Problems

Functions \(u(x), v(x)\) are defined by the equations \begin{align*} u''+u=0, &\quad v''+v=0, \\ u(0)=0, &\quad u'(0)=1; \\ v(0)=1, &\quad v'(0)=0, \end{align*} where and \(u'=\dfrac{du}{dx}\), etc. Without using trigonometrical functions, prove that

1. [(i)] \(u'(x)=v(x), \quad v'(x)=-u(x)\),
2. [(ii)] \(u^2+v^2=1\),
3. [(iii)] \(v(x+y)=v(x)v(y)-u(x)u(y)\).

If \(f(x)\) is a function defined in the interval \((a

A chain consists of two portions \(AC, CB\), each of length \(l\), and of uniform densities \(w, w'\) respectively. \(A\) and \(B\) are attached to two points and the chain hangs under gravity in such a way that \(C\) is at the lowest point. Prove that, if the heights of \(A\) and \(B\) above \(C\) are \(h\) and \(h'\), \[ w\left(h - \frac{l^2}{h}\right) = w'\left(h' - \frac{l^2}{h'}\right). \] Prove also that the curvature at \(C\) of the portions \(AC, CB\) are in the ratio \(w:w'\).

Two equal, perfectly elastic, smooth spheres are suspended by vertical strings so that they are in contact, with their centres at the same level. A third equal sphere falls vertically and strikes the other two spheres symmetrically with velocity \(u\), in such a way that at the moment of impact the three centres are in a vertical plane. Shew that the falling sphere rebounds with velocity \(\dfrac{5u}{7}\) and that the other two spheres begin to move outward with velocity \(\dfrac{2\sqrt{3}u}{7}\). Find the impulsive tension in the strings, given that \(w\) is the weight of each sphere.

A reel consists of a cylinder of radius \(r\) and two rims of radius \(R (>r)\). The mass of the reel is \(M\) and its radius of gyration about its axis is \(k\). It is placed on a perfectly rough horizontal table and the thread is drawn out horizontally from beneath it and passes over a light pulley at the edge of the table. The free end is then attached to a mass \(m\) which is allowed to descend under gravity. Shew that the mass \(m\) descends with acceleration \[ \frac{mg(R-r)^2}{M(R^2+k^2)+m(R-r)^2}. \]

Give a geometrical construction for the circle passing through a given point and coaxal with two given circles which do not meet in real points. \(U\) and \(V\) are two fixed circles, \(P\) is a given point; \(Q\) is the inverse of \(P\) with respect to \(U\), \(R\) is the inverse of \(Q\) with respect to \(V\), \(S\) the inverse of \(R\) with respect to \(U\), and so on. Prove that all these points are concyclic.

Six equal uniform rods, each of weight \(W\), are smoothly jointed together so as to form a regular hexagon \(ABCDEF\) which hangs from the point \(A\) and is kept in shape by strings \(AC, AD, AE\). Find the tensions in these strings.

Prove that the circumcentre \(O\), the centroid \(G\), and the orthocentre \(H\), of a triangle \(ABC\) are collinear, and that \(OH=3OG\). A triangle \(ABC\) is inscribed in a fixed circle with centre \(O\), and varies so that \(A\) is fixed and \(BC\) passes through a fixed point \(P\). Prove that the locus of the orthocentre of \(ABC\) is a circle whose radius is equal to \(OP\).

Shew that, if \(x^4 + ax + b\) has a factor \(x^2 + px + q\), then \[ p^6 - 4bp^2 - a^2 = 0 \quad \text{and} \quad q^6 - bq^4 - a^2q^3 - b^2q^2 + b^3 = 0. \] Solve the equation \[ x(x-1)(x-2)(x-3) = a(a-1)(a-2)(a-3), \] and find for what values of \(a\) the roots are all real.

A bag contains \(n\) balls, three red and the rest white. They are drawn out one by one. Find the probability that no two red balls will be drawn consecutively, and shew that this probability is less than \(\frac{1}{2}\) unless \(n\) is greater than 10.