ISI B.Stat 2005 Subjective Paper| Problems & Solutions

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

Problem 1:

Let $a,b$ and $c$ be the sides of a right angled triangle. Let $ \displaystyle{\theta } $ be the smallest angle of this triangle. If $ \displaystyle{ \frac{1}{a}, \frac{1}{b} } $ and $ \displaystyle{ \frac{1}{c} }$ are also the sides of a right angled triangle then show that $ \displaystyle{ \sin\theta=\frac{\sqrt{5}-1}{2}}$

Discussion

Problem 2:

Let $ \displaystyle{f(x)=\int_0^1 |t-x|t , dt }$ for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

Problem 3:
Let $f$ be a function defined on $ \displaystyle{ {(i,j): i,j \in \mathbb{N}} } $ satisfying $ \displaystyle{ f(i,i+1)=\frac{1}{3} } $ for all i $f(i, j)$ = $f(i, k)$+ $f(k, j)$ - $2 f(i, k) f(k, j)$ for all $k$ such that $i <k$. Find all real solutions of the equation $ \displaystyle{ \sin^{5}x+\cos^{3}x=1 } $.

Discussion

Problem 4:

Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $ \displaystyle{ \angle QCR, \angle QIR }$ and $ \displaystyle{ \angle QOR } $, measured in degrees, are $ \displaystyle{ \alpha, \beta and \gamma } $ respectively. Show that $ \displaystyle{ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} } $ > $ \displaystyle{ \frac{1}{45} }$

Discussion

Problem 5:

Let $f$ be a function defined on $ \displaystyle{ (0, \infty ) }$ as follows: $ \displaystyle{ f(x)=x+\frac1x } $ . Let $h$ be a function defined for all $ \displaystyle{ x \in (0,1) } $ as $ \displaystyle{h(x)=\frac{x^4}{(1-x)^6} }$. Suppose that $g(x)=f(h(x))$ for all $ \displaystyle{x \in (0,1)}$. Show that $h$ is a strictly increasing function. Show that there exists a real number $ \displaystyle{x_0 \in (0,1)}$ such that $g$ is strictly decreasing in the interval $ \displaystyle{ (0,x_0] } $ and strictly increasing in the interval $ \displaystyle{[x_0,1)}$.

Problem 6:

For integers $ \displaystyle{ m,n\geq 1 }$, Let $ \displaystyle{ A_{m,n} , B_{m,n} }$ and $ \displaystyle{ C_{m,n}}$ denote the following sets:

a) $A_{m, n}$=$\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ : $1 \leq \alpha_{1} \leq \alpha_{2} \leq \ldots \leq \alpha_{m} \leq n$ given that $\alpha_{i} \in \mathbb{Z}$ for all $i$

b) $B_{m, n}$=$\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ : $\alpha_{1}+\alpha_{2}+\ldots+\alpha_{m}$=$n$ given that $\alpha_{i} \geq 0$ and $\alpha_{i} \in \mathbb{Z}$ for all $i$

c) $C_{m, n}$=$\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ : $1 \leq \alpha_{1} \alpha_{2} \ldots ; \alpha_{m} \leq n$ given that $\alpha_{i} \in \mathbb{Z}$ for all $i$

d) Define a one-one onto map from $ \displaystyle{A_{m,n}}$ onto $ \displaystyle{B_{m+1,n-1}}$.

e) Find the number of elements of the sets $ \displaystyle{A_{m,n}}$ and $ \displaystyle{B_{m,n}}$

Problem 7:

A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer.

Problem 8:

$ \displaystyle{ g(n)=5^k }$ where k is the number of distinct primes which divide $n$. $ \displaystyle{ h(n)= 0}$ if $n$ is divisible by $k^2$ for some integer $k>1$ ...., 1 otherwise.

Problem 9:

Suppose that to every point of the plane a colour, either red or blue, is associated. Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points $A,B$ and $C$ of the same colour such that $B$ is the midpoint of $AC$. Show that there must be an equilateral triangle with all vertices of the same colour.

