Prove the formula \(\rho=r\frac{dr}{dp}\) for the radius of curvature of a curve at a point \(P\) where the length of the radius vector is \(r\) and the length of the perpendicular from the origin on to the tangent at \(P\) is \(p\). Sketch roughly the curve for which \(\frac{p}{r}\) is a constant and shew that \(\rho\) also is in constant ratio to \(r\) and \(p\). Prove that the radius of curvature can never be less than the corresponding radius vector.
Shew that if \(f(x)\) and \(\phi(x)\) are functions of \(x\) having derivatives \(f'(x), \phi'(x)\) in the range \((a,b)\), then \(\frac{f'(\xi)}{\phi'(\xi)} = \frac{f(b)-f(a)}{\phi(b)-\phi(a)}\) for some value \(\xi\) of \(x\) between \(a\) and \(b\). What restrictions (if any) are to be placed on the behaviour in the interval \((a,b)\) of \(f'(x)\) and \(\phi'(x)\)? Hence or otherwise prove that \(x^\lambda \log_e x \to 0\) as \(x \to 0\), if \(\lambda > 0\).
Find a reduction formula for \(f(m,n) = \int_0^1 x^{n-1}(1-x)^{m-1}dx\) and shew that \[ f(m,n) = \frac{(m-1)!}{n(n+1)\dots(n+m-1)}, \] \(m\) and \(n\) being positive integers. Shew further that if \(I(n) = \int_0^\infty e^{-x} x^{n-1}dx\), then (i) \(I(n+1) = n.I(n)\), (ii) \(f(m,n) = \frac{I(n).I(m)}{I(m+n)}\).
If \(x,y,z\) are three variables each of which may be regarded as a function of the other two, shew that (i) \(\left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{(\frac{\partial y}{\partial x})_z}\), where \(\left(\frac{\partial x}{\partial y}\right)_z\) denotes the partial differential coefficient of \(x\) with respect to \(y\), \(z\) being constant. (ii) \(\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y + 1 = 0\). By changing the independent variables to \(\xi=x+y, \eta=x-y\) prove that the solution of \[ \frac{\partial^2 z}{\partial x^2} = \frac{\partial^2 z}{\partial y^2} \] is given by \(z=F_1(x+y)+F_2(x-y)\), where \(F_1(t), F_2(t)\) denote arbitrary functions of \(t\).
Prove that the volume enclosed by rotating a closed plane curve about a non-intersecting coplanar axis is given by the product of the area enclosed by the curve, and the length of the path traced by the centroid of the area. Shew that the volume of the surface formed by rotating the larger part of an ellipse about the latus rectum which is its boundary, is given by \[ 2\pi\frac{l^3}{(1-e^2)^3}\left\{ e\cos^{-1}(-e) + \sqrt{1-e^2}\left(\frac{2+e^2}{3}\right) \right\}, \] where \(e\) is the eccentricity and \(l\) the semi-latus rectum of the ellipse.
If \(a, b, c, d\) are four coplanar lines, prove that
(i) The equation of a central conic referred to rectangular axes is \[ S = ax^2+2hxy+by^2+2gx+2fy+c=0. \] Shew that, if \(y=mx\) and \(y=m'x\) are parallel to conjugate diameters, then \[ a+h(m+m') + bmm' = 0. \] Hence, or otherwise, shew that the axes of the conic are given by \[ h\xi^2+(b-a)\xi\eta - h\eta^2 = 0, \] where \(\xi = ax+hy+g\), and \(\eta = hx+by+f\), and shew that the axes bisect the angles between the asymptotes. (ii) Shew that the equation of the director circle of the conic \(S=0\) is \[ C(x^2+y^2)-2Gx-2Fy+A+B=0, \] where \(A, B, C, F, G\) are the minors of \(a, b, c, f, g\) respectively in the discriminant of the conic.
If \(\alpha, \beta, \gamma\) are the roots of \(x^3+bx+c=0\), find an expression for \[ (\alpha-\beta)^2(\beta-\gamma)^2(\gamma-\alpha)^2. \] Hence shew that the roots of \(x^3+px^2+qx+r=0\) are all real provided \[ p^2q^2+18pqr-4q^3-4p^3r-27r^2 \ge 0. \] If this condition is satisfied, shew that the necessary and sufficient condition for all the roots to be positive is that \(p\) and \(r\) should be negative and \(q\) positive.
The polynomials \(f(x)\) and \(\phi(x)\) are of degrees \(n\) and \(m\) respectively, \(n\) being greater than \(m\). Shew that, if \(f(x)\) and \(\phi(x)\) have no common factor, it is possible to find polynomials \(F(x)\) and \(\Phi(x)\) such that \[ f(x)F(x) + \phi(x)\Phi(x) = 1. \] Shew further that it is always possible to find an \(F\) whose degree is less than \(m\), and that the polynomials \(F\) and \(\Phi\) are then unique. Hence prove that, if the polynomials \(A\) and \(B\) have no common factor, the rational function \(C/AB\) can be expressed in only one way as \[ D + \frac{P}{A} + \frac{Q}{B}, \] where \(P\) and \(Q\) are polynomials whose degrees are less than those of \(A\) and \(B\) respectively. If \(A\) is \((x-a)^n\), and \((x-a)\) is not a factor of \(C\), then \(P/A\) can be expressed as \[ \frac{\lambda_1}{x-a} + \frac{\lambda_2}{(x-a)^2} + \dots + \frac{\lambda_n}{(x-a)^n}. \] Shew that \[ \lambda_{n-r} = \frac{1}{n!}\frac{d^r}{dx^r}\left[\frac{C(x)}{B(x)}\right]_{x=a}. \]
The function \(f_n(x)\) is defined to be \[ \frac{d^n}{dx^n}\{(x^2-1)^n\}. \] Shew by integration by parts, or otherwise, that \[ \int_{-1}^1 x^m f_n(x) dx = 0, \] if \(m\) is an integer less than \(n\), and is equal to \[ \frac{2^{2n+1}(n!)^3}{(2n+1)!} \] if \(m=n\). Shew that, if \(\phi(x)\) is a polynomial of degree less than \(n\), \[ \int_{-1}^1 \phi(x) f_n(x) dx = 0, \] and by using this last result, or otherwise, prove that \(f_n(x)\) has exactly \(n\) zeros in the range \(-1 \le x \le 1\).