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.