ISI Entrance 2011 - B.Math Subjective Paper| Problems & Solutions

Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1 :

Let $a \geq 0$ be a constant such that $\sin (\sqrt{x+a})=\sin (\sqrt{x})$ for all $x \geq 0 .$ What can you say about $a$ ? Justify your answer.

Problem 2 :

Let $f(x)=e^{-x}$ for $x>0$. Define a function $g$ on the nonnegative real numbers as follows: for each integer $k>0$, the graph of the function $g$ on the interval $[k, k+1]$ is the straight line segment connecting the points $(k, f(k))$ and $(k+1, f (k+1) )$. Find the total area of the region which lies between the curves of $f$ and $g$.

Problem 3 :

For any positive integer $n$, show that $\frac{1}{2}\cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \cdot \frac{(2 n-1)}{2 n}<\frac{1}{\sqrt{2 n+1}}$.

Problem 4 :

If $a_{1}, \ldots, a_{7}$ are not necessarily distinct real numbers such that $1<a_{i}<13$ for all $i$, then show that we can choose three of them such that they are the lengths of the sides of a triangle.

Problem 5 :

For any real number $x,$ let $[x]$ denote the largest integer which is less than or equal to $x$. Let $N_{1}=2, N_{2}=3, N_{3}=5, \ldots$ be the sequence of non-square positive integers. If the $n$ th non-square positive integer satisfies $m^{2}<N_{n}<$ $(m+1)^{2},$ then show that $m=\left[\sqrt{n}+\frac{1}{2}\right]$

Problem 6 :

Let $R$ and $S$ be two cubes with sides of lengths $r$ and $s$ respectively, where $r$ and $s$ are positive integers. Show that the difference of their volumes equals the difference of their surface areas, if and only if $r=s$.

Problem 7 :

Let $A B C$ be any triangle and let $O$ be a point on the line segment $B C .$ Show that there exists a line parallel to $A O$ which divides the triangle $A B C$ into two equal parts of equal area.

Problem 8 :

Let $t_{1}<t_{2}<\cdots<t_{99}$ be real numbers, and consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x)=\left|x-t_{1}\right|+\left|x-t_{2}\right|+\cdots+\left|x-t_{99}\right| .$ Show that $\min _{x \in R} f(x)=f\left(t_{50}\right)$

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ISI Entrance 2010 - B.Math Subjective Paper| Problems & Solutions

Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Prove that in each year, the $13$ th day of some month occurs on a Friday.

Problem 2:

In the accompanying figure, $y=f(x)$ is the graph of a one-to-one continuous function $f$. At each point $P$ on the graph of $y=2 x^{2}$, assume that the areas $O A P$ and $O B P$ are equal. Here $P A \cdot P B$ are the horizontal and vertical segments. Determine the function $f$.

Problem 3:

Show that. for any positive integer $n,$ the sum of $8 n+4$ consecutive positive integers cannot be a perfect square.

Problem 4:

If $a, b, c \in(0,1)$ satisfy $a+b+c=2,$ prove that

$\frac{a b c}{(1-a)(1-b)(1-c)} \geq 8$.

Problem 5:

Let $a_{1}>a_{2}>\cdots>a_{r}$ be positive real numbers. Compute $\lim _{n \rightarrow \infty}\left(a_{1}^{n}+a_{2}^{n}+\cdots+a_{r}^{n}\right)^{1 / n}$.

Problem 6:

Let each of the vertices of a regular $9$-gon (polygon of $9$ equal sides and equal angles) be coloured black or white.
(a) Show that there are two adjacent vertices of the same colour.
(b) Show there are 3 vertices of the same colour forming an isosceles triangle.

Problem 7:

Let $a, b, c$ be real numbers and. assume that all the roots of $x^{3}+a x^{2}+b x+c=0$ have the same absolute value, Show that $a=0$ if, and only if, $b=0$.

Problem 8:

I et $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}$ exists, and is finite. Prove that $f^{\prime}(0)=0$.

Problem 9:

Let $f(x)$ be a polynomial with integer coefficients. Assume that 3 divides the value f(n) for each integer $n$. Prove that when $f(x)$ is divided by $x^{3}-x$ the remainder is of the form $3 r(x)$, where $r(x)$ is a polynomial with integer coefficients.

Problem 10:

Consider a regular heptagon (polygon of 7 equal sides and equal angles) ABCDEFG.

(a) Prove $\frac{1}{\sin \frac{\pi}{7}}=\frac{1}{\sin \frac{2 \pi}{7}}+\frac{1}{\sin \frac{3 \pi}{7}}$.
(b) Using (a) or otherwise, show that $\frac{1}{A G}=\frac{1}{A F}+\frac{1}{A E}$. (See the figure appearing in the next page.)