Let \(p\) be a prime number, and let \(C\) denote the set of all complex \(p'\)th-power roots of unity (that is, the set of all \(\exp(2\pi in/p^r)\) with \(n\) and \(r\) positive integers). Show that \(C\) is a commutative group with respect to multiplication of complex numbers. Identify all the subgroups of \(C\). [It may be helpful to use the fact that, for integers \(m\) and \(n\), with no common factors other than \(\pm 1\), there are (not necessarily positive) integers \(a\) and \(b\) such that \(am + bn = 1\).]
(i) Show that every group all of whose non-identity elements have order 2 is commutative. (ii) Let \(G\) be the set of \(3 \times 3\) matrices of the form \(\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}\) with entries integers modulo 3. Show that, with respect to matrix multiplication, \(G\) is a non-commutative group all of whose non-identity elements have order 3. [The order of an element \(x\) of a group is the least integer \(n \geq 1\) such that \(x^n\) is the identity element. You may assume that \((AB)C = A(BC)\) for \(3 \times 3\) matrices \(A, B\) and \(C\).]
\(ABC\) is a triangle, and \(BCA', CAB', ABC'\) are equilateral triangles; \(A, A'\) being on opposite sides of \(BC\), \(B, B'\) on opposite sides of \(CA\) and \(C, C'\) on opposite sides of \(AB\). Prove that the lines \(AA', BB', CC'\) are of equal length and meet in a point.
\(ABCD\) is a square, whose opposite vertices \(A,C\) lie, respectively, on the lines \(y = mx, y = -mx\). If the equation of \(AC\) is \(\frac{x}{a} + \frac{y}{b} = 1\), find the coordinates of \(B\) and \(D\). Hence, or otherwise, show that if \(AC\) varies in such a way that \(B\) lies on the line \(px + qy + r = 0\), then the locus of \(D\) is a straight line, and find its equation.
Show that the four points \((at_i^2, 2at_i)\), for \(i = 1,2,3,4\), of the parabola \(y^2 = 4ax\) are concyclic if, and only if, \(t_1 + t_2 + t_3 + t_4 = 0\). \(PP'\) is a chord of a parabola perpendicular to the axis. A circle touches the parabola in \(P\) and meets it again in \(Q\) and \(R\). Show that \(QR\) is parallel to the tangent at \(P'\).
\(P\) is a variable point that moves so that the sum of its distances from fixed points \(S, S'\) is constant. By finding the equation of the locus of \(P\), or otherwise, show that the tangent to this locus at \(P\) bisects the angle \(SPS'\) externally.
Let \(I_n(z) = \int_0^1 (1-y)^n(e^{yz}-1)dy\), for all \(n \geq 0\). Prove that for all \(n \geq 1\), \(I_{n-1}(z) = \frac{z}{n}I_n(z) + \frac{z}{n(n+1)}\). Deduce that for all \(n \geq 1\), \(e^z = \sum_{r=0}^n \frac{z^r}{r!} + \frac{z^n}{(n-1)!}I_{n-1}(z)\).
The graph of \(y = f(x)\) for \(x \geq 0\) is a continuous smooth curve passing through the origin and lying in the first quadrant. It is such that, to each value of \(y\) there corresponds exactly one value of \(x\). Solving for \(x\), we obtain the equation \(x = g(y)\). Let \(F(X) = \int_0^X f(x)dx\), \(G(Y) = \int_0^Y g(y)dy\). Give a geometric argument to show that \(F(a) + G(b) \geq ab\) for any positive \(a, b\). When is equality obtained? Prove that \(u\log u - u + e^v - uv \geq 0\) if \(u \geq 1\) and \(v \geq 0\).
Sketch the graph of the function \(f(x) = -x\textrm{cosec} x\) in the range \(0 < x < 2\pi\). Prove that there is a unique value \(\bar{x}\) of \(x\) which minimises \(f(x)\) in the range \(\pi < x < 2\pi\), and show that the corresponding minimum value of \(f\) is \(\sqrt{(1+\bar{x}^2)}\).
Sketch the plane curve \(C\) whose polar equation is \(r = a\textrm{cosec}^2\frac{1}{2}\theta\), where \(0 < \theta < 2\pi\). Calculate: (i) the length of the arc \(C_1\) consisting of those points of \(C\) such that \(\frac{1}{2}\pi \leq \theta \leq \pi\); (ii) the area enclosed by the arc \(C_1\) and the radii \(\theta = \frac{1}{2}\pi\) and \(\theta = \pi\).