Select Page

ISI Entrance Paper – from Indian Statistical Institute’s B.Stat Entrance 2005

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 $\displaystyle{ \sin\theta=\frac{\sqrt{5}-1}{2}}$
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)?
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 $\displaystyle{ f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j) }$ for all k such that i <k
4. Find all real solutions of the equation $\displaystyle{ \sin^{5}x+\cos^{3}x=1 }$.
5. 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 $\displaystyle{ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} }$ > $\displaystyle{ \frac{1}{45} }$
6. 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.
1. 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 $\displaystyle{[x_0,1)}$.
7. For integers $\displaystyle{ m,n\geq 1 }$, Let $\displaystyle{ A_{m,n} , B_{m,n} }$ and $\displaystyle{ C_{m,n}}$ denote the following sets:
$\displaystyle{A_{m,n}={(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n}}$ given that $\displaystyle{\alpha _i \in \mathbb{Z}}$ for all i
$\displaystyle{B_{m,n}={(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n}}$ given that $\displaystyle{\alpha _i \geq 0}$ and $\displaystyle{\alpha_ i \in \mathbb{Z}}$ for all i
$\displaystyle{C_{m,n}={(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1 \alpha_2 \ldots; \alpha_m \leq n}}$ given that $\displaystyle{\alpha _i \in \mathbb{Z}}$ for all i

1. Define a one-one onto map from $\displaystyle{A_{m,n}}$ onto $\displaystyle{B_{m+1,n-1}}$.
2. Define a one-one onto map from $\displaystyle{A_{m,n}}$ onto $\displaystyle{C_{m,n+m-1}}$.
3. Find the number of elements of the sets $\displaystyle{A_{m,n}}$ and $\displaystyle{B_{m,n}}$.
8. 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.
1. $\displaystyle{ g(n)=5^k }$ where k is the number of distinct primes which divide n.
2. $\displaystyle{ h(n)= 0} \text{if} n \text{is divisible by} k^2 \text{for some integer} k>1$ …., 1 otherwise
9. Suppose that to every point of the plane a colour, either red or blue, is associated.
1. 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.
2. Show that there must be an equilateral triangle with all vertices of the same colour.
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?