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

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  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 } .
    1. Discussion
  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} }
    1. Discussion
  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?