By means of the equation \((x+b)(x+c)-f^2=0\), prove that the equation in \(x\) \[ \begin{vmatrix} x+a & h & g \\ h & x+b & f \\ g & f & x+c \end{vmatrix} = 0 \] has three real roots which are separated by the two roots of the first equation. It may be assumed that \(a,b,c,f,g,h\) are all real and different from zero.
(i) Shew that \[ x - \frac{x^3}{3} + \dots + \frac{x^{4r+1}}{4r+1} > \tan^{-1}x > x - \frac{x^3}{3} + \dots - \frac{x^{4r+3}}{4r+3}, \] where \(r\) is a positive integer and \(0 < \tan^{-1}x < \frac{\pi}{4}\). \par (ii) Shew that, for real quantities, \[ (a_1^2+a_2^2+\dots+a_n^2)(b_1^2+b_2^2+\dots+b_n^2)\dots(k_1^2+k_2^2+\dots+k_n^2) \] is not less than the square of \[ (a_1b_1\dots k_1) + (a_2b_2\dots k_2) + \dots + (a_nb_n\dots k_n). \]
By expanding the function \(x^{n-r}(e^x-1)^r\), prove that for positive integral values of \(s\) less than \(r\) \[ c_0(n-r)^s + c_1(n-r+1)^s + \dots + c_r n^s = 0, \] \(r\) being a positive integer and \(c_0, c_1, \dots c_r\) the coefficients in the expansion of \((1-x)^r\). \par Hence, or otherwise, prove that a series whose \(n\)th term is a polynomial of degree \(r-1\) in \(n\) is a recurring series whose scale of relation is given by \(c_0, c_1, \dots c_r\).
If the circumradius \(R\), and the area \(\Delta\), of a triangle \(ABC\) are regarded as functions of \(b, c, A\), prove that \[ \frac{\partial R}{\partial b}\frac{\partial R}{\partial c} = 4R\sin A \frac{\partial R}{\partial A}, \] \[ \frac{\partial\Delta}{\partial b}\frac{\partial\Delta}{\partial c} + \frac{\partial\Delta}{\partial c}\frac{\partial\Delta}{\partial b} = \frac{R\sin A}{2}. \]
The ordinate of any point on a curve is equal to a cubic polynomial in the abscissa. The curve touches \(Ox\) at the origin and intersects that axis again at the point \((2,0)\). Prove that the tangent to the curve at its point of inflexion cuts \(Ox\) at the point \((\frac{2}{3},0)\). \par Sketch the curve.
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 \[ \int_{-1}^{+1} y_m(x)y_n(x)dx=0. \]