A small bead can move without friction on a smooth wire in the form of a circle of radius \(a\) which is made to rotate about a fixed vertical diameter with constant angular velocity \(\omega\). If the radius through the bead makes an angle \(\theta\) with the downward vertical at time \(t\), express \(d^2\theta/dt^2\) in terms of \(\theta\); hence prove that the bead can rest at the lowest point of the wire and that this configuration is stable provided that \(a\omega^2 < g\).
(i) Prove that the sum of the cubes of the first \(n\) integers is equal to the square of the sum of the integers. (ii) Prove that if \(n\) is a positive integer, \[ \frac{a}{b} + \frac{a(a+x)}{b(b+x)} + \dots + \frac{a(a+x)\dots(a+\overline{n-1}x)}{b(b+x)\dots(b+\overline{n-1}x)} = \frac{a}{a-b} \left[ \frac{(a+x)(a+2x)\dots(a+nx)}{b(b+x)\dots(b+nx)} - 1 \right]. \]
By using the identity \(\frac{1}{1-x} + \frac{x}{x-1} = 1\), show that % The identity is 1 + x/(1-x) = 1/(1-x). I will transcribe what is on the paper. \[ \sum_{r=1}^n \frac{r}{[(n+1-r)!r!]^2} = \frac{(2n+1)!}{[(n+1)!n!]^2} - \frac{(n+1)}{[(n+1)!]^2}. \]
Form the equation whose roots are the reciprocals of the roots of the equation \[ x^3+ax^2+bx-c=0. \] Hence solve the equation \[ 35x^3 - 18x^2 + 1 = 0. \]
Find the condition that the two equations \begin{align*} x^2+2ax+b^2 &= 0, \\ x^3+3p^2x+q^3 &= 0 \end{align*} should have a common root. Verify your condition from first principles
Find the values of \(x\) for which \(y=x^2(x-2)^3\) has maximum and minimum values, and evaluate for these values of \(x\) the curvature at the points of the curve given by the equation above.
(i) If \(I_n\) denote \(\int_0^{2\pi} \frac{\cos(n-1)x - \cos nx}{1-\cos x} dx\), show that \(I_n\) is independent of \(n\), where \(n\) is a positive integer. Hence evaluate \(I_n\) and prove that \[ \int_0^{2\pi} \left( \frac{\sin nx}{\sin x} \right)^2 dn = 2n\pi. \] (ii) Evaluate \(\int_0^\infty \frac{dx}{(1+x^2)^n}\), where \(n\) is a positive integer.
Prove that the mean distance of points on the surface of a sphere of radius \(a\) from an external point distant \(c\) from the centre is \(c+\frac{1}{3}\frac{a^2}{c}\). What is the value for an internal point?
Sketch the curve whose polar equation is \(r^2 = a^2(1+3\cos\theta)\) and find the area it encloses.
Justify Newton's method of approximation to the roots of the equation \(f(x)=0\) in the form \(\alpha - f(\alpha)/f'(\alpha)\), where \(\alpha\) is the first approximation, explaining by a diagram the importance of the equality of the signs of \(f(\alpha)\) and \(f''(\alpha)\), both assumed to be non-zero. Through a point on the circumference of a circle two chords are drawn to divide the area of the circle into three equal parts. Prove that the angle between the chords is approximately \(30^\circ 44'\).