Write \(f_n(x)\) for the polynomial \(d^n/dx^n (x^2-1)^n\). Prove that if \(k < n\) $$\int_{-1}^{1} x^k f_n(x) dx = 0.$$ Deduce that, if \(g_n(x) = d/dx\{(x^2-1)f_n'(x)\}\) and \(k < n\), $$\int_{-1}^{1} x^k g_n(x) dx = 0.$$ Hence show that, if \(\lambda\) is the constant such that the coefficient of \(x^n\) in \(h_n(x) = g_n(x) - \lambda f_n(x)\) vanishes, \(h_n(x)\) is identically zero.
\begin{align} a(t) &= a_1 t + a_2 t^2/2! + \ldots + a_n t^n/n! + \ldots, \\ b(t) &= 1 + b_1 t + b_2 t^2/2! + \ldots + b_n t^n/n! + \ldots \end{align} are two power series such that \(b(t) = \exp\{a(t)\}\). Prove that the \(b_i\) are all integers if and only if the \(a_i\) all are.
Two great circles on a sphere of radius \(r\) meet at an angle \(A\). Find the areas of the four regions into which the surface of the sphere is divided. By considering the areas into which the sides of the triangle (produced) divide the sphere, or otherwise, show that the area of a spherical triangle with angles \(A\), \(B\), \(C\) is \(r^2(A + B + C - \pi)\). [The sides of a spherical triangle are arcs of great circles.]
Prove that the binomial coefficient \(\binom{a+b}{b}\) is odd if and only if, when \(a\) and \(b\) are expressed in binary notation and added, there is no `carrying over'.
The rhesus factor in blood is determined by two genes, one inherited from each parent, each to be either of the parent's two genes. There are two sorts of genes \(R\) and \(r\), and \(r\) is recessive, \(RR\) and \(Rr\) are positive. If the proportion of genes \(R\) and \(r\) in the population is \(55:9\), calculate the proportion of genes \(R\) and \(r\) positive to negative in the population is stable and they have 4 children. What are the odds that at least 2 of their children are positive? [You may assume that the proportion of positive to negative in the population is stable and that a man takes no account of rhesus factors in choosing a wife.]
Explain what is meant by an involution of pairs of points on a line. A line \(p\) meets the sides \(BC\), \(CA\), \(AB\) of a triangle \(ABC\) in \(L\), \(M\), \(N\), respectively. If \((L, L')\), \((M, M')\), \((N, N')\) are pairs of an involution on \(p\), prove that the lines \(AL'\), \(BM'\), \(CN'\) are concurrent.
A non-singular conic \(S\) and two points \(A_1\), \(A_2\) are in general position in a plane, and \(P\) is a variable point such that the lines \(A_1 P\), \(A_2 P\) are conjugate with respect to \(S\). Prove that the locus of \(P\) is a non-singular conic \(S'\) through \(A_1\), \(A_2\) and that the line \(A_1 A_2\) has the same pole with respect to \(S\) and \(S'\). Discuss the special cases when (i) \(A_1\), \(A_2\) are conjugate with respect to \(S\); (ii) the line \(A_1 A_2\) touches \(S\).
A billiard ball \(A\) is at rest when it is struck obliquely by another billiard ball \(B\). The collision is perfectly elastic and the balls are smooth. Show that it is possible to determine the ratio of the masses of \(A\) and \(B\) solely from measurements of the angles between the final paths of \(A\) and \(B\) and the initial path of \(B\).
Metal of uniform density is to be made into a body of externally cylindrical shape, symmetric about any plane through the axis of the cylinder, but possibly hollow. The body is to be rolled down a rough inclined plane of coefficient of friction \(\mu\) with its axis of symmetry horizontal. The inclination of the plane to the horizontal can be adjusted but the cylinder must roll. How should the cylinder be designed if it is to roll down the plane as quickly as possible?
A particle of mass \(m\) slides on a long smooth helical wire which can rotate freely about its vertical axis. The wire is of mass \(\beta m\) and its gradient is \(\alpha\). The projection of the wire onto a horizontal plane is a circle of radius \(a\). If the particle is released from rest, show that it falls through a height $$z = \frac{\alpha^2 \beta^2(1+\beta)}{2\{\alpha^2(1+\beta)+\beta\}}$$ in time \(t\).