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Learn MoreHere, you will find all the questions of ISI Entrance Paper 2012 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

A rod $A B$ of length 3 rests on a wall as follows:

$P$ is a point on $AB$ such that $AB : PB = 1 : 2$. If the rod slides along the wall, then the locus of $P$ lies on

(A) $2 x+y+x y=2$

(B) $4 x^{2}+y^{2}=4$

(C) $4 x^{2}+x y+y^{2}=4$

(D) $x^{2}+y^{2}-x-2 y=0$

**Problem 2: **

Consider the equation $x^2 + y^2 = 2007$. How many solutions $(x,y)$ exist such that $x$ and $y$ are positive integer?

(A) None

(B) Exactly two

(C) More than two but finitely many

(D) Infinitely many.

**Discussion: **

**Problem 3:**

Consider the functions $f_{1}(x)=x, f_{2}(x)=2+\log _{e} x, x>0$ (where $e$ is the base of natural logarithm ). The graphs of the functions intersect

(A) once in (0,1) and never in $(1, \infty)$

(B) once in (0,1) and once in $\left(e^{2}, \infty\right)$

(C) once in (0,1) and once in $\left(e, e^{2}\right)$

(D) more than twice in $(0, \infty)$.

**Problem 4:**

Consider the sequence

$$

u_{n}=\sum_{r=1}^{n} \frac{r}{2^{r}}, n \geq 1

$$

Then the limit of $u_{n}$ as $n \rightarrow \infty$ is

(A) $1$

(B) $2$

(C) $e$

(D) $1 / 2$.

**Problem 5: **

Suppose that $z$ is any complex number which is not equal to any of $\{3,3 \omega, 3 \omega^{2}\}$ where $\omega$ is a complex cube root of unity. Then

$$

\frac{1}{z-3}+\frac{1}{z-3 \omega}+\frac{1}{z-3 \omega^{2}}

$$

equals

(A) $\frac{3 z^{2}+3 z}{(z-3)^{3}}$

(B) $\frac{3 z^{2}+3 \omega z}{z^{3}-27}$

(C) $\frac{3 z^{2}}{z^{3}-3 z^{2}+9 z-27}$

(D) $\frac{3 z^{2}}{z^{3}-27}$.

**Problem 6:**

Consider all functions $f:\{1,2,3,4\} \rightarrow \{1,2,3,4\}$ which are one-one, onto and satisfy the following property:

if $f(k)$ is odd then $f(k+1)$ is even, $k=1,2,3$.

The number of such functions is

(A) $4$

(B) $8$

(C) $12$

(D) $16$ .

**Problem 7:**

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by

$f(x)=\left\{\begin{array}{cl}e^{-\frac{1}{x}}, & x \geq 0 \\ 0 & x \leq 0\end{array}\right.$

Then

(A) $f$ is not continuous

(B) $f$ is differentiable but $f^{\prime}$ is not continuous

(C) $f$ is continuous but $f^{\prime}(0)$ does not exist

(D) $f$ is differentiable and $f^{\prime}$ is continuous.

**Problem 8:**

The last digit of $9 !+3^{9966}$ is

(A) $3$

(B) $9$

(C) $7$

(D) $1$ .

**Problem 9:**

Consider the function $f(x)=\frac{2 x^{2}+3 x+1}{2 x-1}, 2 \leq x \leq 3$. Then

(A) maximum of $f$ is attained inside the interval $(2,3)$

(B) minimum of $f$ is $\frac{28}{5}$

(C) maximum of $f$ is $\frac{28}{5}$

(D) $f$ is a decreasing function in $(2,3)$ .

**Problem 10:**

A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^{2}=4 a x$ is always $90^{\circ} .$ The locus of $P$ is

(A) a parabola

(B) a circle

(C) an ellipse

(D) a straight line.

**Problem 11:**

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x)=\left|x^{2}-1\right|, x \in \mathbb{R}$. Then

(A) $f$ has a local minima at $x=\pm 1$ but no local maximum

(B) $f$ has a local maximum at $x=0$ but no local minima

(C) $f$ has a local minima at $x=\pm 1$ and a local maximum at $x=0$

(D) none of the above is true.

**Problem 12:**

The number of triples $(a, b, c)$ of positive integers satisfying $2^{a}-5^{b} 7^{c}=1$ is

(A) infinite

(B) $2$

(C) $1$

(D) $0$ .

**Problem 13:**

Let $a$ be a fixed real number greater than $-1 .$ The locus of $z \in \mathbb{C}$ satisfying $|z-i a|=Im(z)+1$ is

(A) parabola

(B) ellipse

(C) hyperbola

(D) not a conic.

**Problem 14:**

Which of the following is closest to the graph of $\tan (\sin x), x>0 ?$

**Problem 15:**

Consider the function $f: \mathbb{R} \backslash{1} \rightarrow \mathbb{R} \backslash{2}$ given by $f(x)=\frac{2 x}{x-1}$. Then

(A) $f$ is one-one but not onto

(B) $f$ is onto but not one-one

(C) $f$ is neither one-one nor onto

(D) $f$ is both one-one and onto.