Problem 10:

Let $ABC$ be a triangle. Take n point lying on the side $AB$ (different from $A$ and $B$) and connect all of them by straight lines to the vertex $C$. Similarly, take n points on the side $AC$ and connect them to $B$. Into how many regions is the triangle $ABC$ partitioned by these lines? Further, take $n$ points on the side $BC$ also and join them with $A$. Assume that no three straight lines meet at a point other than $A,B$ and $C$. Into how many regions is the triangle $ABC$ partitioned now?

some useful link

Let $a,b$ and $c$ be the sides of a right angled triangle. Let $ \displaystyle{\theta } $ be the smallest angle of this triangle. If $ \displaystyle{ \frac{1}{a}, \frac{1}{b} } $ and $ \displaystyle{ \frac{1}{c} }$ are also the sides of a right angled triangle then show that $ \displaystyle{ \sin\theta=\frac{\sqrt{5}-1}{2}}$

Discussion

Problem 2:

Let $ \displaystyle{f(x)=\int_0^1 |t-x|t , dt }$ for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

Problem 3:
Let $f$ be a function defined on $ \displaystyle{ {(i,j): i,j \in \mathbb{N}} } $ satisfying $ \displaystyle{ f(i,i+1)=\frac{1}{3} } $ for all i $f(i, j)$ = $f(i, k)$+ $f(k, j)$ - $2 f(i, k) f(k, j)$ for all $k$ such that $i <k$. Find all real solutions of the equation $ \displaystyle{ \sin^{5}x+\cos^{3}x=1 } $.

Discussion

Problem 4:

Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $ \displaystyle{ \angle QCR, \angle QIR }$ and $ \displaystyle{ \angle QOR } $, measured in degrees, are $ \displaystyle{ \alpha, \beta and \gamma } $ respectively. Show that $ \displaystyle{ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} } $ > $ \displaystyle{ \frac{1}{45} }$

Discussion

Problem 5:

Let $f$ be a function defined on $ \displaystyle{ (0, \infty ) }$ as follows: $ \displaystyle{ f(x)=x+\frac1x } $ . Let $h$ be a function defined for all $ \displaystyle{ x \in (0,1) } $ as $ \displaystyle{h(x)=\frac{x^4}{(1-x)^6} }$. Suppose that $g(x)=f(h(x))$ for all $ \displaystyle{x \in (0,1)}$. Show that $h$ is a strictly increasing function. Show that there exists a real number $ \displaystyle{x_0 \in (0,1)}$ such that $g$ is strictly decreasing in the interval $ \displaystyle{ (0,x_0] } $ and strictly increasing in the interval $ \displaystyle{[x_0,1)}$.

Problem 6:

For integers $ \displaystyle{ m,n\geq 1 }$, Let $ \displaystyle{ A_{m,n} , B_{m,n} }$ and $ \displaystyle{ C_{m,n}}$ denote the following sets:

a) $A_{m, n}$=$\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ : $1 \leq \alpha_{1} \leq \alpha_{2} \leq \ldots \leq \alpha_{m} \leq n$ given that $\alpha_{i} \in \mathbb{Z}$ for all $i$

b) $B_{m, n}$=$\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ : $\alpha_{1}+\alpha_{2}+\ldots+\alpha_{m}$=$n$ given that $\alpha_{i} \geq 0$ and $\alpha_{i} \in \mathbb{Z}$ for all $i$

c) $C_{m, n}$=$\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ : $1 \leq \alpha_{1} \alpha_{2} \ldots ; \alpha_{m} \leq n$ given that $\alpha_{i} \in \mathbb{Z}$ for all $i$

d) Define a one-one onto map from $ \displaystyle{A_{m,n}}$ onto $ \displaystyle{B_{m+1,n-1}}$.

e) Find the number of elements of the sets $ \displaystyle{A_{m,n}}$ and $ \displaystyle{B_{m,n}}$

Problem 7:

A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer.

Problem 8:

$ \displaystyle{ g(n)=5^k }$ where k is the number of distinct primes which divide $n$. $ \displaystyle{ h(n)= 0}$ if $n$ is divisible by $k^2$ for some integer $k>1$ ...., 1 otherwise.

Problem 9:

Suppose that to every point of the plane a colour, either red or blue, is associated. Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points $A,B$ and $C$ of the same colour such that $B$ is the midpoint of $AC$. Show that there must be an equilateral triangle with all vertices of the same colour.

Problem 10:

Let $ABC$ be a triangle. Take n point lying on the side $AB$ (different from $A$ and $B$) and connect all of them by straight lines to the vertex $C$. Similarly, take n points on the side $AC$ and connect them to $B$. Into how many regions is the triangle $ABC$ partitioned by these lines? Further, take $n$ points on the side $BC$ also and join them with $A$. Assume that no three straight lines meet at a point other than $A,B$ and $C$. Into how many regions is the triangle $ABC$ partitioned now?