Find the linear factors of \[ a^3(b-c) + b^3(c-a) + c^3(a-b). \] Show that if \(x^3+y^3+z^3 = 3mxyz\), and \begin{align*} ax^2+by^2+cz^2&=0 \\ ayz+bzx+cxy&=0 \end{align*} then \[ a^3+b^3+c^3 = 3mabc. \]
Thirty balls of which twelve are alike and black, and eighteen are alike and white, are dropped into three boxes, so that there are ten balls in each box. Taking into account permutations among the boxes, find the number of ways in which the balls can be distributed between the boxes so that no box contains more than five black balls.
Find the sum of the series \(a_0+a_1x+a_2x^2+\dots\), whose coefficients satisfy the relation \[ 3a_n - 7a_{n-1} + 5a_{n-2} - a_{n-3} = 0, \text{ and } a_0=1, a_1=8, a_2=17 \] proving that \[ 2a_n = 20n - 7 + 3^{2-n}. \]
Find the only value of \(x\) which satisfies the equation \[ 3\tan^{-1}x + \tan^{-1}3x = \frac{1}{4}\pi, \] where \(\tan^{-1}x\) and \(\tan^{-1}3x\) each lie between \(0\) and \(\frac{1}{2}\pi\).
An aeroplane is travelling in a horizontal line with constant velocity. Explain how two observers at the same level and at a known distance apart may calculate the speed and direction of motion when two pairs of simultaneous observations of the bearing of the aeroplane are made at observed times.
The real quantities \(x,y,u,v\) are connected by the equation \[ \cosh(x+iy) = \cot(u+iv). \] Prove that \[ \frac{\sinh 2y}{\sin 2u} = -\tanh x \tan y, \] and that \[ \coth 2v = -(\cosh 2x + \cos 2y + 2)/4 \sinh x \sin y. \]
Expand \((x^2+1)^{\frac{1}{2}}\sinh^{-1}x\) in a series of ascending powers of \(x\), and if \(a_n\) is the coefficient of \(x^n\), prove that \[ a_{n+2} = -\left(\frac{n-1}{n+2}\right)a_n. \quad (n>1.) \]
Prove that if \(s\) is the arc of the curve \(3ay^2 = x(x-a)^2\) from the origin to the point \((x,y)\), then \[ s^2=y^2+\frac{4}{9}x^2. \] If \(S\) is the whole length of the loop of the curve, \(A\) its area and \(B\) its greatest breadth parallel to the \(y\) axis, prove that \(A=\frac{4}{15}BS\).
Evaluate \[ \int \frac{dx}{\sqrt{(1+\sin x)(2-\sin x)}}. \] Prove that \[ \int_0^1 \frac{(4x^2+3)dx}{8x^2+4x+5} = \frac{1}{4} - \frac{1}{8}\log\frac{17}{5} + \frac{1}{4}\tan^{-1}\frac{6}{7}, \] and \[ \int_0^{\frac{\pi}{2}} \frac{dx}{1+2\cos x} = \frac{1}{\sqrt{3}}\log(2+\sqrt{3}). \]
Prove that \[ \sin A + \sin B + \sin C - \sin(A+B+C) = 4\sin\frac{1}{2}(B+C)\sin\frac{1}{2}(C+A)\sin\frac{1}{2}(A+B). \] If \(A+B+C=180^\circ\), shew that \begin{align*} \cos 3A \sin(B-C) + \cos 3B \sin(C-A) + \cos 3C \sin(A-B) \\ + 4\sin(B-C)\sin(C-A)\sin(A-B) = 0. \end{align*}