Problems

Trace the curve given by the equation \[ a^3(y+x) - 2a^2x(y+x) + x^5 = 0. \]

Establish the \((p,r)\) formula for the radius of curvature of a plane curve. \par For a certain curve it is known that the radius of curvature is given by \(a^n/r^{n-1}\) where \(n>-1\) and \(a\) is positive, and also that \(p=a/(n+1)\) when \(r=a\); shew that it is possible to express the curve by the polar equation \(r^n = (n+1)a^n \cos n\theta\).

If \(y = \sin(a\sin^{-1}x)\), shew that \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + a^2y = 0, \] where \(-\frac{\pi}{2} \le \sin^{-1}x \le \frac{\pi}{2}\), and \(a \neq 1\). \par Hence or otherwise, obtain a series of increasing powers of \(x\) for \(\sin(a\sin^{-1}x)\). \par Deduce that if \(a\) is an odd integer, the function is a polynomial in \(x\).

Shew that the area contained between a complete arc of the cycloid \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta) \] and its line of cusps is three times the area of the generating circle. Find also the centroid of this area. \par Determine the volume obtained by rotating the complete arc of the curve about its line of cusps.

(i) By use of a reduction formula, or otherwise, prove that \[ \int_0^\infty x^n e^{-ax} \sin bx \, dx = \frac{n!}{(a^2+b^2)^{\frac{n+1}{2}}} \sin(n+1)\alpha, \] where \(a\) and \(b\) are positive constants, \(n\) is a positive integer, and \(\tan\alpha=b/a\). (It may be assumed that \(\lim_{x\to\infty} x^n e^{-ax} = 0\).) \par (ii) Evaluate \(\int_0^1 x^m(1-x)^n dx\), where \(m\) and \(n\) are positive integers.

(i) Explain fully what is meant by an involution of pairs of points on a straight line and prove that \[ BC' \cdot CA' \cdot AB' + B'C \cdot C'A \cdot A'B = 0 \] is a necessary and sufficient condition that the three pairs \((A, A')\), \((B, B')\), \((C, C')\) should belong to an involution range. \par State the corresponding condition that three pairs of concurrent lines should belong to an involution pencil and deduce (or prove otherwise) that if two of the pairs are at right angles the third pair are also at right angles. \par (ii) \(A, B\) are the ends of a fixed diameter of a conic and \(P\) is a variable point on a fixed line \(\lambda\) through \(B\); if the tangents from \(P\) to the conic meet the tangent at \(A\) at \(Q, Q'\), prove that (a) \(Q, Q'\) are a pair in an involution, (b) \(AQ+AQ'\) is constant, (c) if the line \(\lambda\) cuts the conic again at \(C\) and the tangent at \(A\) at \(D\), then the tangent to the conic at \(C\) bisects \(AD\).

Prove that

1. [(i)] the coordinates \((x,y)\) of any point on any given conic can be written in the form \[ x:y:1 = \alpha t^2 + 2\alpha' t + \alpha'' : \beta t^2 + 2\beta' t + \beta'' : \gamma t^2 + 2\gamma' t + \gamma'', \] where \(t\) is a parameter;
2. [(ii)] the condition that the line \(lx+my+n=0\) should touch the conic is \[ (\alpha l + \beta m + \gamma n)(\alpha'' l + \beta'' m + \gamma'' n) - (\alpha' l + \beta' m + \gamma' n)^2 = 0. \]

Shew that, if \(a_n \to a\) and \(b_n \to b\) as \(n \to \infty\), then

1. [(i)] \(\text{Max}\,(a_n, b_n) \to \text{Max}\,(a,b)\), where \(\text{Max}\,(a,b)\) is equal to \(a\) if \(a \geq b\), and is equal to \(b\) if \(b > a\),
2. [(ii)] if \(b \neq 0\), then \(\frac{a_n}{b_n} \to \frac{a}{b}\),
3. [(iii)] if \(P(x)\) and \(Q(x)\) are polynomials and if \(Q(a) \neq 0\), then \(\frac{P(a_n)}{Q(a_n)} \to \frac{P(a)}{Q(a)}\).

Define \(\log x\) for positive values of \(x\), and prove from your definition that

1. [(i)] \(\log(xy) = \log x + \log y\),
2. [(ii)] \(\log x\) increases strictly from \(-\infty\) to \(\infty\) as \(x\) increases from \(0\) to \(\infty\),
3. [(iii)] \(\frac{d}{dx} \log x = \frac{1}{x}\),
4. [(iv)] if \(n\) is an integer, then \(\frac{(\log x)^n}{x} \to 0\) as \(x \to \infty\).

If \(x=r\cos\theta\), \(y=r\sin\theta\), find the values of \(A, B, C, D\) such that \begin{align*} \delta x &= A\delta r + B\delta\theta + \epsilon (|\delta x| + |\delta\theta|), \\ \delta y &= C\delta r + D\delta\theta + \epsilon' (|\delta x| + |\delta\theta|), \end{align*} where \(\epsilon\) and \(\epsilon'\) both tend to zero as \(|\delta x| + |\delta\theta| \to 0\). \par If \(\phi(x,y)\) is a function of \(x\) and \(y\), then it may be regarded as a function of any two of the variables \(x, y, r\) and \(\theta\); find the values of \(\frac{\partial\phi}{\partial x}\) and \(\frac{\partial^2\phi}{\partial x^2}\) (i) when the independent variables are \(x\) and \(r\), (ii) when they are \(x\) and \(\theta\). These values are to be expressed in terms of the partial derivatives of \(\phi\) when the independent variables are \(x\) and \(y\).