ISI Entrance Paper BMath 2011 - from Indian Statistical Institute's Entrance

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1. Given $latex \mathbf{ a,x\in\mathbb{R}}$ and $latex \mathbf{x\geq 0,a\geq 0}$ . Also $latex \mathbf{sin(\sqrt{x+a})=sin(\sqrt{x})}$ . What can you say about a? Justify your answer.
2. Given two cubes R and S with integer sides of lengths r and s units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that r=s.
3. For $latex \mathbf{n\in\mathbb{N}}$ prove that $latex \mathbf{\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}}$
   Solution
4. Let $latex \mathbf{t_1 < t_2 < t_3 < \cdots < t_{99}}$ be real numbers. Consider a function $latex \mathbf{f: \mathbb{R} to \mathbb{R}}$ given by $latex \mathbf{f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|}$ . Show that f(x) will attain minimum value at $latex \mathbf{x=t_{50}}$
5. Consider a sequence denoted by F_n of non-square numbers . $latex \mathbf{F_1=2,F_2=3,F_3=5}$ and so on . Now , if $latex \mathbf{m^2\leq F_n<(m+1)^2}$ . Then prove that m is the integer closest to $latex \mathbf{\sqrt{n}}$
6. Let $latex \mathbf{f(x)=e^{-x} for all x\geq 0}$ and let g be a function defined as for every integer $latex \mathbf{k \ge 0}$, a straight line joining (k,f(k)) and (k+1,f(k+1)) . Find the area between the graphs of f and g.
7. If $latex \mathbf{a_1, a_2, \cdots, a_7}$ are not necessarily distinct real numbers such that $latex \mathbf{1 < a_i < 13}$ for all i, then show that we can choose three of them such that they are the lengths of the sides of a triangle.
8. In a triangle ABC , we have a point O on BC . Now show that there exists a line l such that l||AO and l divides the triangle ABC into two halves of equal area.

This is a problem number 8 from ISI BMath 2007 based on the Continuity and composition of a function. Try this out.

Problem: Continuity and composition of a function

Let $ \mathbf{P:\mathbb{R} \to \mathbb{R}}$ be a continuous function such that $P(x)=x$ has no real solution. Prove that $P(P(x))=x$ has no real solution.

Discussion:

Hunch: There is no solution of $P(x) = x$ implies the graph of $P(x)$ never 'crosses' the $y=x$ line,
Suppose $P(P(x)) = x$ has a solution at $x = a$ then $P(P(a)) = a$.

Suppose $P(a) = b$ (b is not equal to a as P(x) = x has no solution) then $P(P(a)) = a$ implies $P(b) = a$. Hence we have the following:
$P(a) = b$ and $P(b) = a$.

There fore the points $(a, b)$ and $(b, a)$ are both on the graph of $P(x)$. But these two points are reflections about $y=x$ line implying $P(x)$ is on both side of that line hence continuity implies $P(x)$ must intersect $y=x$ line leading to a solution of $P(x) = x$ hence contradiction. Therefore $P(P(x)) = x$ has not solution.

Final Solution

Consider the auxiliary function $g(x) = P(x) - x$. If $P(P(x)) = x$ has a solution then we have already established that there exists $a, b$ such that $P(a) = b$ and $P(b) = a$. We plug in $a, b$ in $g(x)$.

$g(a) = P(a) - a = b - a$
$g(b) = P(b) - b = a - b$

Now one of $a-b$ and $b-a$ is positive. The other is negative. Since $P(x)$ is continuous so is $g(x)$. So by intermediate value property theorem, $g(x)$ will become $0$ for some value $c$ between $a$ and $b$.

$g(c) = 0$ implies $P(c) - c = 0$ or $P(c) = c$ a contradiction.

Hence $P(P(x)) = x$ has no solution.

Proved

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How to use invariance in Combinatorics – ISI Entrance Problem – Video

This is a problem number 7 from ISI B.Math 2007 based on an inequality related to (sin x)/x function. Try out this problem.

Problem: An inequality related to (sin x)/x function

Let $ \mathbf{0\leq \theta\leq \frac{\pi}{2}}$ . Prove that $\mathbf{\sin \theta \geq \frac{2\theta}{\pi}}$.

Discussion:

We consider the function $ \mathbf{ f(x) = \frac{\sin x }{x} } $. The first derivative of this function is $ \mathbf{ f'(x) = \frac{x \cos x - \sin x} {x^2} }$ In the interval $ \mathbf{[0, \frac{\pi}{2}]}$ the numerator is always negative as x is less than tan x.

