation of the tower at both $B$ and $C$ are also $60^{\circ}$ each, then the height of the tower is

(A) $50$ feet;

(B) $50 \sqrt{3}$ feet;

(C) $100 \sqrt{3}$ feet;

(D) $100$ feet.

**Problem 25:**

A box contains 10 red cards numbered $1, \ldots, 10$ and $10$ black cards numbered $1, \ldots, 10 .$ In how many ways can we choose $10$ out of the $20$ cards so that there are exactly $3$ matches, where a match means a red card and a black card with the same number?

(A) $\left(\begin{array}{c}10 \\ 3\end{array}\right)\left(\begin{array}{l}7 \\ 4\end{array}\right) 2^{4}$

(B) $\left(\begin{array}{c}10 \\ 3\end{array}\right)\left(\begin{array}{l}7 \\ 4\end{array}\right)$;

(C) $\left(\begin{array}{c}10 \\ 3\end{array}\right) 2^{7}$

(D) $\left(\begin{array}{c}10 \\ 3\end{array}\right)\left(\begin{array}{c}14 \\ 4\end{array}\right)$.

**Problem 26:**

Let $P$ be a point on the ellipse $x^{2}+4 y^{2}=4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C: P D$ equals

(A) $2$ ;

(B) $1 / 2$;

(C) $4$ ;

(D) $1 / 4$.

**Problem 27:**

Let $\alpha$ denote the absolute value of the difference between the lengths of the two segments of a focal chord of a parabola. Let $\beta$ denote the length of a chord passing through the vertex and parallel to that focal chord. Then which of the ollowing is always true?

A) $\alpha^{2}=2 \beta$

(B) $\alpha=2 \beta$

(C) $\alpha=\beta$

(D) $\beta^{2}=2 \alpha$

**Problem 28:**

The directrix of the parabola traced out by the centre of a moving circle, which touches both the straight line $y=-x$ and the circle $(x-3)^{2}+(y-4)^{2}=9,$ is

(A) $y=-x+3$

(B) $y=-x-3$

(C) $y=-x+3 \sqrt{2}$

(D) $y=x-3 \sqrt{2}$.

**Problem 29:**

For a real number $x$, let $[x]$ denote the largest integer smaller than or equal to $x$ and $(x)$ denote the smallest integer larger than or equal to $x .$ Let $f(x)=$ $\min (x-[x],(x)-x)$ for $0 \leq x \leq 12$. The volume of the solid obtained by rotating the curve $y=f(x)$ about the $X$ -axis is

(A) $\pi$;

(B) $4 \pi$

(C) $\pi / 2$

(D) $\pi / 4$.

**Problem 30:**

For a real number $x$, let $[x]$ denote the largest integer smaller than or equal to

$x .$ The value of $\int_{-100}^{100}\left[t^{3}\right] d t$ is

(A) $0$ ;

(B) $100$ ;

(C) $-100$ ;

(D) $-100^{3}$.

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