Problems

Shew that there is a unique value of \(\lambda\) for which \(ax^4+6cx^2+4dx+e\) is expressible in the form \(A(x-\alpha)^4 + B(x-\beta)^4\), where \(\lambda, A, B, \alpha, \beta\) are independent of \(x\). If \(a, c, d, e\) are real, find the condition that \(A, B, \alpha, \beta\) shall be real.

Shew that if \(n\) be a positive integer:

1. \(n - \dfrac{n^2(n-1)}{1!2!} + \dfrac{n^2(n^2-1^2)(n-2)}{2!3!} - \dots + (-1)^{n-1} \dfrac{n^2(n^2-1^2)\dots(n^2-(n-2)^2)\cdot 1}{(n-1)!n!} = 0\).
2. \(n - \dfrac{n(n^2-1^2)}{1!2!} + \dfrac{n(n^2-1^2)(n^2-2^2)}{2!3!} - \dots + (-1)^{n-1} \dfrac{n(n^2-1^2)\dots(n^2-(n-1)^2)}{(n-1)!n!} = (-1)^{n-1}\).

1. Find the sum to \(n\) terms of the series \[ \frac{1}{1.3} + \frac{1}{2.4} + \frac{1}{3.5} + \dots. \]
2. Find the general term of the series \[ 3+4+6+10+\dots, \] given that any three successive terms satisfy a given relation of the form \[ u_r+au_{r-1}+bu_{r-2}=0. \]

State a rule for the multiplication of two determinants of the same order. By considering the determinant \[ \begin{vmatrix} x & y & z \\ z & x & y \\ y & z & x \end{vmatrix} \] or otherwise, shew that the product \(x^3+y^3+z^3-3xyz\) and \(a^3+b^3+c^3-3abc\) can be expressed in the form \(X^3+Y^3+Z^3-3XYZ\), and find the values of \(X, Y, Z\) in terms of \(x, y, z, a, b, c\). Prove that the expression \[ \begin{vmatrix} a_1 & a_2 & a_3 & \dots & a_n \\ a_n & a_1 & a_2 & \dots & a_{n-1} \\ a_{n-1} & a_n & a_1 & \dots & a_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_2 & a_3 & a_4 & \dots & a_1 \end{vmatrix} \] has \(n\) factors of the form \((a_1+a_2\omega+a_3\omega^2+\dots+a_n\omega^{n-1})\), where \(\omega\) is one of the \(n\)th roots of unity.

Establish the result for the radius of curvature at any point of a plane curve whose tangential-polar \((p, \psi)\) equation is given. If in such a curve the intercept on any tangent between the point of contact and the foot of the perpendicular from the origin on to the tangent is \(p+a\), where \(a\) is a constant, find the angle between the tangents at the points for which \(p=a\) and \(p=2a\) respectively. Find the radius of curvature at the first of these points.

1. Shew that the coefficient of \(x^{n-1}\) in the expansion in a series of ascending powers of \(x\) of \(\dfrac{\cot\theta-x}{1+(\cot\theta-x)^2}\) is \(\cos n\theta\sin^n\theta\).
2. If \(y=(\tan^{-1}x)^2\), eliminate \(\tan^{-1}x\) between the expressions for \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\). Deduce a linear relation between the values of \(\dfrac{d^ny}{dx^n}, \dfrac{d^{n-2}y}{dx^{n-2}}, \dfrac{d^{n-4}y}{dx^{n-4}}\) when \(x=0\).

If \(x>1\), prove that \begin{align*} x^3+3x+2+6x\log x &> 6x^2, \\ x^4+8x+12x^2\log x &> 8x^3+1. \end{align*}

Find the asymptotes of the curve \(xy^2 = 4(x-a)(x-b)\), where \(b>a>0\). Sketch the curve, and find the area bounded by the curve and the lines \(x=0, x+b=0\).

If \(y = \sin^p x \cos^q x \sqrt{1-k^2\sin^2 x}\) and \(p, q, k\) are constants, find \(\sqrt{1-k^2\sin^2 x}\dfrac{dy}{dx}\). Hence, by taking suitable values for \(p\) and \(q\), express \(I_m = \int_0^{\frac{\pi}{2}} \dfrac{\sin^m x\,dx}{\sqrt{1-k^2\sin^2 x}}\) in terms of \(I_{m-2}\) and \(I_{m-4}\).

If \(r\) denotes distance from a focus of an ellipse, find the mean value of \(r\) with respect to angular distance from the major axis for points on the perimeter of the ellipse. Determine also for the ellipse the mean value of \(r\) with respect to area, stating the result in terms of the eccentricity \(e\) and the semi-latus rectum \(\lambda\).