If \(P\) and \(Q\) are polynomials and if the degree of \(Q\) is less than the degree of \(P\), show that polynomials \(P_0, P_1, P_2, \dots\) all of degree less than \(Q\) can be found such that \[ P = \Sigma P_i Q^i. \] Prove that the polynomials \(P_i\) are unique. \newline If the roots \(\alpha_1, \alpha_2, \dots, \alpha_n\) of the equation \(Q=0\) are all different, find the polynomial of least degree which takes the value \(a\) whenever \(Q=0\) and whose derived polynomial takes the value \(b\) whenever \(Q=0\). \newline [The derived polynomial of \(P(x)\) is the coefficient of \(h\) in the expansion of \(P(x+h)\) in powers of \(h\).]
State a necessary and sufficient condition that \[ z^2+4axz+6byz+4cxy+dy^2+2\lambda(x^2-yz) \] shall be the product of two linear factors. \newline By taking \(z=x^2, y=1\), state briefly the steps to be taken in order to find the roots of the quartic equation \[ x^4+4ax^3+6bx^2+4cx+d=0. \] Hence find the roots of the equation \[ x^4-x^3-4x^2+x+1=0. \]
The function \(f(x)\) is such that \[ \frac{f(c)-f(b)}{c-b} > \frac{f(b)-f(a)}{b-a} \] whenever \(a< b< c\). Interpret this geometrically. \newline Prove that, if \(f(x)\) has the above property and if \(a_1 < A < a_2\), then \[ f(a_1)+f(a_2) > f(A)+f(a_1+a_2-A). \] Deduce that \[ \frac{f(a_1)+f(a_2)+\dots+f(a_n)}{n} > f\left(\frac{a_1+a_2+\dots+a_n}{n}\right). \]
\(P\) is a point on a bar \(AB\) which moves in a plane and returns to its original position after completing exactly one revolution. \(S_P, S_A, S_B\) are the areas of the closed curves described by \(P, A, B\) respectively. Prove that \[ S_P = \frac{aS_B+bS_A}{a+b} - \pi ab, \] where \(a=AP, b=PB\). \newline [The areas are reckoned positive if the bounding curve is described anti-clockwise, otherwise they are reckoned negative.]
Give (with proofs) a method for finding the foci and directrices of a conic whose equation in rectangular cartesian coordinates is known. \newline One focus of an ellipse of eccentricity \(\frac{1}{2}\) is the point \((-3, 0)\), and the corresponding directrix is given by the equation \[ 3x+4y-6=0. \] Find the equation of the ellipse and use your method to find the equations of the other focus and directrix.
Prove Pascal's theorem that, if \(A, B, C, D, E, F\) are six points (assumed distinct) on a conic, then the three points of intersection \((AB, DE), (BC, EF), (CD, FA)\) are collinear. \newline Obtain theorems for a quadrangle \(ACEF\) inscribed in a conic (i) by making \(A, B\) coincide and \(C, D\) coincide; (ii) by making \(A, B\) coincide and \(D, E\) coincide. \newline Give direct proofs of the theorems you obtain.
Four uniform rods each of length \(l\) and weight \(W\) are smoothly jointed to form a rhombus \(ABCD\). The ends of a light elastic string of unstretched length \(l\) are tied to the joints \(A\) and \(C\) and the system hangs in equilibrium from \(A\). The string is such that the tension is \(4W(x-l)/l\) when the length \(x\) exceeds \(l\). Show that there is a position of equilibrium in which each rod makes the same angle with the vertical. What is this angle?
Define the moment of inertia of a plane lamina about an axis in its own plane. Prove that, if \(Ox, Oy\) are rectangular coordinate axes in this plane, then the moment of inertia about a coplanar axis \(OP\), inclined at an angle \(\alpha\) to \(Ox\), can be expressed in the form \[ A \cos^2\alpha - 2H \cos\alpha \sin\alpha + B\sin^2\alpha, \] where \(A, B\), and \(H\) are independent of \(\alpha\). \newline Show that a uniform triangular lamina of mass \(M\) has the same moment of inertia about any axis in its plane as three particles of mass \(M/3\) attached to the mid-points of the sides.
State Newton's laws of motion. \newline A raindrop falls from rest through an atmosphere containing water vapour at rest. The mass of the raindrop, which is initially \(m_0\), increases by condensation uniformly with time in such a way that after a given time \(T\) it is equal to \(2m_0\). The motion is opposed by a frictional force \(\lambda m_0/T\) times the velocity of the drop, where \(\lambda\) is a positive constant. Show that after time \(T\) the velocity is \[ \frac{gT}{2+\lambda}(2-2^{-(1+\lambda)}). \]
A simple pendulum, consisting of a bob of mass \(m\) attached to a fixed point by a light string, executes small damped oscillations in a medium that produces a retarding force given by \(2\mu m\) times the speed of the bob. Show that the equation of motion can be written as \[ \frac{d^2x}{dt^2} + 2\mu\frac{dx}{dt} + \left(\mu^2 + \frac{4\pi^2}{p^2}\right)x = 0, \] where \(x\) is the displacement of the bob from the position of equilibrium and \(p\) is the observed period of the oscillations. \newline Initially the pendulum is set in motion with \(x=0, dx/dt=v\). At every half swing, when \(x=0\), an impulse \(I\) is applied in the direction of motion, and of such an amount that successive swings have the same amplitude. Show that \[ I = mv(1-e^{-\mu p/2}). \]