ISI Entrance Paper 2008 – from Indian Statistical Institute’s B.Stat Entrance

1. Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer
2. 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?
3. Study the derivatives of the function $\mathbf{y=\sqrt{x^3-4x}}$
and sketch its graph on the real line.
4. 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
5. 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 $\mathbf{r^2=\frac{xyz}{x+y+z}}$
6. Evaluate: $\mathbf{\lim_{nto\infty} \frac{1}{2n} \ln\binom{2n}{n}}$
7. 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.
8. 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?
9. 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 $\mathbf{a \in S}$
10. .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.