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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.
May 4, 2014