Express \[ f(x) = \frac{x+1}{(x+2)(x-1)^2} \] in partial fractions. Show that the coefficient of \(x^n\) in the expansion of \(f(x)\) in increasing powers of \(x\) is \[ \frac{1}{9}\{12n+10 - (-)^n\}. \]
\(a, b\) and \(c\) are real numbers. Show that the least of the three expressions \[ (b-c)^2, \quad (c-a)^2, \quad (a-b)^2 \] does not exceed \(\frac{1}{2}(a^2+b^2+c^2)\).
If \(a_r = x+(r-1)y\), show that \[ \begin{vmatrix} a_1 & a_2 & a_3 & \dots & a_n \\ a_n & a_1 & a_2 & \dots & a_{n-1} \\ a_{n-1} & a_n & a_1 & \dots & a_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_2 & a_3 & a_4 & \dots & a_1 \end{vmatrix} = (-ny)^{n-1}\{x+\frac{1}{2}(n-1)y\}. \]
Solve: \begin{align*} y^2+yz+z^2 &= 1, \\ z^2+zx+x^2 &= 4, \\ x^2+xy+y^2 &= 7. \end{align*}
A sequence of non-negative numbers \(u_0, u_1, u_2, \dots\) is defined by the recurrence relations \[ u_n^2 = 3u_{n-1}-2 \quad (n \ge 1) \] in terms of the first member of the sequence \(u_0\). It is given that \(1 < u_0 \le 2\). Show that \(u_n \ge u_0\) and that \[ 0 \le 4 - u_n^2 \le \left(\frac{3}{2+u_0}\right)^n (4-u_0^2) \] for all \(n \ge 0\). Hence, or otherwise, prove that \(u_n\) tends to a definite limit as \(n\) tends to infinity and evaluate this limit for each \(u_0\).
Determine the values of \(x\) giving stationary values of \(\phi(x) = \int_x^{2x} f(t)dt\), in the cases (i) \(f(t)=e^t\), (ii) \(f(t)=\frac{\sin t}{t}\). Distinguish in each case between maxima and minima.
Find \[ \int \frac{(x-1)dx}{x\sqrt{1+x^2}}, \quad \int xe^x\sin x dx. \] Prove that \[ \int_0^\frac{\pi}{2} \log(2\sin x)dx = \int_0^\frac{\pi}{2} \log(2\cos x)dx = \int_\frac{\pi}{2}^\pi \log(2\sin x)dx, \] and that each equals 0.
A point \(P\) varies so that \(PA.PA' = a^2\), where \(A\) and \(A'\) are fixed points with midpoint \(O\) and \(AA'=2a\). Sketch the curve and find its equation in polar co-ordinates, where \(OP=r\). Determine the relation between \(r\) and \(p\), the length of the perpendicular from \(O\) to the tangent at \(P\). Hence, or otherwise, express the radius of curvature as a function of \(r\).
Prove Pappus' Theorem about the volume of a solid of revolution. \(O\) is the centre and \(OA\) a radius of a circle; \(OB\) and \(OC\) are radii inclined at \(\alpha\) to \(OA\). Locate the centre of gravity of the sector between \(OB\) and \(OC\). By considering the solid obtained by rotating this sector about a diameter perpendicular to \(OA\), or otherwise, prove that the volume cut off on a cone of semi-vertical angle \((\frac{\pi}{2}-\alpha)\) by a sphere of radius \(r\) is \(\frac{2}{3}\pi r^3(1-\sin\alpha)\).
We define \(f(x,y) = \frac{x^3-y^3}{x^2+y^2}\), unless \(x=y=0\), and \(f(0,0)=0\). If \(f_x(h,k)\) means the value of \(\frac{\partial f}{\partial x}\) at \(x=h, y=k\), find \(f_x(h, mh)\) for \(h \ne 0\), and \(f_x(0,0)\). For what values of \(m\) is \(\lim_{h \to 0} f_x(h,mh) = f_x(0,0)\)?