Problems

Prove that, if \(n\) and \(r\) are positive integers, the coefficient of \(x^{n+r-1}\) in the expansion of \(\frac{(1+x)^n}{(1-x)^2}\) in ascending powers of \(x\) is \(2^{n-1}(n+2r)\). Find the sum to \(n\) terms of the series \[ \frac{3}{8} + \frac{3.5}{8.10} + \frac{3.5.7}{8.10.12} + \dots. \]

Prove that the number of ways in which \(n\) like things may be distributed among \(r\) people (\(n>r\)) so that everybody may have one at least is \[ \frac{(n-1)!}{(n-r)!(r-1)!}, \] but that if the things are all different the number is \[ r^n - r(r-1)^n + \frac{r(r-1)}{2!}(r-2)^n - \dots. \]

(i) If \[ y=(x+\sqrt{x^2-1})^n, \] prove that \[ (x^2-1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} = n^2y. \] (ii) If \(y=a+x\log\frac{y}{b}\), and \(x\) is so small that \(x^2\) and higher powers of \(x\) are negligible, prove by Maclaurin's theorem or otherwise that \[ y=a+x\log\frac{a}{b} + \frac{x^2}{a}\log\frac{a}{b}. \]

Prove that the radius of curvature at any point of a plane curve is \[ \frac{\{1+(\frac{dy}{dx})^2\}^{\frac{3}{2}}}{\frac{d^2y}{dx^2}}. \] A curve is determined by the property that the tangent to the curve at any point \(P\) meets a fixed straight line in \(R\), so that the length \(PR\) is constant. Prove that the radius of curvature of the curve at \(P\) is \(\frac{PR \cdot RN}{PN}\), where \(PN\) is the perpendicular from \(P\) to the fixed straight line.

In a given sphere of radius \(a\) a right circular cylinder is inscribed. Prove that the whole surface of the cylinder (including the ends) is a maximum when its height is \[ a\sqrt{2-\frac{2}{\sqrt{5}}}. \]

Prove that the curve \(y=e^{-ax}\cos bx\) lies between the curves \(y=e^{-ax}\) and \(y=-e^{-ax}\), touching each in turn, and has, if \(a=b\), points of inflexion at the points of contact.

Integrate

1. [(i)] \(\int \sin 3x . \sin 4x . dx\),
2. [(ii)] \(\int \log_e (x+\sqrt{a^2+x^2}) dx\),
3. [(iii)] \(\int \frac{x^2+2}{x^4+4}dx\),
4. [(iv)] \(\int_0^{\frac{\pi}{2}} \frac{dx}{4+5\cos x}\).

Make a rough sketch of the curve \[ y^2 = x^2(3-x)(x-2), \] and shew that its area is \(\frac{5\pi}{8}\).

Prove that the rectangle contained by the perpendiculars drawn from any point \(P\) on a circle to any two tangents is equal to the square of the perpendicular from \(P\) on the chord of contact.

\(TP\) and \(TQ\) are tangents to a parabola whose focus is \(S\). Prove that the triangles \(PST, TSQ\) are similar. From a point \(R\) on the axis of the parabola tangents are drawn meeting any other tangent in \(U\) and \(V\). Prove that \(SU=SV\).