**Problem 16:**

Consider a real valued continuous function $f$ satisfying $f(x+1)=f(x)$ for all $x \in \mathbb{R} .$ Let $g(t)=\int_{0}^{t} f(x) d x, \quad t \in \mathbb{R}$. Define $h(t)=\lim _{n \rightarrow \infty} \frac{q(t+n)}{n}$, provided the limit exists. Then

(A) $h(t)$ is defined only for $t=0$

(B) $h(t)$ is defined only when $t$ is an integer

(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$

(D) none of the above is true.

**Problem 17:**

Consider the sequence $a_{1}=24^{1 / 3}, a_{n+1}=\left(a_{n}+24\right)^{1 / 3}, n \geq 1$. Then the integer part of $a_{100}$ equals

(A) $2$

(B) $10$

(C) $100$

(D) $24$ .

**Problem 18: **

Let $x, y \in(-2,2)$ and $x y=-1 .$ Then the minimum value of $\frac{4}{4-x^{2}}+\frac{9}{9-y^{2}}$ is

(A) $\frac{8}{5}$

(B) $\frac{12}{5}$

(C) $\frac{12}{7}$

(D) $\frac{15}{7}$.

**Problem 19: **

What is the limit of $\left(1+\frac{1}{n^{2}+n}\right)^{n^{2}+\sqrt{n}}$ as $n \rightarrow \infty$ ?

(A) $e$

(B) $1$

(C) $0$

(D) $\infty$.

**Problem 20:**

Consider the function $f(x)=x^{4}+x^{2}+x-1, x \in(-\infty, \infty) .$ The function

(A) is zero at $x=-1,$ but is increasing near $x=-1$

(B) has a zero in $(-\infty,-1)$

(C) has two zeros in (-1,0)

(D) has exactly one local minimum in (-1,0) .

**Problem 21:**

Consider a sequence of 10 A's and 8 B's placed in a row. By a run we mean one or more letters of the same type placed side by side. Here is an arrangement of $10 A$ 's and $8 B$ 's which contains 4 runs of $A$ and 4 runs of $B:$

AAABBABBBAABAAAABB

In how many ways can $10 A$ 's and $8 B$ 's be arranged in a row so that there are 4 runs of $A$ and 4 runs of $B ?$

(A)$2 {{9} \choose {3}}$ ${{7} \choose {3}}$

(B) ${{9} \choose {3}}$ ${{7} \choose {3}}$

(C) ${{10} \choose {4}}$ ${{8} \choose {4}}$

(D) ${{10} \choose {5}}$ ${{8} \choose {5}}$.

**Problem 22: **

Suppose $n \geq 2$ is a fixed positive integer and $f(x)=x^{n}|x|, x \in \mathbb{R}$. Then

(A) $f$ is differentiable everywhere only when $n$ is even

(B) $f$ is differentiable everywhere except at 0 if $n$ is odd

(C) $f$ is differentiable everywhere

(D) none of the above is true.

**Problem 23:**

The line $2 x+3 y-k=0$ with $k>0$ cuts the $x$ axis and $y$ axis at points $A$ and $B$ respectively. Then the equation of the circle having $A B$ as diameter is

(A) $x^{2}+y^{2}-\frac{k}{2} x-\frac{k}{3} y=k^{2}$

(B) $x^{2}+y^{2}-\frac{k}{3} x-\frac{k}{2} y=k^{2}$

(C) $x^{2}+y^{2}-\frac{k}{2} x-\frac{k}{3} y=0$

(D) $x^{2}+y^{2}-\frac{k}{3} x-\frac{k}{2} y=0$.

**Problem 24:**

Let $\alpha>0$ and consider the sequence $x_{n}=\frac{(\alpha+1)^{n}+(\alpha-1)^{n}}{(2 \alpha)^{n}}, n=1,2, \ldots$

Then $lim_{x \to \infty} x_n$ is

(A) $0$ for any $\alpha>0$

(B) $1$ for any $\alpha>0$

(C) $0$ or $1$ depending on what $\alpha>0$ is

(D) $0,1$ or $\infty$ depending on what $\alpha>0$ is.

