Solve the equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2+2z &= 9, \\ xyz+xy &= -2. \end{align*}
(i) Show that, if \begin{align*} x^3+px+q &= 0, \\ x^3+rx+s &= 0 \end{align*} have a common root, then \[ (q-s)^3 = (ps-qr)(p-r)^2. \] (ii) If \(\alpha, \beta, \gamma\) are the roots of \(x^3+px+q=0\), find the equation with roots \(\alpha^3, \beta^3, \gamma^3\).
If \(x_1, \dots, x_n; y_1, \dots, y_n\) are real numbers, prove that \[ (x_1^2 + \dots + x_n^2)(y_1^2 + \dots + y_n^2) \ge (x_1y_1 + \dots + x_ny_n)^2, \] and state under what conditions the equality sign holds. If \[ C_r = \frac{n!}{r!(n-r)!}, \] prove that \[ \sqrt{C_1} + \sqrt{C_2} + \dots + \sqrt{C_n} \le \sqrt{\{n(2^n-1)\}}. \]
The circumference of a circle, centre \(O\) and radius \(a\), is divided into \(2n+1\) equal arcs by points \(A_1, A_2, \dots, A_{2n+1}\), where \(n \ge 1\). Starting from \(A_1\), with the pencil never leaving the paper and moving always in an anticlockwise direction round \(O\), a sequence of chords is drawn which subtend at \(O\) successively the angles \(\theta, 2\theta, \dots, n\theta, \theta, 2\theta, \dots, n\theta, \theta, 2\theta, \dots\), where \((2n+1)\theta=2\pi\). Show that, if this construction is continued for sufficiently long, a polygonal line is drawn which begins and ends at \(A_1\) and contains each chord \(A_iA_j\) (\(1\le i < j \le 2n+1\)) exactly once; and find the total length of this line.
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.