Prove that a solid right circular cone of given total surface area has the greatest volume when the slant height is three times the radius of the base.
Show that the equation \[ x = 2 + \log x \] has two positive roots. Let these roots be \(A\) and \(B\), where \(A < B\). The sequence \(x_n\) satisfies the relation \[ x_{n+1} = 2 + \log x_n \text{ for } n=1, 2, 3, \dots. \] If \(A < x_1 < B\), prove that
Prove that the four complex numbers \(z_1, z_2, z_3, z_4\) represent concyclic points in the Argand diagram if and only if \[ \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \] is real. (A straight line is regarded as a special case of a circle.) A variable point \(P\) on a fixed circle is represented by \(z\), and to each \(P\) there corresponds a point \(Q\) represented by \[ Z = \frac{1}{az+b}, \] where \(a, b\) are complex constants and \(a \ne 0\). Prove that \(Q\) lies on a fixed circle.
A curve lying above the \(x\)-axis is such that the portion of its tangent between the point of contact and the \(x\)-axis is of constant length \(c\). Give a rough sketch of the curve and show that the area between the curve and the \(x\)-axis is \(\frac{1}{2}\pi c^2\).
An isosceles triangle \(ABC\) with sides \(AB=AC=5a\), \(BC=8a\), lies in the same plane as a line \(l\). The lines \(BC\) and \(l\) are parallel at a distance \(a\) apart, and are such that \(BC\) lies between \(A\) and \(l\). Find the volume and surface area of the solid obtained by rotating the triangle about \(l\) (through an angle \(2\pi\)).
It is given that, for all \(x, y\), \[ f(x)f(y) = f(x+y), \] where \(f(x)\) is differentiable and \(f(0)\ne 0\). Write down the results of differentiating the above identity partially with respect to \(x\) and \(y\), and deduce that \[ f(x) = e^{ax}, \] where \(a\) is a constant.
Evaluate:
If \(m>1\), prove that \[ \int_m^{m+1} \frac{dt}{t} < \frac{1}{m} < \int_{m-1}^m \frac{dt}{t}. \] Hence, or otherwise, prove that, if \(n\) is a positive integer, \[ \log 2 < \frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n-1} < \frac{1}{2n} + \log 2. \]
If \(u=f(x,y)\) is a homogeneous function of degree \(n\) (i.e. \(f(kx, ky) = k^n f(x,y)\) for all positive \(k\)), prove that \[ x^2\frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = n(n-1)u. \] Verify this identity directly for the particular function \(u=xy/\sqrt{x^2+y^2}\).
By considering the differential equation \[ \frac{d^3y}{dx^3}=y \] with appropriate initial conditions, or otherwise, show that the sum of the infinite series \[ 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + \frac{x^9}{9!} + \dots \] is \[ \frac{1}{3} \left\{ e^x + 2e^{-x/2} \cos\left(\frac{x\sqrt{3}}{2}\right) \right\}. \]