Hence f(x) is a monotonically decreasing function in the given interval. Hence f(x) attains least value at $\mathbf{x = \frac{\pi}{2} }$ which equals $ \mathbf{ \frac{\sin\frac{\pi}{2}}{\frac{\pi}{2}} = \frac {2}{\pi}}$

Therefore $\mathbf{\frac{\sin \theta}{\theta} \ge \frac{2}{\pi}}$ in the given interval.

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How to use invariance in Combinatorics – ISI Entrance Problem – Video

Here, you will find all the questions of ISI Entrance Paper 2005 from the Indian Statistical Institute's B.Math Entrance. You will also get the solutions soon to all the previous year's problems.

Problem 1 :

For any \( k \in\mathbb{Z}^+ \) , prove that:-
$$ \displaystyle{ 2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<\\2(\sqrt{k}-\sqrt{k-1})}$$
Also compute integral part of \(\displaystyle{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}}\).

Problem 2 : 

Let \(\displaystyle{a_1=1 }\) and \(\displaystyle{a_n=n(a_{n-1}+1)}\) for all \(\displaystyle{n\ge 2}\). Define : \(\displaystyle{P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)}\) Compute \(\displaystyle{\lim_{n\to \infty} P_n}\)

Problem 3 :

Let \( ABCD \) be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides \( (AB+CD=AD+BC) \). Prove that the circles inscribed in triangles \( ABC \) and \( ACD \) are tangent to each other.

Problem 4 :

For a set S we denote its cardinality by |S|. Let \(\displaystyle{e_1,e_2,\ldots,e_k }\) be non-negative integers. Let \(\displaystyle{A_k}\) (respectively \(\displaystyle{B_k}\) be the set of all k-tuples \(\displaystyle{(f_1,f_2,\ldots,f_k)}\) of integers such that \(\displaystyle{0\leq f_i\leq e_i}\) for all i and \(\displaystyle{\sum_{i=1}^k f_i }\) is even (respectively odd). Show that \(\displaystyle{|A_k|-|B_k|=0 \textrm{ or } 1}\).

Problem 5 :

Find the point in the closed unit disc \(\displaystyle{D={ (x,y) | x^2+y^2\le 1 }}\) at which the function \(f(x,y)=x+y\) attains its maximum.

Problem 6 :

Let \( a_0=0<a_1<a_2<... \) \(\displaystyle{\int_{a_j}^{a_{j+1}} p(t),dt = 0 \forall 0 \le j\le n-1}\) Show that, for \(\displaystyle{0\le j\le n-1}\) , the polynomial p(t) has exactly one root in the interval \(\displaystyle{(a_j,a_{j+1})}\)

Problem 7 :

Let \( M \) be a point in the triangle \( ABC \) such that \(\displaystyle{\text{area}(ABM)=2 \cdot \text{area}(ACM)}\)
Show that the locus of all such points is a straight line.

Problem 8 : 

In how many ways can one fill an \( n*n \) matrix with \( +1 \) and \( -1 \) so that the product of the entries in each row and each column equals \( -1 \)?

some useful link

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Try this problem from Test of Mathematics, TOMATO Objective problem number 44, useful for ISI B.Stat and B.Math.

Problem: TOMATO Objective 44

Suppose that $\mathbf{ x_1 , \cdots , x_n}$ (n> 2) are real numbers such that x $\mathbf{x_i = -x_{n-i+1}}$ for $\mathbf{1\le i \le n}$ . Consider the sum $\mathbf{ S = \sum \sum \sum x_i x_j x_k }$ where the summations are taken over all i, j, k: $\mathbf{ 1\le i, j, k \le n }$ and i, j, k are all distinct. Then S equals:

(A) $\mathbf{n!x_1 x_2 \cdots x_m }$ ; (B) (n-3)(n-4); (C) (n-3)(n-4)(n-5); (D) none of the foregoing expressions;

Discussion:

$\mathbf {( x_1 + x_2 + ... + x_n )^3} $

$\mathbf{= \sum \sum \sum {x_i x_j x_k }+ \sum x_i ^2 ( \sum x_j ) + \sum x_i^3} $

Since $\mathbf {x_1 = - x_n} $

Hence $\mathbf {x_1 ^3 = -x_n ^3} $

Since $\mathbf{\sum {x_i} = 0 } $ and $\mathbf{\sum {x_i}^3 = 0} $

Therefore $\mathbf{\sum \sum \sum {x_i x_j x_k } = 0} $.

