Problems

Let \(G\) be a group and let \(g \in G\). Let \[C(g) = \{x \in G : xg = gx\}.\] Show that \(C(g)\) is a subgroup of \(G\). Now let \(G\) be the group of symmetries of the square \(ABCD\). Let \(a\) be the rotation through \(\pi /2\) about an axis through the centre and perpendicular to the square. Let \(b\) be the rotation through \(\pi\) about an axis through the mid-points of \(AB\) and \(CD\). Show that every element of \(G\) can be written in one of the forms \(a^i\) or \(ba^i\) for \(i = 0, 1, 2, 3\). Determine those elements whose square is the identity. Show further that \(C(a^2) = G\) and that \(C(b) \neq G\).

A triangle inscribed in the parabola \(y^2 = x\) has fixed centroid \((\xi, \eta)\) (where \(\eta^2 < \xi\)). Show that the area of the triangle is greatest when one vertex is at the point \((\eta^2, \eta)\). [The area of the triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) is given by the absolute value of \[\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}.\] The product of the differences of the roots of the cubic equation \(t^3 + \alpha t^2 + \beta t + \gamma = 0\) is the square root of \(\alpha^2\beta^2 - 4\beta^3 + (18\alpha\beta - 4\alpha^3)\gamma - 27\gamma^2\).]

Show that there is an infinite number of rectangles circumscribing a given ellipse and that their vertices lie on a circle. Hence find the circumscribing rectangle of greatest area.

Show that it is not always possible to inscribe a circle within a convex quadrilateral with sides (taken consecutively) of given lengths, but that if it is possible then a circle may be inscribed in any convex quadrilateral with sides (taken consecutively) of these lengths.

Evaluate the following.

1. [(i)] \(\displaystyle \int_{-\pi}^{\pi} |\sin x + \cos x| dx,\)
2. [(ii)] \(\displaystyle \int_{-\pi}^{\pi} x^5 \cos x \, dx.\)

Show that if \(e(x)\) is a differentiable function with \(e'(x) = e(x)\) and \(e(0) = 1\) then, if \(a\) is any fixed real number, \[\frac{d}{dx}[e(a-x)e(x)] = 0.\] Deduce that \(e(x)e(y) = e(x+y)\) for all \(x\) and \(y\). Let \(c(x)\), \(s(x)\) be differentiable functions such that \(c'(x) = -s(x)\), \(s'(x) = c(x)\), \(s(0) = 0\) and \(c(0) = 1\). Show that \(c(x+y) = c(x)c(y) - s(x)s(y)\).

1. [(i)] Given that \(e^x = \sum_{r=0}^{\infty} \frac{x^r}{r!}\) \([0! = 1]\) prove that \(|e^{1.9} - 1\cdot 405| < 10^{-3}/2\).
2. [(ii)] The following is a table of the kinematic viscosity \(\nu\) of dry air at a pressure of one atmosphere at a given temperature \(T\), correct to three decimal places:
   \begin{tabular}{c|ccccc} \(T\) & 0 & 10 & 20 & 30 & 40 \\ \hline \(\nu\) & 0.132 & 0.141 & 0.150 & 0.160 & 0.159 \end{tabular}
   Would you be prepared to guess the value of \(\nu\) when \(T = 18\), \(T = 25\), \(T = 42\), \(T = 50\), \(T = 100\) or \(T = 500\)? What guess would you make if you did guess? How good would you expect your guesses to be? Give your reasons briefly but clearly.

Are the following statements true or false? If they are true give an example of a function \(f(x)\) defined for all real \(x\) with the stated behaviour. If they are false prove that no such function exists.

1. [(i)] Given \(g(x) > 0\) we can find \(f(x) > 0\) such that \(g(x)f(x) \to \infty\) as \(x \to \infty\).
2. [(ii)] There exists a function \(f(x)\) with an infinite number of maxima.
3. [(iii)] There exists a function \(f(x)\) with \(1 \geq f(x) \geq 0\) for all \(x\) such that \[\int_{-1}^{1} f(x) dx = \frac{1}{2} \quad \text{and} \quad \int_{-1}^{1} [f(x)]^2 dx = 1.\]
4. [(iv)] There exists a strictly increasing function \(f(x)\) [i.e. a function such that \(f(y) > f(z)\) whenever \(y > z\)] with no zeros [i.e. the equation \(f(x) = 0\) has no solution].

1. Show that \(\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}\) for \(|x| < 1\).
2. Deduce that \(\sum_{n=0}^{\infty} (n+r)(n+r-1)\ldots(n+1)x^n = \frac{r!}{(1-x)^{r+1}}\).
3. \(K\) brothers inherit \(\mathcal{L}n\). Show that the number \(N\) of ways of sharing this sum between them so that each receives a whole number of pounds (possibly zero) is equal to the coefficient of \(x^n\) in \((1+x+x^2+\ldots)^K\).
4. What is the value of \(N\) in terms of \(K\) and \(n\)? [Except in part (a) you need not justify your manipulations of power series.]

Let \(C\) be the arc of the parabola \(y = \frac{1}{2}x^2\) between \(x = 0\) and \(x = a\). Calculate the length of \(C\) and the area swept out when \(C\) is rotated about the \(x\)-axis.