The sides \(AB\), \(BC\), \(CD\), \(DA\) of a plane quadrilateral are of lengths \(a\), \(b\), \(c\), \(d\), respectively, and the lengths of the two diagonals \(AC\) and \(BD\) are \(x\) and \(y\), respectively. Prove that the area of the quadrilateral is \[\tfrac{1}{4}\{4x^2y^2 - (b^2 + d^2 - a^2 - c^2)^2\}^{\frac{1}{2}}.\] State the particular form of the result when a circle can be inscribed within the quadrilateral.
(i) Solve the equation \[2\cos 5\theta + 10\cos 3\theta + 20\cos \theta - 1 = 0.\] (ii) Prove that if \(\theta_1\), \(\theta_2\), and \(\theta_3\) are values of \(\theta\) which satisfy the equation \[\tan(\theta + \alpha) = \kappa \tan 2\theta\] and are such that no two of them differ by an integral multiple of \(\pi\), then \(\theta_1 + \theta_2 + \theta_3\) is an integral multiple of \(\pi\).
Find the maximum and minimum values of \(\cos\theta + \cos(z - \theta)\), where \(z\) is fixed and \(\theta\) is variable. Hence, or otherwise, show that, if \(A\), \(B\), \(C\) are the angles of any triangle, then $$\cos A + 2(\cos B + \cos C) \leq 2.$$
The coordinates of any point on a curve are given by \(x = \phi(t)\), \(y = \psi(t)\), where \(t\) is a parameter; prove that the equation of the tangent is $$\begin{vmatrix} x & \phi(t) & \phi'(t) \\ y & \psi(t) & \psi'(t) \\ 1 & f(t) & f'(t) \end{vmatrix} = 0.$$ Prove that the condition that the tangents at the points of the curve $$x = at/(t^2 + bt^2 + ct + d), \quad y = a/(t^2 + bt^2 + ct + d),$$ whose parameters are \(t_1\), \(t_2\), \(t_3\) may be concurrent is $$3(t_2 t_3 + t_3 t_1 + t_1 t_2) + 2b(t_1 + t_2 + t_3) + b^2 = 0.$$
Specify the loci in the complex plane given by $$|z - 1| = a|z + 1| + b,$$ when \((a, b)\) take the values \((1, 0)\), \((1, 1)\), \((0, 3)\), \((3, 0)\), \((1, 3)\).
If the substitutions \(x = \frac{1}{2}(u^2 - v^2)\), \(y = uv\) transform \(f(x, y)\) into \(F(u, v)\), show that $$u^2 \frac{\partial F}{\partial v} - v \frac{\partial F}{\partial u} = 2\left(x \frac{\partial f}{\partial y} - y \frac{\partial f}{\partial x}\right)$$ and that $$\frac{\partial^2 F}{\partial u^2} + \frac{\partial^2 F}{\partial v^2} = (u^2 + v^2)\left(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\right).$$
The sides \(a\), \(b\), \(c\) of a triangle are measured with a possible small percentage error \(\epsilon\) and the area is calculated. Prove that the possible percentage error in the area is approximately \(2\epsilon\) or \(2\cot B \cot C\) according as the triangle is acute-angled or obtuse-angled at \(A\).
Obtain indefinite integrals of the functions
Prove (using tables if you wish) that $$1 < \int_0^{1\pi} \sqrt{\sin x} \, dx < \sqrt{1 \cdot 25}.$$
Prove that a loop of the curve \(r = 2a\cos k\theta\) (\(k > 1\)) has the same area and perimeter as an ellipse with semi-axes \(a\) and \(a/k\). If \(k = 1 + \epsilon\), where \(\epsilon\) is small, obtain an approximate expression for the perimeter of the loop in powers of \(\epsilon\) including the term in \(\epsilon^2\).