\(f(x)\) is a polynomial of the fifth degree, the coefficient of \(x^5\) being 3. \(f(x)\) leaves the same remainder when divided by \(x^2+1\) or \(x^2+3x+3\). It leaves the remainder \(4x+5\) when divided by \((x-1)^2(x+1)\). Find \(f(x)\).
Prove that, if \(n\) is a positive integer, \((1+x)^n\) can be expressed in the form \[ c_0+c_1x+\dots+c_nx^n, \] where \(c_r\) depends only on \(n\) and \(r\), and find the value of \(c_r\). Find the sums of the series \[ \text{(i) } \sum_{r=0}^{r=n-k} c_r c_{r+k}; \quad \text{(ii) } \sum_{r=0}^{r=n} \frac{c_r}{(r+1)(r+2)}. \]
\(f(x)\) is a polynomial of degree \(n\). If \(a_1, \dots, a_n\) are distinct and \[ \frac{f(x)}{(x-a_1)^2(x-a_2)\dots(x-a_n)} = \frac{A_0}{(x-a_1)} + \frac{A_1}{(x-a_1)^2} + \frac{A_2}{(x-a_2)} + \dots + \frac{A_n}{(x-a_n)}, \] find \(A_0, \dots, A_n\). Find the polynomial of the fourth degree such that \(f(0)=f(1)=1, f(2)=13, f(3)=73, f'(0)=0\).
\(ABCD\) is a convex quadrilateral, with \(AB=a, BC=b, CD=c, DA=d\) and the sum of the interior angles at \(A\) and \(C\) equal to \(2\alpha\). Express the area of \(ABCD\) as a function of \(a,b,c,d\) and \(\alpha\) and prove that if \(a,b,c,d\), are given, the area is a maximum when \(ABCD\) is cyclic.
The sequence \(A_0, A_1, \dots, A_n, \dots\) is defined by \[ A_0=0, \quad A_{n+1}\cos n\theta - A_n \cos(n+1)\theta = 1. \] If \(\cos n\theta \neq 0\) for any integral value of \(n\), prove that \[ A_{n+2} - 2A_{n+1}\cos\theta + A_n = 0 \] and hence find \(A_n\). Hence, or otherwise, sum the series \[ \sum_{r=1}^{r=n} \sec r\theta \sec(r+1)\theta. \]
Express \(\cos 3\theta\) in terms of \(\cos\theta\). Show that, for any real \(\theta\), \[ \cos\theta - \tfrac{1}{2}(1-\cos 2\theta) < \cos 3\theta < \cos 2\theta + \tfrac{1}{4}(1-\cos\theta). \]
Prove the formula \[ \frac{1}{(x^2+1)^n} = \frac{1}{2n-2}\frac{d}{dx}\left(\frac{x}{(x^2+1)^{n-1}}\right) + \frac{2n-3}{2n-2}\frac{1}{(x^2+1)^{n-1}} \] for \(n \ge 2\), and hence obtain a recurrence relation for the indefinite integral \[ I_n = \int \frac{dt}{(1+t^2)^n}. \] Evaluate \[ \int_0^1 \frac{dt}{(1+t^2)^3}. \]
Given that \(f_0(x)>0\) for \(x \ge 0\), and that \[ f_n(x) = \int_0^x f_{n-1}(t)dt \quad (n=1,2,3,\dots), \] prove that \[ \frac{f_n(x)}{x^n} > \frac{f_{n+1}(x)}{x^{n+1}} \] for \(n \ge 1\) and \(x > 0\). By repeated integration by parts, verify the formula \[ f_n(x) = \frac{1}{(n-1)!} \int_0^x (x-u)^{n-1} f_0(u)du \] for \(n \ge 1\) and \(x \ge 0\).
A curve is given parametrically by the equations \[ x = a\cos^3 t \quad y = a\sin^3 t. \] Find the parametric equations of the locus of its centre of curvature.
Sketch the curve \[ x=t^2+1 \quad y=t(t^2-4). \] Show that it has a loop, and find the area of this loop.