Shew that the necessary and sufficient conditions that both roots of the equation \[ x^2+ax+b=0 \] should have their modulus equal to unity, are \[ |a|\leq 2, \quad |b|=1, \quad am b = 2 \text{ am } a. \] % "am" likely means "argument" or "amplitude"
Solution: (\(\Rightarrow\)) Suppose the roots are \(z_1, z_2\) then \(|b| = |z_1z_2| = |z_1||z_2| = 1\). \(|a| = |-(z_1+z_2)| \leq |z_1|+|z_2| = 2\) \(\arg(b) = \arg(z_1)+ \arg(z_2)\). \(\arg(a) = \arg(\cos \theta_1+\cos \theta_2 + i(\sin \theta_1 + \sin \theta_2)) = \arg (2 \cos\frac{\theta_1+\theta_2}{2}\cos \frac{\theta_1-\theta_2}{2} + 2i \sin \frac{\theta_1+\theta_2}{2} \cos \frac{\theta_1 - \theta_2}{2}) = \frac{\theta_1+\theta_2}{2}\)
Discuss the convergence of the series \[ \sum \frac{1 \cdot 3 \cdot 5 \dots (2n+1)}{2 \cdot 4 \cdot 6 \dots (2n+2)} p^n x^n \] for all real values of \(x\) and \(p\).
Prove that, if \(f(x)\) is continuous for \(a \leq x \leq b\), then \[ \frac{1}{n} \sum_{\nu=0}^{n-1} f\left\{a+\frac{\nu}{n}(b-a)\right\} \] tends to a limit as \(n\to\infty\). Taking \(a=0, b=1\), and \(f(x)=x\log x\), deduce that \[ 1^1 2^2 \dots (n-1)^{n-1} = n^{\frac{1}{2}n^2} e^{-\frac{1}{4}n^2(1+\epsilon_n)}, \] where \(\epsilon_n \to 0\) as \(n \to \infty\).
Shew that the integral \[ \int_0^\pi \cos(x\sin\phi-n\phi)d\phi \] is a solution of the equation \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-n^2)y=0, \] if and only if \(n\) is an integer.
Shew that the circles \[ (x-a)^2+(y-b)^2=r^2, \] where \(a, b\) and \(r\) are functions of a parameter \(t\), will be the circles of curvature of their envelope if \[ r'^2 = a'^2+b'^2, \] dashes denoting differentiations with respect to \(t\); and that the envelope will be given by \[ x = a - \frac{a'r}{r'}, \quad y = b - \frac{b'r}{r'}. \]
Shew that the equation of the osculating plane at the point \((x,y,z)\) of the sphero-conic in which the cone \(a\xi^2+b\eta^2+c\zeta^2=0\) cuts the sphere \(\xi^2+\eta^2+\zeta^2=1\) is \[ \frac{a(\xi-x)x^3}{b-c} + \frac{b(\eta-y)y^3}{c-a} + \frac{c(\zeta-z)z^3}{a-b} = 0. \]
Two particles of masses \(m\) and \(M\) are connected by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\). Shew that it is possible for the particles to describe concentric circles with uniform angular velocities each equal to \(\omega\), the distance between them being \[ \frac{(m+M)l\lambda}{(m+M)\lambda - mMl\omega^2}; \] and that the period of small oscillations about the state of steady motion is \(\pi/\omega\).
Calculate the principal moments of inertia at the vertex of a uniform right circular cone of semivertical angle \(\alpha\) and of mass \(M\). Such a cone has its vertex \(O\) and a point \(P\) on the rim of its base fixed, and rotates under the action of no forces with angular velocity \(\omega\) about \(OP\). Shew that the stress at \(O\) is \(\frac{1}{4}Mr\omega^2(5\sin^2\alpha+1)\), and at \(P\) is \(\frac{1}{4}Mr\omega^2(5\cos^2\alpha-1)\); where \(r\) is the radius of the circle described by the centroid of the cone.
A light string of length \(6l\) is stretched between two fixed points with tension \(T\); two particles, each of mass \(m\), are attached at the points of trisection, and a particle of mass \(M\) at the middle point. Shew that in small transverse oscillations one period is \(2\pi\sqrt{\left(\frac{2}{3}\frac{ml}{T}\right)}\); and that the other two periods cannot lie between this value and \(2\pi\sqrt{\left(\frac{Ml}{2T}\right)}\).
A sphere of S.I.C. \(K\) is introduced into a uniform field of electric force. Obtain expressions for the electric potential at points inside and outside the sphere; and shew that the greatest discontinuity in the direction of a line of force at the surface of the sphere is \[ \frac{\pi}{2} - 2(\text{arc cot}\sqrt{K}). \]