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Explore the Back-StoryHere, 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 $\deg(f)=2^{k}$

- ISI BStat and B.Math 2011 Problems and Solutions
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Please publish the solutions to the problems

which one?

this one..plz publish the ans