Hence option D

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How to use invariance in Combinatorics – ISI Entrance Problem – Video

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

Problem 1:

Let $\mathbf{a_1,a_2,\cdots, a_n }$ and $\mathbf{ b_1,b_2,\cdots, b_n }$ be two permutations of the numbers $\mathbf{1,2,\cdots, n }$. Show that $ {\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2 }$

Problem 2:

Let $a,b,c,d$ be distinct digits such that the product of the $2$-digit numbers $ \mathbf{ab}$ and $\mathbf{cb}$ is of the form $ \mathbf{ddd}$. Find all possible values of $a+b+c+d$.

Problem 3:

Let $\mathbf{I_1, I_2, I_3}$ be three open intervals of $\mathbf{\mathbb{R}}$ such that none is contained in another. If $\mathbf{I_1\cap I_2 \cap I_3}$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.

Problem 4:

A real valued function $f$ is defined on the interval ($-1,2$). A point $ \mathbf{x_0}$ is said to be a fixed point of $f$ if $\mathbf{f(x_0)=x_0}$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval ($0,1$).

Problem 5:

Let $A$ be the set of all functions $\mathbf{f:\mathbb{R} \to \mathbb{R}}$ such that $f(xy)=xf(y)$ for all $\mathbf{x,y \in \mathbb{R}}$.(a) If $\mathbf{f \in A}$ then show that $f(x+y)=f(x)+f(y)$ for all $x,y \mathbf{\in \mathbb{R}}$(b) For $\mathbf{g,h \in A}$, define a function \( \mathbf{g \circ h} \) by $\mathbf{(g \circ h)(x)=g(h(x))}$ for $\mathbf{x \in \mathbb{R}}$. Prove that $\mathbf{g \circ h}$ is in $A$ and is equal to $\mathbf{h \circ g}$.

Problem 6:

Consider the equation $\mathbf{n^2+(n+1)^4=5(n+2)^3}$(a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation.(b) Does the equation have a solution in positive integers?

Problem 7:

Consider a rectangular sheet of paper $ABCD$ such that the lengths of $AB$ and $AD$ are respectively $7$ and $3$ centimetres. Suppose that $B'$ and $D'$ are two points on $AB$ and $AD$ respectively such that if the paper is folded along $B'D'$ then $A$ falls on $A'$ on the side $DC$. Determine the maximum possible area of the triangle $AB'D'$.

Problem 8:

Take $r$ such that $\mathbf{1\le r\le n}$, and consider all subsets of $r$ elements of the set $\mathbf{{1,2,\ldots,n}}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: $ \mathbf{F(n,r)={n+1\over r+1}}$.

Problem 9:

Let $\mathbf{f: \mathbb{R}^2 \to \mathbb{R}^2}$ be a function having the following property: For any two points $A$ and $B$ in $ \mathbf{\mathbb{R}^2}$, the distance between $A$ and $B$ is the same as the distance between the points $f(A)$ and $f(B)$.Denote the unique straight line passing through $A$ and $B$ by $l(A,B)$(a) Suppose that $C,D$ are two fixed points in $\mathbf{\mathbb{R}^2}$. If $X$ is a point on the line $l(C,D)$, then show that $f(X)$ is a point on the line $l(f(C),f(D))$.(b) Consider two more point $E$ and $F$ in $\mathbf{\mathbb{R}^2}$ and suppose that $l(E,F)$ intersects $l(C,D)$ at an angle $\mathbf{\alpha}$. Show that $l(f(C),f(D))$ intersects $l(f(E),f(F))$ at an angle $\alpha$. What happens if the two lines $l(C,D)$ and $l(E,F)$ do not intersect? Justify your answer.

Problem 10:

There are $100$ people in a queue waiting to enter a hall. The hall has exactly $100$ seats numbered from $1$ to $100$. The first person in the queue enters the hall, chooses any seat and sits there. The $n$-th person in the queue, where $n$ can be $2 \cdot 100$, enters the hall after $(n-1)$-th person is seated. He sits in seat number $n$ if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which $100$ seats can be filled up, provided the $100$-th person occupies seat number $100$.

