Show that with \(n\) rods of lengths \(1, 2, 3, \ldots, n\) it is possible to form exactly \(\frac{1}{24}n(n-2)(2n-5)\) triangles if \(n\) is even, and find the corresponding number if \(n\) is odd.
In a euclidean plane a point \(P'\) is said to be the reflection of a point \(P\) in a point \(A\) if \(A\) is the mid-point of \(PP'\); \(P'\) is the reflection of \(P\) in a line \(l\) if the line \(l\) bisects \(PP'\) at right angles. The operation of reflection in a point or a line will be denoted by the same symbol as the point or line itself. A symbol such as \(\ln B/l\) denotes the operation of successive reflection in \(l\), then in \(B\), then in \(A\) etc., taken in this order. If \(A\) is any operation of this type, \(R\) is denoted by \(R^{-1}\) and so on. The identity operation, in which every point of the plane is left unaltered, is denoted by \(I\). Show that every point of the plane is left unaltered if and only if \(R = I\) and only if
A uniform solid hemisphere is balanced in equilibrium with its curved surface in contact with a sufficiently rough inclined plane. Find the greatest possible value of the inclination of the plane to the horizontal, and less inclination there may be two positions of equilibrium, one stable and one unstable. Show also that the coefficient of friction has to be greater than \(3/\sqrt{(55)}\) for the plane to be sufficiently rough.
A uniform rod of length \(l\) lies horizontally on a rough plane inclined to the horizontal at an angle \(\alpha\). The coefficient of friction \(\mu\) is greater than \(\tan \alpha\). A gradually increasing force is applied upwards along the line of greatest slope at one end of the rod. Show that when the rod begins to move, the length of rod which moves upwards is less than \(l/\sqrt{2}\).
List clearly and concisely the main dynamical principles and problems involved in designing (i) an earth satellite, and (ii) a moon rocket. Where relevant express your answers also in mathematical form.
A photon of momentum \(k_0\) is absorbed by an electron initially at rest which instantly recoils and emits a second photon of momentum \(k\) in a direction making an angle \(\theta\) with the direction of \(k_0\). The electron at rest has an energy \(m_0\), and when moving with momentum \(p\) has an energy \(\sqrt{(m_0^2 + p^2)}\). The photons have energies \(k_0\) and \(k\) respectively. Prove that \[ \frac{1}{k} - \frac{1}{k_0} = \frac{1}{m_0}(1 - \cos \theta). \] [It is to be assumed that total energy is conserved, and that total momentum is conserved in the direction of \(k_0\) and perpendicular to this direction. Gravitational effects can be neglected.]
Prove that if \(x_1, x_2, \ldots, x_n\) and \(y_1, y_2, \ldots, y_n\) are two sets of positive quantities, both in increasing order of magnitude, then \[\frac{1}{n} \sum_{r=1}^{n} x_r y_r > \left(\frac{1}{n} \sum_{r=1}^{n} x_r\right) \left(\frac{1}{n} \sum_{r=1}^{n} y_r\right).\] Prove that if \(a\), \(b\), \(c\) are three unequal positive quantities, then \[a^3 + b^3 + c^3 > abc(a^2 + b^2 + c^2).\]
(i) Find the equation whose roots are the cubes of the roots of the equation \[x^3 + ax^2 + bx + c = 0.\] (ii) Show how to obtain the equation whose roots are the roots of a given algebraic equation multiplied by the same constant quantity. Hence, or otherwise, prove that an algebraic equation with integer coefficients and unit coefficient for the highest power cannot have a real rational root which is not integral.
Discuss the recurring series which is such that each term above the second is equal to the sum of the previous term and twice the term previous to that. Write down the scale of relation and determine the general term if the values of the first two terms are given. Examine the case when \(u_0 = 1\) and \(u_1 = 2\).
Stating without proof any properties of determinants used, express as a product of two linear terms and one quadratic term the determinant: \[\begin{vmatrix} x+a & b & c & d \\ b & x+c & d & a \\ c & d & x+a & b \\ d & a & b & x+c \end{vmatrix}.\]