Express \[ \frac{x}{(x-2)^5(x+1)(x-1)} \] in partial fractions, and verify by taking \(x=3\).
Solution: \begin{align*} && \frac{x}{(x-2)^5(x+1)(x-1)} &=\sum_{i=1}^5 \frac{A_i}{(x-2)^i} + \frac{B}{x+1} + \frac{C}{x-1} \\ \Rightarrow && \frac{x}{(x-2)^5(x-1)}&=(x+1)\sum_{i=1}^5 \frac{A_i}{(x-2)^i} + B+ \frac{C(x+1)}{x-1} \\ x = -1: && \frac{-1}{(-3)^5(-2)} &= B \\ \Rightarrow && B &= -\frac{1}{2 \cdot 3^5} \\ \Rightarrow && \frac{x}{(x-2)^5(x+1)}&=(x-1)\sum_{i=1}^5 \frac{A_i}{(x-2)^i} + \frac{B(x-1)}{x+1}+ C \\ x = 1: && \frac{1}{(-1)^5 \cdot 2} &= C \\ \Rightarrow && C &= -\frac{1}{2} \\ && \frac{x}{(x-2)^5(x+1)(x-1)} + \frac{1}{2(x-1)} + \frac{1}{2\cdot3^5(x+1)} &= \frac{x+\frac12(x+1)(x-2)^5+\frac1{2\cdot3^5}(x-1)(x-2)^5}{(x-2)^5(x+1)(x-1)} \\ &&&= \frac{}{} \end{align*}
Find necessary and sufficient conditions for \(ax^2+2bx+c\) to be positive for all real values of \(x\). \(a, b, c\) are real. \par If \(a,b,c\) are positive, find conditions such that \[ (a-c)x^2 + 2(b-c)^2 x + (a-c)^3 \] shall be positive for all real values of \(x\).
(i) If \((1+x)^{1+x}=1+p\), where \(p\) is small, find the expansion in terms of \(p\) correct to the term in \(p^4\) for the value of \(x\) which is approximately \(p\). \par (ii) If \(a+b+c=0\), express \(\frac{a^5+b^5+c^5}{a^2+b^2+c^2}\) as a quadratic polynomial in \(a,b,c\).
If the equation \(x^5+5a_4x^4+10a_3x^3+10a_2x^2+5a_1x+a_0=0\) has three equal roots each equal to the arithmetic mean of the other roots, prove that \(a_0 = 10a_3 a_4^3 - 9a_4^5\) and obtain similar expressions for \(a_1\) and \(a_2\) in terms of \(a_3\) and \(a_4\). \par State and prove the converse theorem.
If \(u_{n+1}=\frac{1}{2}(u_n+1/u_n)\), and if \(u_1\) is positive, shew that, for \(n>1\), \[ 1 \le u_n, \] \[ u_{n+1} \le u_n, \] also that \[ (u_{n+1}-1) \le \frac{1}{2}(u_n-1)^2; \] \[ \frac{1}{2} \le u_n - u_{n+1}, \quad \text{if } u_n \ge 2. \] Hence shew that \(u_n\) tends to a limit as \(n\) tends to infinity, and state the value of the limit.
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.