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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 $x$ 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) $-100^{3}$.

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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