Problems

Particles in a certain system can only have certain given energies \(E_1\), \(E_2\) or \(E_3\). If \(n_i\) particles have energy \(E_i\) (\(i = 1, 2, 3\)) write down conditions that

1. [(i)] the total number of particles is \(N\)
2. [(ii)] the total energy is \(U\).

Hence express two of the \(n_i\) as linear functions of the third. The 'entropy' \(S\) of the system is defined as the stationary value of the function \[W = \log(n_1! n_2! n_3!)\] subject to the above two conditions. Assuming that the \(n_i\) are so large that they may be regarded as continuously varying, show that \(W\) is stationary when \[\log n_i = \alpha+E_i/T\] where \(T\) is the root of the equation \[N\sum E_ie^{E_i/T} = U\sum e^{E_i/T},\] and where \[Ne^{-\alpha} = \sum e^{E_i/T}.\] If we define \(Z(T) = \sum e^{E_i/T}\), show that \[U = -NT^2\frac{d}{dT}\log Z\] and \[S = N\left(\log \frac{N}{Z}-1\right)+\frac{U}{T}.\] If \(U\) varies while \(N\) is fixed, show that \(\frac{dU}{dS} = T\). \(\left[\frac{d}{dn}\log n! \approx \log n \text{ for large } n, \log n! \approx n(\log n-1).\right]\)

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:

1. [(a)] \(|z+i| < 1\),
2. [(b)] \(|\Re(z)| < 1\),
3. [(c)] \(\Im(z^2) > 0\).

Show that the condition \(\Re(z + z^{-1}) > 1\), \(z \neq 0\), is equivalent to either \(x \geq 1\), or \(0 < x < 1\) and \(y^2 < x^3(1 - x)^{-1} + x\). Hence sketch the region determined by the condition \(\Re(z + z^{-1}) > 1\), \(z \neq 0\). (\(\Re(z)\) and \(\Im(z)\) denote the real and imaginary part, respectively, of a complex number \(z\).)

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:

1. [(i)] that, when \(\varepsilon = 0\), the value of \(v\) is independent of \(t\);
2. [(ii)] that, for a given value of \(\varepsilon > 0\), \(R\) is at the vertex of the parabola for just one value of \(t > 0\).