Form the equation whose roots are \(\omega^{-1}p+\omega q, p+q, \omega p+\omega^{-1}q\), where \(\omega^3=1\) (\(\omega\ne 1\)). Solve the equation \[ x^3+6x^2-12x+32=0, \] by reducing it to the form of equation so obtained.
Find the sum to \(n\) terms of the recurring series \[ 1+2x+3x^2+9x^3+\dots, \] for which the scale of relation is \[ u_n - u_{n-1}x + 10u_{n-2}x^2 - 8u_{n-3}x^3. \]
\(ABCD\) is a quadrilateral inscribed in a conic \(S\), and circumscribed to a conic \(\Sigma\). \(AD, BC\) meet in \(Y\); \(AB, CD\) in \(Z\); \(AC, BD\) in \(X\). Taking \(XYZ\) as triangle of reference, shew that the equations of \(S\) and \(\Sigma\) can be written \begin{align*} S &\equiv x^2+y^2+z^2=0, \\ \Sigma &\equiv ax^2+by^2+cz^2+2fyz=0, \end{align*} where \[ a(b+c) = bc - f^2. \]
Interpret the equations \[ S+\lambda S'=0, \quad S+\lambda L^2=0, \quad S+\lambda LT=0, \] where \(S\) and \(S'\) are conics, \(L\) a straight line, and \(T\) a tangent to the conic \(S\). \par A circle meets a conic in four points \(A, B, C\) and \(D\). Shew that there are two parabolas and one rectangular hyperbola through these four points, and that the tangents at any one point \(A, B, C\) or \(D\) to the circle, hyperbola and parabolas form a harmonic pencil.
Find the maximum and minimum values of the function \[ u=x^3+y^3+z^3, \] where \(x,y\) and \(z\) are connected by the relations \begin{align*} x+y+z &= a, \\ x^2+y^2+z^2 &= a^2. \end{align*}
State and prove the formula for integration by parts, and shew that \[ \int_0^1 x^n(1-x)^m dx = \frac{m!n!}{(m+n+1)!}. \]
A particle of mass \(m\) is suspended by an elastic string of natural length \(l\), and is in equilibrium at a depth \(2l\) below the point of suspension. The particle is set in motion downwards with velocity \(\frac{an^2}{n-p}\), where \(n^2=\frac{g}{l}\), and simultaneously the point of suspension begins to oscillate with a motion given by \(z=a\sin pt\), \(z\) being measured vertically downwards. Prove that at time \(t\) the depth of the particle below its initial position is \[ \frac{an^2}{n^2-p^2}(\sin nt + \sin pt). \]
A particle moves in a plane under a central force \(\frac{\mu}{r^2}\) towards a point \(O\). Prove that the orbit is a conic. \par Find the equation of this conic if the particle is projected with velocity \(v\) from a point \(P\) at a distance \(a\) from \(O\), if the initial direction of the particle makes an angle \(\alpha\) with \(OP\) produced. Find also the velocity of the particle at each end of the major axis.
A uniform solid circular disc rests, with its plane vertical, on a planar lamina whose angle with the horizontal is \(\alpha\). The disc is released, and simultaneously the lamina begins to move downwards (without rotation) in its own plane, with uniform acceleration \(f\). If the disc rolls without slipping, shew that the velocity \(v\) of the centre of the disc, when it has moved through a distance \(l\), is given by \[ v^2 = \frac{2l}{3}(f+2g\sin\alpha). \]