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• ISI BStat and B.Math 2011 Problems and Solutions
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Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Two train lines intersect each other at a junction at an acute angle $ \mathbf{\theta}$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\mathbf{\alpha}$ at a station on the other line. It subtends an angle $\mathbf{\beta (<\alpha)}$ at the same station, when its rear is at the junction. Show that $\mathbf{ \tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}}$

Problem 2:

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $\mathbf{[0,2\pi]}$ and $\mathbf{f''(x) \geq 0 }$ for all $x$ in $ \mathbf{[0,2\pi]}$. Show that
$\mathbf{\int_{0}^{2\pi} f(x)\cos x dx \geq 0}$

Problem 3:

Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR$, $APQ$ and $PQCR$. Find the minimum possible value of $M$.

Problem 4:

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $ \mathbf{k\geq 2}$, a sequence is called an arithmetic progression of the $k$-th order if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.
The numbers $4,6,13,27,50,84$ are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.

Problem 5:

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Problem 6:

Let $f(x)$ be a function satisfying $xf(x)=\ln x$ for $x>0$
Show that $\mathbf{f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)}$ where $ \mathbf{f^{(n)}(x) }$ denotes the $n$-th derivative evaluated at $x$.

Problem 7:

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $ \mathbf{\frac{x}{2}cosec 36^{\circ}}$.

Problem 8:

Find the number of ways in which three numbers can be selected from the set $ \mathbf{{1,2,\cdots ,4n}}$, such that the sum of the three selected numbers is divisible by $4$.

Problem 9:

Consider $6$ points located at $\mathbf{P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)}$. Let $R$ be the region consisting of all points in the plane whose distance from $P_0$ is smaller than that from any other $ \mathbf{P_i, i=1,2,3,4,5}$. Find the perimeter of the region $R$.

Problem 10:

Let $\mathbf{x_n}$ be the $n$-th non-square positive integer. Thus $ \mathbf{x_1=2, x_2=3, x_3=5, x_4=6}$ , etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$ . If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle=1,\langle 2.8\rangle=3,\langle 3.5\rangle=3 . \text { Show that } x_{n}=n+\langle\sqrt{n}\rangle$

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Here, you will find all the questions of ISI Entrance Paper 2008 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

Problem 2:

A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

Problem 3:

Study the derivatives of the function
$\mathbf{y=\sqrt[3]{x^3-4x}}$
and sketch its graph on the real line.

Problem 4:

Suppose $P$ and $Q$ are the centres of two disjoint circles $\mathbf{C_1}$ and $\mathbf{C_2}$ respectively, such that $P$ lies outside $\mathbf{C_2}$ and $Q$ lies outside $\mathbf{C_1}$. Two tangents are drawn from the point $P$ to the circle $\mathbf{C_2}$, which intersect the circle $\mathbf{C_1}$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $ \mathbf{C_1}$, which intersect the circle $\mathbf{C_2}$ at points $M$ and $N$. Show that $AB=MN$

Problem 5:

Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC$, $CA$, and $AB$ at $D,E$ and $F$ respectively. If $BD=x$, $CE=y$ and $AF=z$, then show that $\mathbf{r^2=\frac{xyz}{x+y+z}}$

Problem 6:

Evaluate: $\lim_{n \to\infty} \frac{1}{2n} \ln {{2n} \choose{n}}$

Problem 7:

Consider the equation $\mathbf{x^5+x=10}$. Show that
(a) the equation has only one real root;
(b) this root lies between $1$ and $2$;
(c) this root must be irrational.

Problem 8:

In how many ways can you divide the set of eight numbers $ \mathbf{{2,3,\cdots,9}}$ into $4$ pairs such that no pair of numbers has $ \mathbf{\text{gcd} }$ equal to $2$?

Problem 9:

Suppose $S$ is the set of all positive integers. For $\mathbf{a,b \in S}$, define
$\mathbf{a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}}$
For example $8 * 12=6$.
Show that exactly two of the following three properties are satisfied:
(i) If $\mathbf{a,b \in S}$, then $\mathbf{a * b \in S}$.
(ii) $\mathbf{(a*b)*c=a*(b*c)}$ for all $\mathbf{a,b,c \in S}$.
(iii) There exists an element $ \mathbf{i \in S}$ such that $\mathbf{a *i =a}$ for all $\mathbf{a \in S}$

Problem 10:

Two subsets $A$ and $B$ of the ($x,y$)-plane are said to be equivalent if there exists a function $f: A$ to $B$ which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.

Some useful links :

• ISI BStat and B.Math 2009 Problems and Solutions
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