**Problem 25:**

If $0<\theta<\pi / 2$ then

(A) $\theta<\sin \theta$

(B) $\cos (\sin \theta)<\cos \theta$

(C) $\sin (\cos \theta)<\cos (\sin \theta)$

(D) $\cos \theta<\sin (\cos \theta)$

**Problem 26:**

Consider a cardboard box in the shape of a prism as shown below. The length of the prism is 5 . The two triangular faces $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are congruent and isosceles with side lengths 2,2,3 . The shortest distance between $B$ and $A^{\prime}$ along the surface of the prism is

(A) $\sqrt{29}$

(B) $\sqrt{28}$

(C) $\sqrt{29-\sqrt{5}}$

(D) $\sqrt{29-\sqrt{3}}$

**Problem 27:**

Assume the following inequalities for positive integer $k : \frac{1}{2 \sqrt{k+1}}<\sqrt{k+1}-\sqrt{k}<\frac{1}{2 \sqrt{k}}$. The integer part of $\sum_{k=2}^{9099} \frac{1}{\sqrt{k}}$

equals

(A) $198$

(B) $197$

(C) $196$

(D) $195$.

**Problem 28:**

Consider the sets defined by the inequalities

$A=\{(x, y) \in \mathbb{R}^2: x^4+y^2 \leq 1\}, B=\{(x, y) \in \mathbb{R}^2: x^6+y^4\leq 1\}$

Then

(A) $B \subseteq A$

(B) $A \subseteq B$

(C) each of the sets $A-B, B-A$ and $A \cap B$ is non-empty

(D) none of the above is true.

**Problem 29:**

The number $\left(\frac{2^{10}}{11}\right)^{11}$ is

(A) strictly larger than ${{10} \choose {1}}^2$ ${{10} \choose {2}}^2$ ${{10} \choose {3}}^2$ ${{10} \choose {4}}^2$ ${{10} \choose {5}}^2$

(B) strictly larger than ${{10} \choose {1}}^2$ ${{10} \choose {2}}^2$ ${{10} \choose {3}}^2$ ${{10} \choose {4}}^2$ but strictly smaller than

${{10} \choose {1}}^2$ ${{10} \choose {2}}^2$ ${{10} \choose {3}}^2$ ${{10} \choose {4}}^2$ ${{10} \choose {5}}$

(C) less than or equal to ${{10} \choose {1}}^2$ ${{10} \choose {2}}^2$ ${{10} \choose {3}}^2$ ${{10} \choose {4}}^2$

(D) equal to ${{10} \choose {1}}^2$ ${{10} \choose {2}}^2$ ${{10} \choose {3}}^2$ ${{10} \choose {4}}^2$ ${{10} \choose {5}}$

**Problem 30:**

If the roots of the equation $x^{4}+a x^{3}+b x^{2}+c x+d=0$ are in geometric progression then

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

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

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

(D) $c^{2}=a^{2} d$.

**Problem 1:**

Let $X, Y, Z$ be the angles of a triangle.

(1) Prove that $\tan \frac{X}{2} \tan \frac{Y}{2}+\tan \frac{X}{2} \tan \frac{Z}{2}+\tan \frac{Z}{2} \tan \frac{Y}{2}=1$

(2) Using (1) or otherwise prove that $\tan \frac{X}{2} \tan \frac{Y}{2} \tan \frac{Z}{2} \leq \frac{1}{3 \sqrt{3}}$

**Problem 2:**

Let $\alpha$ be s real number. Consider the function $g(x)=(\alpha+|x|)^{2} e^{(5-|x|)^{2}},-\infty<x<\infty \cdot \infty .$

(i) Determine the values of $\alpha$ for which $g$ is continuous at all $x$.

(ii) Determine the values of $\alpha$ for which $g$ is differentiable at all $x$.

**Problem 3:**

Write the set of all positive integers in triangular array as

Find the row number and column number where $20096$ occurs. For example $8$ appears in the third row and second column.

**Problem 4:**

Show that the polynomial $x^{8}-x^{7}+x^{2}-x+15$ has no real root.

**Problem 5:**

Let $m$ be a natural number with digits consisting entirely of 6 's and 0 's. Prove that $m$ is not the square of a natural number.

**Problem 6:**

Let $0<a<b$.

(i) Show that amongst the triangles with base $a$ and perimeter $a+b$ the maximum area is obtained when the other two sides have equal length $\frac{b}{2}$.

(ii) Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter the square has maximum area.

**Problem 7:**

Let $0<a<b$. Consider two circles with radii $a$ and $b$ and centers $(a, 0)$ and $(0, b)$ respectively with $0<a<b$. Let $c$ be the center of any circle in the crescent shaped region $M$ between the two circles and tangent to both (See figure below). Determine the locus of $c$ as its circle traverses through region $M$ maintaining tangency.

**Problem 8:**

Let $n \geq 1$, and $S={1,2, \ldots, n}$.For a function $f: S \rightarrow S$, a subset $D \subset S$ is said $t$ be invariant under $f,$ if $f(x) \in D$ for all $x \in D$. Note that the empty set and $S$ are invariant for all $f $. Let $\deg(f)$ be the number of subsets of $S$ invariant under $f$.

(i) Show that there is a function $f: S \rightarrow S$ such that $\deg(f)=2$.

(ii) Further show that for any $k$ such that $1 \leq k \leq n$ there is a function $f: S \rightarrow S$ such that $\deg(f)=2^{k}$

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