I.S.I. B.STAT ENTRANCE SUBJECTIVE 2008

    1. Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

 

    1. A 40 feet high screen is put on a vertical wall 10 feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

 

    1. Study the derivatives of the function
      \(\mathbf{y=\sqrt[3]{x^3-4x}}\)
      and sketch its graph on the real line.

 

    1. Suppose P and Q are the centres of two disjoint circles \(\mathbf{C_1}\) and \(\mathbf{C_2}\) respectively, such that P lies outside \(\mathbf{C_2}\) and Q lies outside \(\mathbf{C_1}\). Two tangents are drawn from the point P to the circle \(\mathbf{C_2}\), which intersect the circle \(\mathbf{C_1}\) at point A and B. Similarly, two tangents are drawn from the point Q to the circle \(\mathbf{C_1}\), which intersect the circle \(\mathbf{C_2}\) at points M and N. Show that AB=MN

 

    1. Suppose ABC is a triangle with inradius r. The incircle touches the sides BC, CA, and AB at D,E and F respectively. If BD=x, CE=y and AF=z, then show that \(\mathbf{r^2=\frac{xyz}{x+y+z}}\)

 

    1. Evaluate: \(\mathbf{\lim_{nto\infty} \frac{1}{2n} \ln\binom{2n}{n}}\)

 

    1. Consider the equation \(\mathbf{x^5+x=10}\). Show that
      (a) the equation has only one real root;
      (b) this root lies between 1 and 2;
      (c) this root must be irrational.

 

    1. In how many ways can you divide the set of eight numbers \(\mathbf{{2,3,\cdots,9}}\) into 4 pairs such that no pair of numbers has \(\mathbf{\text{gcd} }\) equal to 2?

 

    1. Suppose S is the set of all positive integers. For \(\mathbf{a,b \in S}\), define
      \(\mathbf{a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}}\)
      For example 8*12=6.
      Show that exactly two of the following three properties are satisfied:
      (i) If \(\mathbf{a,b \in S}\), then \(\mathbf{a*b \in S}\).
      (ii) \(\mathbf{(a*b)*c=a*(b*c)}\) for all \(\mathbf{a,b,c \in S}\).
      (iii) There exists an element \(\mathbf{i \in S}\) such that \(\mathbf{a *i =a}\) for all \(\mathbf{a \in S}\)

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  1. Two subsets A and B of the (x,y)-plane are said to be equivalent if there exists a function f: Ato B which is both one-to-one and onto.
    (i) Show that any two line segments in the plane are equivalent.
    (ii) Show that any two circles in the plane are equivalent.

2 Replies to “I.S.I. B.STAT ENTRANCE SUBJECTIVE 2008”

    1. Solution to 6: Let y be the given limit. Then e^y = (2nCn)^(1/2n). Now 2nCn = (2n)!/(n!)^2 . We divide both Nr and Dr by (2n)^2n. Hence e^y = [ {(2n)!/(2n)^2n}^(1/2n)]/[{n!/n^n}^2]^(1/2n)*2. Now limit n tends to infinity (n!/n^n)^(1/2n) is e^-1. Hence on simplification limit n tends to infinity e^y =2 and finally result is log 2. Do it yourself to understand the solution.

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