Show that $$x^{2n} - 2x^n \cos n\theta + 1 = \prod_{r=0}^{n-1} \left[ x^2 - 2x \cos \left( \theta + \frac{2\pi r}{n} \right) + 1 \right].$$ By taking a particular value of \(x\), or otherwise, show that $$\sin n\phi = 2^{n-1} \prod_{r=0}^{n-1} \sin \left( \phi + \frac{r\pi}{n} \right).$$ Hence or otherwise show that $$\prod_{r=1}^{r=n-1} \sin \frac{r\pi}{n} = \frac{n}{2^{n-1}}.$$
If \(y = \sin(k \sin^{-1} x)\), show that $$(1 - x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + k^2 y = 0.$$ Assuming that \(y\) can be expanded in the form \(\sum_{n=0}^{\infty} a_n x^n\), find the coefficients \(a_n\). When does the expansion reduce to a polynomial?
Evaluate the integral $$I_n = \int_0^{\pi/2} \sin^n x \, dx$$ by using a reduction formula. By comparing the integrals \(I_{2n}\), \(I_{2n+1}\) and \(I_{2n+2}\), or otherwise, show that $$\frac{4}{3} \cdot \frac{16}{15} \cdots \frac{4n^2}{4n^2 - 1} \to \frac{\pi}{2} \quad \text{as} \quad n \to \infty.$$
Sketch the cubic curve $$(xy - 12)(x + y - 9) = a$$
The function \(f(x)\) is such that \(f'(t) \geq f'(u)\) whenever \(t \leq u\). By applying the Mean Value Theorem to the function \(f\) over suitable intervals, or otherwise, show that $$f(\lambda x + \mu y) \geq \lambda f(x) + \mu f(y)$$ whenever \(\lambda \geq 0\), \(\mu \geq 0\), \(\lambda + \mu = 1\). By taking a suitable function \(f\) (or otherwise) show that if \(x\), \(y\) are positive and \(\lambda\), \(\mu\) are as above, we have $$\lambda x + \mu y \geq x^{\lambda} y^{\mu}.$$
Find all the solutions of the equations \begin{align} Bx + 7y + 8z &= A \\ 4x - 2y - 3z + 9 &= 0 \\ x + y + z &= 11 \end{align} for all possible values of the constants \(A\) and \(B\).
Express in partial fractions $$\frac{(x+1)(x+2)\ldots(x+n+1)-(n+1)!}{x(x+1)(x+2)\ldots(x+n)}.$$ Hence, or otherwise, prove that $$\frac{C_1}{1} - \frac{C_2}{2} + \frac{C_3}{3} - \ldots + (-1)^{n-1} \frac{C_n}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n},$$ where \(C_r = n!/r!(n-r)!\).
If \(a\), \(b\) and \(c\) are the roots of the equation $$x^3 + px^2 + qx + r = 0,$$ form the equation with the roots \(a^3 - bc\), \(b^3 - ca\), and \(c^3 - ab\). Prove that one of the roots is the geometric mean of the other two if, and only if, \(rp^3 = q^3\). Find in a similar way a condition for one root to be the arithmetic mean of the other two. What can be said about \(a\), \(b\) and \(c\) if both these conditions hold?
By considering the solution of the recurrence relation $$U_{n+2} - 6U_{n+1} + 4U_n = 0 \quad \text{with} \quad U_0 = 2, U_1 = 6,$$ or otherwise, prove that the whole number next greater than \((3+\sqrt{5})^n\) is, for \(n \geq 1\), divisible by \(2^n\). Is it possible for this number to be divisible by \(2^{n+1}\) for all \(n \geq N\) (for some integer \(N\))?