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I.S.I. B.STAT ENTRANCE SUBJECTIVE 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 \(\displaystyle{ \sin^{5}x+\cos^{3}x=1 } \).
    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?
May 2, 2014

4 comments

  1. doubt question no. 4and 5

  2. 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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