Explain how complex numbers may be represented as points in the Argand diagram. If \(P_1, P_2\) represent the numbers \(z_1, z_2\) respectively, give constructions for the points representing the numbers \(z_1+z_2\) and \(z_1z_2\). The point \(P_n\) represents the number \(z^n\), where \(z\) is a non-zero complex number. Show that the points \(P_n\), for \(n=0, \pm 1, \pm 2, \dots\) must either be collinear or concyclic, or lie on a certain equiangular spiral, that is, a curve such that the tangent at the general point \(P\) makes a constant angle with \(OP\); and in the latter case, find the constant angle in terms of \(z\).
Give a geometrical interpretation of the definite integral \(\int_a^b f(x)\,dx\) and deduce that, if \(f(x) \ge g(x)\) whenever \(a \le x \le b\), then \[ \int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx. \] Prove that, if \(0 \le x < 1\), \[ \left(1+\frac{x^2}{2}\right)^2 \le \frac{1}{1-x^2} \] and deduce that \[ x + \frac{x^3}{6} < \sin^{-1} x, \] if \(0 < x < 1\).
Evaluate
If \[ I_n = \int_\alpha^\beta \frac{x^n \,dx}{\sqrt{\{(\beta-x)(x-\alpha)\}}}, \] where \(\beta > \alpha, n \ge 0\), show that \begin{align*} 2I_1 &= (\alpha+\beta)I_0, \\ 2nI_n &= (2n-1)(\alpha+\beta)I_{n-1} - 2(n-1)\alpha\beta I_{n-2}, \quad (n \ge 2). \end{align*} Evaluate \[ \int_{-1}^2 \frac{x^3 \,dx}{\sqrt{\{2+x-x^2\}}}. \]
A beaker of thick glass is in the form of a circular cylinder, one end, the base, being closed. The walls of the beaker and the base are of uniform thickness \(t\). If the volume of glass used in making the beaker is fixed, find the ratio of the height of the beaker to the radius of the base which makes the internal volume of the beaker a maximum.
Sketch the curve \[ r(1-2\cos\theta) = 3a\cos 2\theta, \] and find the equations of its asymptotes.
What conditions must the positive integer \(n\) and the constants \(a\) and \(b\) satisfy in order that the \(n+1\) equations \begin{gather*} x_k - x_{k-1} + x_{k-2}=0, \quad (k=2,3,\dots,n) \\ x_0=a, \quad x_n=b \end{gather*} for the unknowns \(x_0, \dots, x_n\) shall have (i) a solution, (ii) one and only one solution?
Explain briefly how to find the H.C.F. of two integers or two polynomials. If \(m\) and \(n\) are positive integers whose H.C.F. is \(k\), prove that the H.C.F. of the integers \(2^m-1\) and \(2^n-1\) is \(2^k-1\) and that the H.C.F. of the polynomials \(x^{2^m}-x\) and \(x^{2^n}-x\) is \(x^{2^k}-x\).
A pack contains \(n\) cards numbered \(1, 2, \dots, n\). Two cards are drawn from the pack at random and a score is made equal to the product of the numbers on the two cards drawn. What will be the average score for all possible drawings (i) when the two cards are drawn simultaneously, (ii) when the first card is replaced and the pack shuffled before the second card is drawn?
Prove that \[ (1+x)^n - (1-x)^n = 2nx \prod_{k=1}^m \left(1+x^2\cot^2\frac{k\pi}{n}\right), \] where \(n\) is any positive integer and \(m\) is the greatest integer less than \(\frac{1}{2}n\).