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I.S.I. and C.M.I. Entrance

ISI Entrance Paper – B.Stat Subjective 2005

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

Also see: ISI and CMI Entrance Course at Cheenta

  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. 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.
    • \displaystyle{ g(n)=5^k } where k is the number of distinct primes which divide n.
    • \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.
    • 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.
  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?

By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

4 replies on “ISI Entrance Paper – B.Stat Subjective 2005”

Do you please help me giving solutions of I.S.I. B.STAT ENTRANCE SUBJECTIVE PAPERS of 2
005, 2006,2007,2008,2009,2010,2011,2012

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