- Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer
- 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?
- Study the derivatives of the function
and sketch its graph on the real line.
- Suppose P and Q are the centres of two disjoint circles and respectively, such that P lies outside and Q lies outside . Two tangents are drawn from the point P to the circle , which intersect the circle at point A and B. Similarly, two tangents are drawn from the point Q to the circle , which intersect the circle at points M and N. Show that AB=MN
- 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
- Consider the equation . Show that
(a) the equation has only one real root;
(b) this root lies between 1 and 2;
(c) this root must be irrational.
- In how many ways can you divide the set of eight numbers into 4 pairs such that no pair of numbers has equal to 2?
- Suppose S is the set of all positive integers. For , define
For example 8*12=6.
Show that exactly two of the following three properties are satisfied:
(i) If , then .
(ii) for all .
(iii) There exists an element such that for all
- 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.