The four roots \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) of \(x^4 -px^2+qx-r = 0\) satisfy \(\alpha\beta+\gamma\delta = 0\); by considering the two quadratic equations satisfied by \(\alpha\beta\), \(\gamma\delta\) and by \(\alpha+\beta\), \(\gamma+\delta\), or otherwise, prove that \(q^2 = 4pr\). Solve \(x^4-12x^2 + 12x- 3 = 0\).
Prove that, if \(0 \leq r \leq n\), then \(\sum_{i=r}^n \binom{i}{r} = \binom{n+1}{r+1}\). Hence or otherwise show that, for \(n \geq 4\), \[\sum_{i=0}^n i^4 = 24 \binom{n+1}{5}+36 \binom{n+1}{4}+ 14 \binom{n+1}{3} +\binom{n+1}{2}\] (The binomial coefficient \(\binom{n}{r}\) is defined by \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\); by convention, \(0! = 1\).)
Explain briefly how complex numbers may be represented geometrically as points of the complex plane. Describe geometrically the regions of the plane determined by each of the following conditions:
For a positive integer \(N\), \(\sigma(N)\) denotes the sum of all the positive integers which divide \(N\) (including 1 and \(N\)). If \(N = p^n\) where \(p\) is prime, show that \[\sigma(N) = \frac{p^{n+1} - 1}{p - 1}.\] Show further that, for an arbitrary positive integer \(N\) which factorizes as \(p_1^{n_1}p_2^{n_2}\ldots p_s^{n_s}\), where \(p_1,\ldots, p_s\) are distinct primes, \[\sigma(N) = \sigma(p_1^{n_1}) \ldots \sigma(p_s^{n_s}).\] Deduce that if \(N\) is an odd integer such that \(\sigma(N)\) is also odd, then \(N\) is a square.
For a positive integer \(N\) we write \(N = a_n a_{n-1} \ldots a_1 a_0\), where \(0 \leq a_i \leq 9\) for \(i = 0, \ldots, n\), to mean \(N = 10^n a_n + 10^{n-1} a_{n-1} + \ldots + 10 a_1 + a_0\). Show that any integer less than 1000, say \(N = a_2 a_1 a_0\), is divisible by 7 if and only if \(a_0 + 3a_1 + 2a_2\) is divisible by 7. Hence, or otherwise, show that an arbitrary integer \(N\) is divisible by 7 if and only if \(S\) is divisible by 7, where \[S = (a_0-a_3+a_6- \ldots) + 3(a_1-a_4+a_7- \ldots) + 2(a_2-a_5+a_8- \ldots).\]
Let \(G\) be a finite group of order \(n\) with identity element \(e\). For every integer \(m\) dividing \(n\) the subset \(G_m\) of \(G\) is defined by \[G_m = \{g \in G; g^m = e\}.\] Show that if \(G\) is Abelian then \(G_m\) is a subgroup of \(G\). What is the order of \(G_m\) when \(G\) is cyclic? If \(G\) is the group of rotations and reflections of an equilateral triangle, show that \(G_2\) is not a subgroup. Explain this.
Let \(E^{(ij)}\) be the \(3 \times 3\) real matrix with 1 in the \((i,j)\)th position and zeros everywhere else. Let \(F^{(ij)}(\lambda) = I + \lambda E^{(ij)}\) where \(I\) is the identity \(3 \times 3\) matrix. Show that for an arbitrary \(3 \times 3\) matrix \(A\), \(F^{(ij)}(\lambda)A\) (for \(i \neq j\)) is the matrix obtained from \(A\) by replacing the \(i\)th row \(A^{(i)}\) by \(A^{(i)} + \lambda A^{(j)}\) where \(A^{(j)}\) is the \(j\)th row of \(A\). Let \(A = \begin{pmatrix} 1 & -1 & 1 \\ 3 & 1 & 4 \\ 0 & 3 & 1 \end{pmatrix}\). Find a matrix \(Q\), which is the product of several \(F^{(ij)}(\lambda)\) for suitable \(i\), \(j\) and \(\lambda\), such that \(QA\) is of the form \(\begin{pmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{pmatrix}\). Hence solve the equation \[A \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}.\]
The tangents at points \(A\) and \(B\) of a circle \(\Gamma\) meet at a point \(O\). A chord of \(\Gamma\) passes through \(O\) and intersects \(\Gamma\) at \(P\) and \(Q\). The lines \(AB\) and \(PQO\) meet at \(R\). Prove that \[\frac{1}{OP} + \frac{1}{OQ} = \frac{2}{OR}.\]
In the Cartesian plane a point \(P\) on a parabola has parametric coordinates \((at^2, 2at)\). The points \(Q\) and \(R\) have coordinates \((at^2 + k, 2at + \varepsilon k)\) and \((a(1 + v), 0)\) respectively, where \(t > 0\), \(k > 0\), \(\varepsilon > 0\), and \(Q\) lies inside the parabola. The lines \(PQ\) and \(PR\) make equal angles with the inward normal to the parabola at \(P\). Show that \[(\varepsilon+t)(t^2 + 1 + v) = t(1 - \varepsilon t)(t^2 + 1 - v).\] Show further:
A solid is constructed by cutting the corners off a cube in such a way that its set of faces consists of six identical squares and eight identical equilateral triangles. At each vertex two triangular faces and two square faces meet. Find the cosine of the angle \(\theta\) between any two triangular faces which meet at a vertex.