Problems

Show that, if the polynomial \[f(x) = x^3+3ax+b \quad (a \neq 0)\] can be expressed in the form \[A(x-p)^3+B(x-q)^3,\] where \(A\) and \(B\) are constants, then \(p \neq q\), and \(p, q\) are the roots of the equation \[at^2+bt-a^2 = 0.\] Prove, conversely, that if this equation has unequal roots then \(f(x)\) can be written in the form (1). Hence or otherwise find the real root of the equation \[x^3+54x+54 = 0.\]

Show that, if \(a > b > 0\) and \(m\) is a positive integer, then \[a^{m+1}- b^{m+1} \leq (a-b)(a+b)^m.\] Deduce that \[1^m+ 3^m+ ... + (2n - 1)^m \geq n^{m+1}.\] Interpret this result in terms of the position of the centre of mass of equal particles suitably placed on the curve \(y = x^m\). For what values of \(m\) other than positive integers do you think the result (2) is true?

Show that, if \(z_0\) is any non-zero complex number, then there is a complex number \(w_0\) such that \(z_0 w_0 = 1\). Let \(A\) be the set of all complex numbers \(z = x+iy\) such that \(x\) and \(y\) are integers, and let \(B\) be the set of all complex numbers such that \(x\) and \(y\) are rational. Let \(z_3 \neq 0\). Show that, if \(z_1, z_2\) belong to \(A\), then \(z_1/z_2\) need not belong to \(A\); but that if \(z_1, z_2\) belong to \(B\), then \(z_1/z_2\) must belong to \(B\). Find a number \(z_3\) of \(A\) such that \[\left|\frac{2+7i}{3+i}-z_3\right| \leq \frac{1}{\sqrt{2}},\] and show, more generally, that for any \(z_1, z_2\) of \(A\) with \(z_2 \neq 0\) there is a \(z_3'\) of \(A\) such that \[\left|\frac{z_1}{z_2}-z_3'\right| \leq \frac{1}{\sqrt{2}}.\] Find also a member \(z_4\) of \(A\) such that \[\left|\frac{z_4}{3+i}-z\right| \geq \frac{1}{\sqrt{2}}\] for all \(z\) of \(A\).

Let \(g\) be an element of a group \(G\), and let \(\langle g \rangle\) denote the set of elements \(g^i\) for all integers \(i\) (positive, negative or zero); let \(e\) be the identity element of \(G\). Prove the following results.

1. [(i)] \(\langle g \rangle\) is a subgroup of \(G\).
2. [(ii)] If \(g\) has order \(N\) (that is, if \(g^N = e\) but \(g^i \neq e\) for \(1 \leq i < N\)), then \(g^n = e\) if, and only if, \(N\) is a factor of \(n\).
3. [(iii)] If \(g\) has order \(N\), where \(N\) is prime, and if \(g^r \neq e\), then \(\langle g^r \rangle = \langle g \rangle\).

The pattern of work of an employee consists of \(m\) days on duty, followed by one day off; \(m\) is always the same, and all days of the week, including Bank Holidays, are treated alike in the rota; his annual holiday is to be ignored. By considering a suitable group, show that if his day off is not always on the same day of the week it must sometimes fall on a Sunday.

The rectangular Cartesian coordinates of \(P'\) are \((x', y')\) and a mapping \(\alpha\) of the plane into itself sends \(P\) to \(P' = (x', y')\), where \[\begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix},\] \(A\) being a \(2 \times 2\) matrix. The \(2 \times 2\) matrix \(B\) yields a mapping \(\beta\) in the same way. Show how the mapping determined by the matrix \(AB\) is related to \(\alpha\) and \(\beta\). Hence or otherwise show that, if \(l, m\) are two distinct fixed lines through the origin, and if \(P'\) is the projection of \(P\) onto \(l\) parallel to \(m\) (i.e. \(P'\) is on \(l\) and \(PP'\) is parallel to \(m\)), then \(A^2 = A\). It may be assumed without proof that this mapping is of the type (3). Now let \(B\) be any \(2 \times 2\) matrix which is neither the zero matrix nor the identity matrix, and which satisfies \(B^2 = B\). Show that \(B\) must take one or other of the forms \[\text{(i)} \begin{pmatrix} a & \lambda a \\ c & \lambda c \end{pmatrix} \text{ with } a+\lambda c = 1, \quad \text{(ii)} \begin{pmatrix} 0 & b \\ 0 & 1 \end{pmatrix}.\] Deduce that the mapping \(\beta\) is a projection onto one line parallel to another.

\(ABCDE\) is a regular pentagon of side 1. \(BD\) and \(CE\) meet in \(A'\), and \(DA\) and \(BC\) meet in \(C'\). Find the length of \(A'C'\). (Your answer should not contain trigonometric functions, but may contain square roots.)

The point \((x', y')\) is exterior to the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0.\] Establish a basic geometric property of the line \[\frac{xx'}{a^2}+\frac{yy'}{b^2}-1 = 0.\] Show that \[\lambda \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1 \right) - \left(\frac{xx'}{a^2}+\frac{yy'}{b^2}-1 \right)^2 = 0\] when \[\lambda = \frac{x'^2}{a^2}+\frac{y'^2}{b^2}-1\] is the equation of a conic which touches the ellipse at two points. Identify this conic.

A parabola rolls symmetrically on an equal fixed parabola. Find the locus of its focus.

Each of three circles \(C_1\), \(C_2\) and \(C_3\) meets the other two, but they do not have a common interior area. Show that there is a circle which meets each of them orthogonally. What happens if the three circles have a common area?

Let \(x_1, x_2, x_3\) be independent vectors in a vector space. Say whether each of the following statements is true, and justify your answers.

1. [(i)] The set \(\{x_1, x_2, x_3, y\}\) is linearly independent provided that every subset of three vectors is independent.
2. [(ii)] The vectors \(x_1+y, x_2+y, x_3+y\) are independent provided that they are all non-zero.
3. [(iii)] If \(y_1, y_2, y_3\) are independent, then so are \(x_1 + y_1, x_2 + y_2, x_3 + y_3\).
4. [(iv)] The vectors \(x_2 + x_3, x_3 + x_1, x_1 + x_2\) are independent.