Problems

The coordinates \((x,y)\) of any point on a given plane curve are expressed as functions of a parameter \(\theta\). Obtain expressions for the coordinates of the centre and for the radius of curvature in terms of \(x,y\) and their differential coefficients with respect to \(\theta\). \par Apply these results to find the equation of the evolute of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) in the form \((ax)^{2/3}+(by)^{2/3}=(a^2-b^2)^{2/3}\).

The solid angle subtended at a point \(O\) by a plane area may be defined as the area cut off on a sphere of unit radius whose centre is \(O\) by the straight lines joining \(O\) to the perimeter of the plane area. Find the solid angle subtended by a circle at a point on the line through its centre and perpendicular to its plane in terms of \(\alpha\), the angle subtended at the point by a radius of the circle. \par Shew also that a rectangle of sides \(2a, 2b\) subtends a solid angle \[ 4\sin^{-1}\frac{ab}{\sqrt{(a^2+h^2)(b^2+h^2)}} \] at a point on the line through its centre and perpendicular to its plane, where \(h\) is the perpendicular distance of the point from the plane.

Evaluate \[ \int \frac{dx}{x^4+a^4}, \quad \int_a^b \sqrt{(b-x)(x-a)}\,dx. \] If \(I(m,n) = \int \sin^m\theta\cos^n\theta d\theta\), express \(I(m,n)\) in terms of \(I(m-2, n-2)\).

If \(y_r(x)\) satisfies the equation \[ \frac{d}{dx}\left((1-x^2)\frac{dy}{dx}\right) + r(r+1)y=0, \] shew that if \(m \ne n\) then \[ \int_{-1}^{+1} y_m(x)y_n(x)dx=0. \]

If \(ABC\) is a triangle self-polar with respect to a conic \(S\), and if \(\alpha\) is the polar of another point \(A'\) with respect to \(S\), prove that the double points of the involution cut out on \(\alpha\) by conics through \(A, B, C\) and \(A'\) are the intersections of \(\alpha\) with \(S\). \par Hence, or otherwise, prove that, if each of two triangles is self-polar with respect to a conic, their six vertices lie on a conic. \par Deduce that if a triangle is self-polar with respect to a rectangular hyperbola, its circumcircle passes through the centre of the rectangular hyperbola.

The equation of a conic referred to rectangular Cartesian coordinates is \[ S \equiv ax^2+2hxy+by^2+2gx+2fy+c=0; \] if \(u \equiv ax+hy+g\) and \(v \equiv hx+by+f\), prove that

1. [(i)] chords of the conic parallel to the line \(mx=ly\) are bisected by the line \(lu+mv=0\);
2. [(ii)] the lines \(mx=ly, m'x=l'y\) are parallel to conjugate diameters of the conic, if \[ all' + h(lm'+l'm)+bmm'=0; \]
3. [(iii)] the asymptotes of the conic are \(bu^2-2huv+av^2=0\);
4. [(iv)] the principal axes of the conic are \(h(u^2-v^2)=(a-b)uv\);
5. [(v)] the director circle of the conic is \((a+b)S = u^2+v^2\).

The equation \(x^n+p_1x^{n-1}+p_2x^{n-2}+\dots+p_n=0\) has roots \(\alpha_1, \alpha_2, \dots, \alpha_n\); and \[ S_m = \alpha_1^m + \alpha_2^m + \dots + \alpha_n^m. \] Obtain equations of the type \[ S_m + p_1 S_{m-1} + p_2 S_{m-2} + \dots + p_n S_{m-n} = 0, \text{ etc.,} \] connecting the sums of the powers of the roots with the coefficients \(p_r\). \par If \(\alpha, \beta, \gamma\) are the roots of the cubic equation \(x^3-px-q=0\), shew, by expanding \(\log(1-px^2-qx^3)\) or otherwise, that \[ \alpha^m+\beta^m+\gamma^m = m \sum \frac{(\lambda+\mu-1)!}{\lambda!\mu!}p^\lambda q^\mu, \] where the summation is over all values of \(\lambda\) and \(\mu\) such that \(2\lambda+3\mu=m\).

Two polynomials, \(P\) and \(Q\), have no factor in common. Shew that the maximum and minimum values of \(P/Q\) are the values of \(\lambda\) for which \(P-\lambda Q=0\) has a real root of even multiplicity. \par Shew further that the value \(\lambda\) is a maximum or a minimum according as \[ \{P^{(2v)}(\alpha)Q(\alpha) - P(\alpha)Q^{(2v)}(\alpha)\} \le 0, \] \(\alpha\) being the root of \(P-\lambda Q=0\) of multiplicity \(2v\). \par Find the turning values of \[ \frac{x+2}{x^2+x+2}, \] distinguishing between maxima and minima. \par [The condition that the cubic \(y^3+gy+h=0\) should have a pair of equal roots is that \(4g^3+27h^2=0\).]

``If \(\xi\) is an approximate root of the equation \(f(x)=0\), then in general \(\xi - f(\xi)/f'(\xi)\) is a better approximation.'' \par Discuss this statement graphically, pointing out cases when repeated application of the approximation will give the value of the root to any desired degree of accuracy and cases when it will not. \par If \(a\) is small the equation \(\sin x = ax\) has a root \(\xi\), nearly equal to \(\pi\). Shew that \[ \xi = \pi\left\{1-a+a^2-\left(\frac{\pi^2}{6}+1\right)a^3\right\} \] is a better approximation, if \(a\) is sufficiently small.

Explain the use of the ``angle of friction'' in the determination of the positions of equilibrium of a system containing imperfectly rough rigid bodies in contact. \par A uniform circular cylinder of radius \(a\) rests on a horizontal plane, with its axis parallel to and at a distance \((a+2a\sin 2\theta)\) from a vertical wall, where \(2\theta\) is a given acute angle. A similar cylinder is gently placed in contact with the first cylinder and with the wall, touching each along a generator. The coefficient of friction between the upper cylinder and the wall is \(\mu_1\), between the two cylinders \(\mu_2\), and between the lower cylinder and the plane \(\mu_3\). Shew that equilibrium is not possible unless \(\mu_1 \ge 1\). \par Assuming now \(\mu_1 \ge 1, \mu_2 \ge 1\), shew that equilibrium is possible for all values of \(\theta\) if \(\mu_3 \ge \frac{1}{4}\). If \(\mu_3 < \frac{1}{4}\), determine the greatest value of \(\theta\) for which equilibrium is possible.