1. Suppose a is a complex number such that \(\mathbf{ a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0 }\) If m is a positive integer, find the value of \(\mathbf{a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}}\)
  2. Use calculus to find the behaviour of the function \(\mathbf{ y=e^x\sin{x} -\infty <x< +\infty}\) and sketch the graph of the function for \(\mathbf{-2\pi \le x \le 2\pi}\). Show clearly the locations of the maxima, minima and points of inflection in your graph.
  3. Let f(u) be a continuous function and, for any real number u, let [u] denote the greatest integer less than or equal to u. Show that for any x>1, \(\mathbf{\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }\)
  4. Show that it is not possible to have a triangle with sides a,b, and c whose medians have length \(\mathbf{\frac{2}{3}a, \frac{2}{3}b \text{and} \frac{4}{5}c}\).
  5. Show that \(\mathbf{-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2 }\) for all values of \(\mathbf{theta}\).
  6. Let \(\mathbf{S={1,2,\cdots ,n}}\) where n is an odd integer. Let f be a function defined on \(\mathbf{{(i,j): i\in S, j \in S}}\) taking values in S such that
    (i) \(\mathbf{f(s,r)=f(r,s) \text{for all} r,s \in S}\)
    (ii) \(\mathbf{{f(r,s): s\in S}=S \text{for all} r\in S}\)Show that \(\mathbf{{f(r,r): r\in S}=S}\)
  7. Consider a prism with triangular base. The total area of the three faces containing a particular vertex A is K. Show that the maximum possible volume of the prism is \(\mathbf{\sqrt{\frac{K^3}{54}}}\) and find the height of this largest prism.
  8. The following figure shows a \(\mathbf{3^2 \times 3^2 }\) grid divided into \(\mathbf{3^2}\) subgrids of size \(\mathbf{3 \times 3}\). This grid has 81 cells, 9 in each subgrid.
    Now consider an \(\mathbf{n^2 times n^2}\) grid divided into \(\mathbf{n^2}\) subgrids of size \(\mathbf{n \times n}\). Find the number of ways in which you can select n^2 cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.
  9. Let X \(\mathbf{\subset \mathbb{R}^2}\) be a set satisfying the following properties:
    (i) if \(\mathbf{(x_1,y_1)}\) and \(\mathbf{(x_2,y_2)}\) are any two distinct elements in X, then
    \(\mathbf{\text{ either, } x_1 > x_2 \text{ and } y_1 > y_2 \text{ or, } x_1 < x_2 \text{ and } y_1 < y_2}\)
    (ii) there are two elements \(\mathbf{(a_1,b_1)}\) and \(\mathbf{(a_2,b_2)}\) in X such that for any \(\mathbf{(x,y) in X}\),
    \(\mathbf{a_1\le x \le a_2 \text{ and } b_1\le y \le b_2 }\)
    (iii) if \(\mathbf{(x_1,y_1) \text{and} (x_2,y_2)}\) are two elements of X, then for all \(\mathbf{\lambda \in [0,1], \left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X }\)
    Show that if \(\mathbf{(x,y) \in X}\), then for some \(\mathbf{\lambda in [0,1], x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2 }\)
  10. Let A be a set of positive integers satisfying the following properties:
    (i) if m and n belong to A, then m+n belong to A;
    (ii) there is no prime number that divides all elements of A.(a) Suppose \(\mathbf{ n_1 \text{and} n_2 }\) are two integers belonging to A such that \(\mathbf{n_2-n_1 > 1}\). Show that you can find two integers \(\mathbf{m_1 \text{and} m_2 }\) in A such that \(\mathbf{0 < m_2-m_1 < n_2-n_1}\)
    (b) Hence show that there are two consecutive integers belonging to A.
    (c) Let \(\mathbf{n_0 \text{and} n_0+1 }\) be two consecutive integers belonging to A. Show that if \(\mathbf{n\geq n_0^2 }\) then n belongs to A.