Find the real linear and quadratic factors of \(z^n-1\) when \(n\) is an odd positive integer. Deduce that \[ \sin\frac{\pi}{n}\sin\frac{2\pi}{n}\dots\sin\frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}. \]
Draw the graph of the function \(a\csc x + b\sec x\) for values of \(x\) between \(0\) and \(2\pi\), taking \(a,b\) to be positive. Determine the number of real roots between \(0\) and \(2\pi\) of the equation \(a\csc x + b\sec x = 1\), distinguishing the cases \[ a^{2/3}+b^{2/3} > , =, < 1. \]
Arrange the following numbers in order so that as \(x\) increases without limit the ratio of each number to the preceding may tend to infinity: \[ x^2, 2^x, x^x, e^x, x^{\log x}, (\log x)^x, 2^{\log x}. \] Find the limiting values of \[ (\cos x)^{1/x}, \quad (\cos x)^{1/x^2}, \quad (\cos x)^{1/x^3} \] as \(x\) tends to zero through positive or negative values.
Find a formula of reduction for the integral \[ \int\sin^m\theta\cos^n\theta\,d\theta \] when \(n\) is an odd positive integer. Find the indefinite integrals \[ \int \left(1-\frac{a}{x^2}\right)^{1/2}\,dx, \quad \int (x^2+x+1)^{-3/2}\,dx. \]
Draw a rough sketch of the curve \[ (x+2)^2y^2 - x(x+2)y + \frac{1}{4}(2x^2-1) = 0, \] and prove that the area enclosed by the curve is equal to \((2-\sqrt{3})\pi\).
Show that \((ay-bx)^2-(bz-cy)(cz-az)\) is the product of two linear factors which are real if \(c^2 > 4ab\). If \(x+y+z+w=0\), prove that \[ wx(w+x)^2+yz(w-x)^2+wy(w+y)^2+zx(w-y)^2+wz(w+z)^2+xy(w-z)^2+4xyzw=0. \]
Prove that the arithmetic mean of \(n\) positive quantities is greater than their geometric mean. If \(s=a_1+a_2+...+a_n\) where all the quantities are positive, prove that \[ \frac{s}{s-a_1} + \frac{s}{s-a_2} + \dots + \frac{s}{s-a_n} \ge \frac{n^2}{n-1}. \]
Sum the series
Prove that if \(p_n/q_n\) is the \(n\)th convergent of \(\displaystyle\frac{a_1}{b_1+}\frac{a_2}{b_2+}\frac{a_3}{b_3+}\dots\), then \[ p_n = b_np_{n-1}+a_np_{n-2}. \] Find the value of \[ \frac{1}{1+}\frac{x}{1-x+}\frac{x}{2-x+}\dots\frac{x}{n+1-x}. \]
Find the \(n\)th differential coefficients of