Show that the simultaneous equations \begin{align*} ax+y+z&=p, \\ x+ay+z&=q, \\ x+y+az&=r \end{align*} have an unique solution if \(a\) has neither of the values 1 or \(-2\). Show also that, if \(a = -2\), there is no solution unless \(p, q\) and \(r\) satisfy a certain condition, and that there are then an infinite number of solutions. Discuss the solution of the equations when \(a=1\). Find the most general solution (if any) in the following cases: (i) \(a=3, p=q=r=1\), (ii) \(a=-2, p=q=r=1\), (iii) \(a=-2, p=1, q=-1, r=0\), (iv) \(a=1, p=q=r=0\).
If \(\alpha\) is a complex root of the equation \(x^7-1=0\), express the other six roots in terms of \(\alpha\). Show that \(\alpha+\alpha^2+\alpha^4\) is a root of a quadratic equation whose coefficients do not involve \(\alpha\). Prove that \[ \cos\frac{\pi}{7} - \cos\frac{2\pi}{7} + \cos\frac{3\pi}{7} = \frac{1}{2}, \quad -\sin\frac{\pi}{7} + \sin\frac{2\pi}{7} + \sin\frac{3\pi}{7} = \frac{\sqrt{7}}{2}. \]
Define the function \(e^y\), and deduce from your definition that, for all values of \(n\), \(y^n e^{-y} \to 0\) as \(y\to\infty\). Examine the behaviour of the following functions as \(x\) varies through real values, and in particular discuss their gradients for small positive and negative values of \(x\). Illustrate your results by sketch-graphs. \[ \text{(i) } \tanh\frac{1}{x}, \quad \text{(ii) } x\tanh\frac{1}{x}. \]
\(f(x)\) is a continuous function with continuous first, second and third derivatives, and \[ R(x) = \frac{1}{2} \int_0^x (x-t)^2 f'''(t) \,dt. \] Prove by integration by parts that \[ f(x) = f(0) + xf'(0) + \frac{x^2}{2}f''(0) + R(x). \] Show also that \[ R(x) = \frac{x^3}{3!}f'''(\theta x), \] where \(0 < \theta < 1\). State and prove a more general result applicable to a function with continuous derivatives up to and including the \(n\)th.
Show that the series \[ 1 + \frac{1}{2^k} + \frac{1}{3^k} + \dots \] is convergent if \(k>1\) but divergent if \(k=1\). Discuss the convergence of the series \[ 1 - \frac{1}{2^k} + \frac{1}{3^k} - \dots \] for real values of \(k\).
Prove that, if the joins of corresponding vertices of two coplanar triangles are concurrent, the intersections of corresponding sides are collinear. Five of the sides of a complete quadrangle \(PQRS\) pass through five of the vertices of a complete quadrilateral \(pqrs\), in such a way that the side \(PQ\) passes through the vertex \(rs\), and so on. Prove that the sixth side of the quadrangle passes through the sixth vertex of the quadrilateral.
Two conics \(S\) and \(S'\) have double contact at the points \(L\) and \(M\); \(A, B, C\) and \(D\) are the common points of \(S\) and a third conic \(S''\); \(p\) and \(q\) are a pair of lines through the four points of intersection of \(S'\) and \(S''\). Prove that there is a conic which passes through \(A, B, C\) and \(D\) and touches the lines \(p\) and \(q\) at their points of intersection with the line \(LM\).
A plank \(AB\), of uniform weight \(w\) per unit length and of length \(l\), rests in a horizontal position upon supports at its two ends. A man of weight \(W\) stands on the plank at a distance \(x\) from the end \(A\). Calculate the shearing force and the bending moment in the plank at a distance \(y\) from \(A\). If \(wl=3W\) and \(x=\frac{1}{4}l\), show in a diagram how the shearing force and the bending moment vary with \(y\), and calculate the greatest value of the bending moment.
A particle moves in a plane under a force of magnitude \(\omega^2 r\) per unit mass directed towards a fixed point \(O\) in the plane, where \(r\) is the distance of the particle from \(O\) and \(\omega\) is constant. \(O\) is taken as the origin of a system of rectangular Cartesian coordinates. The particle is projected from the point \((a, b)\) with velocity \((u, v)\) at time \(t=0\). Find the coordinates of the particle after a time \(t\). Verify that the moment of momentum about \(O\) is constant. Show that the particle is moving at right angles to the radius vector at times given by \[ \tan 2\omega t = \frac{2(au+bv)\omega}{(a^2+b^2)\omega^2 - (u^2+v^2)}. \]
Calculate the moment of inertia of a uniform circular disc of mass \(M\) and radius \(a\) about (i) an axis through its centre and normal to its plane, (ii) a parallel axis through a point \(A\) of its rim. If any general theorem is quoted it should be proved. The disc is mounted on bearings so as to rotate freely in a vertical plane about the axis through \(A\). It is held at rest with the diameter \(AB\) horizontal and is then released. Calculate the horizontal and vertical components of the force of reaction at \(A\) between the disc and its support (i) immediately after the disc is released, and (ii) when \(AB\) is vertical. When \(AB\) is vertical the disc is suddenly brought to rest by a horizontal impulsive force applied at \(B\). Calculate the magnitude of this impulse.