Problems

Find the area and centroid (centre of mass) of the plane region whose boundary is given in polar co-ordinates by \(r=a(1+\cos\theta)\).

Differentiate the following expressions:

1. \(\cos\log x\);
2. \((2+\cos x)\sin x\);
3. \(\int_1^{1+x^2} \frac{\sin t}{t} dt\).

Explain how a knowledge of the solutions of the equation \(f'(x)=0\) may give information about the roots of \(f(x)=0\), where \(f'(x)\) is the derivative of \(f(x)\). Show that the equation \[ 1-x+\frac{x^2}{2}-\frac{x^3}{3}+\dots+(-1)^n\frac{x^n}{n}=0 \] has one and only one real root if \(n\) is odd and no real root if \(n\) is even.

Let \(P(t)\) denote the point \[ (\cos t, f(t)\sin t), \] where \(f(t)\) is a strictly positive continuous function of \(t\) in \(0 \le t \le 2\pi\) with \(f(2\pi)=f(0)\); and let \(\mathcal{C}\) be the closed curve described by \(P(t)\) as \(t\) varies from \(0\) to \(2\pi\). Show that the area \(A\) enclosed by \(\mathcal{C}\) is \[ A = \int_0^{2\pi} f(t)\sin^2 t \,dt. \] Find an expression for the area \(T(t_1, t_2, \dots, t_n)\) of the polygon with vertices \(P(t_1), P(t_2), \dots, P(t_n)\), where \[ t_1 < t_2 < \dots < t_n < t_1+2\pi, \] and show that \[ \int_0^{2\pi} T\left(t, t+\frac{2\pi}{n}, t+\frac{4\pi}{n}, \dots, t+(n-1)\frac{2\pi}{n}\right) dt = nA\sin\frac{2\pi}{n}. \] Deduce that \[ T(t_1, t_2, \dots, t_n) \ge \frac{n\sin\frac{2\pi}{n}}{2\pi} A \] for some \(t_1, t_2, \dots, t_n\).

Prove that, if \(a, b\) are real, \[ ab \le \left(\frac{a+b}{2}\right)^2, \] and deduce that, if \(a, b, c, d\) are positive, \[ abcd < \left(\frac{a+b+c+d}{4}\right)^4, \] with equality only when the numbers are all equal. By giving \(d\) a suitable value in terms of \(a, b, c\), or otherwise, prove that, if \(a, b, c\) are positive, \[ abc \le \left(\frac{a+b+c}{3}\right)^3. \]

Prove what you can about the number of real roots of each of the equations

1. [(i)] \((x-a_1)(x-a_2)\dots(x-a_n) + (x-b_1)(x-b_2)\dots(x-b_n) = 0\), where \[ a_1 > b_1 > a_2 > b_2 \dots > a_n > b_n; \]
2. [(ii)] \((x-a)^m+(x-b)^m=0\) where \(a>b\) and \(m\) is a positive integer.

If \(u_n\) denotes the number of ways in which \(n\) men and their wives can pair off at a dance so that no man dances with his wife, prove that \[ u_n = (n-1)(u_{n-1}+u_{n-2}). \] Deduce that \[ \frac{u_n}{n!} - \frac{u_{n-1}}{(n-1)!} = \frac{(-1)^n}{n!}, \] and hence find an expression for \(u_n\).

Prove that, if \((1+x)^n = c_0 + c_1 x + \dots + c_n x^n\), then

1. [(i)] \(\dfrac{c_0}{1} - \dfrac{c_1}{2} + \dfrac{c_2}{3} - \dots + (-1)^n \dfrac{c_n}{n+1} = \dfrac{1}{n+1}\);
2. [(ii)] \(c_0^2 - c_1^2 + c_2^2 - \dots + (-1)^n c_n^2\) is equal to \((-1)^m (2m)!/(m!)^2\) if \(n\) is an even integer \(2m\). Find its value when \(n\) is an odd integer.

Resolve \(x^{2n}+1\) into real quadratic factors, where \(n\) is a positive integer. Express \[ \frac{1}{x^{2n}+1} \] in partial fractions with these factors as denominators.

If three straight lines do not all lie in one plane, prove that, in general, there are infinitely many straight lines which intersect them. Point out any exceptional cases. Of three straight lines \(ABC, DEF, GHK\), no two are in the same plane. They are all met by each of the straight lines \(ADG, BEH, CFK\). Prove that, in general, the lines \(BD, CG, FH\) are concurrent.