Simplify the fraction \((cos 3\theta + \cos 4\theta)/(2\cos 3\theta - 2\cos 2\theta + 2\cos\theta - 1)\), and solve the equation \(2\cos 3\theta - 2\cos 2\theta + 2\cos\theta - 1 = 0\).
Determine the constants so that the equation \[ \frac{(x^2+ax+b)^2-c}{x-2} - \frac{(x^2+fx+g)^2-c}{x+2} \] may be an identity.
\(A, B, C\) are three points on a circle, and a line through the pole of \(BC\) meets \(AB, AC\) in \(P\) and \(Q\). Prove that \(Q\) lies on the polar of \(P\).
Defining an ellipse as the orthogonal projection of a circle, deduce its properties with respect to the foci.
Draw a rough graph of the curve \(8y = x(x-3)(x+5)\) between the points \(x=\pm 5\), and hence determine for what range of values of \(a\) the equation \(x^3+2x^2-15x-8a=0\) has three real roots.
Prove by induction that the square of the sum of the cubes of the first \(n\) integers is the arithmetic mean of the sum of the 5th powers and the sum of the 7th powers.
Differentiate with regard to \(x\) \[ 2\sqrt{3}\tan^{-1}(2x+1)/\sqrt{3} - 3x/(x^3-1) - \log(x-1)^2/(x^2+x+1) \]
Prove that, under certain conditions \[ f(x+h) = f(x) + hf'(x) + \frac{1}{2}h^2f''(x+\theta h), \quad 0 < \theta < 1. \] Give examples of cases in which the theorem does not hold. Expand \(y\) in terms of \(x\) by Maclaurin's Theorem, knowing that \((1-x^2)y'' - xy' - y = 0\) and that, when \(x=0\), \(y=1\) and \(y'=1\).
Perform the integrations: \[ \int \frac{(6x^3+3x)\,dx}{(x^2-1)(x-1)}, \quad \int \frac{dx}{\sqrt[4]{\{(7-x)(x-3)\}}}, \quad \int \sqrt{(\sec x+1)}\,dx. \] Transform the last integral by writing \(\sec x = \sec^2 \frac{1}{2}y\).
\(AB, AC\) are tangents to a circle and \(D\) is the middle point of the chord \(BC\). Prove that, if \(P\) be any point on the circumference, the ratio \(AP:PD\) is constant.