Sketch the curves \(\cosh x = \dfrac{y\cosh\alpha}{\sin y}\) for different values of the parameter \(\alpha\) (\(\alpha \ge 0\)), and for values of \(y\) between \(-\pi\) and \(\pi\). Show that, on the curve of parameter \(\alpha\), the function \[ \sinh(x+iy) - (x+iy)\cosh\alpha \] is purely real, and indicate its direction of increase along the curve.
Prove that if the real part of the polynomial \[ a_0+a_1z+\dots+a_nz^n, \quad z=x+iy, \] where \(a_0, a_1, \dots, a_n\) are complex numbers, is never negative for any value (real or complex) of \(z\) then \(a_1=a_2=\dots=a_n=0\). Deduce that if the real part of a polynomial is always greater than the imaginary part then the polynomial is a constant.
Show that \[ (1+x)^\lambda = 1 + \lambda x + \frac{\lambda(\lambda-1)}{2!}x^2 + \dots + \frac{\lambda(\lambda-1)\dots(\lambda-n+1)}{n!}x^n \] \[ + \frac{\lambda(\lambda-1)\dots(\lambda-n)}{n!}(1+x)^\lambda \int_0^x t^n (1+t)^{-\lambda-1} dt \] for \(x>-1\), and \(\lambda\) rational. Find the first four terms in the expansion of \(\left(\dfrac{1}{1+x}\right)^\lambda\) in powers of \(x\).
Show that if \(P\) is a homogeneous polynomial in the three variables \(x, y, z\) of degree \(n\) then \[ x\frac{\partial P}{\partial x} + y\frac{\partial P}{\partial y} + z\frac{\partial P}{\partial z} = nP. \] Deduce that if \(P, Q, R\) are homogeneous polynomials in \(x, y, z\) all of the same degree and if, for all \(x, y, z\), \[ P\left(\frac{\partial Q}{\partial z} - \frac{\partial R}{\partial y}\right) + Q\left(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right) + R\left(\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}\right) = 0, \] then \[ \frac{\partial}{\partial y}\left(\frac{P}{xP+yQ+zR}\right) = \frac{\partial}{\partial x}\left(\frac{Q}{xP+yQ+zR}\right). \]
Integrate \[ \int_0^1 \frac{x^2 dx}{(x^2+1)^2}, \quad \int \frac{dx}{(x-a)\sqrt{x^2+1}}, \quad \int_0^{\frac{\pi}{4}} \tan^3 x\, dx. \]
\(f(x)\) 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 \[ \sum_{r=1}^{r=n} \sec r\theta \sec(r+1)\theta. \]