Problems

Prove that, if \(bc+p^2 \neq 0\), then the equations \begin{align*} ax + qy - rz &= a, \\ -qx + by + pz &= 0, \\ rx - py + cz &= 0 \end{align*} have an infinite number of solutions if, and only if, \[ a=0 \quad \text{and} \quad abc + ap^2 + br^2 + cq^2 = 0. \]

The roots of the equation \[ x^3+px^2+qx+r=0 \] are \(\alpha, \beta, \gamma\). Shew that \(\alpha^3, \beta^3, \gamma^3\) are the roots of the equation \[ \begin{vmatrix} p & q & x+r \\ q & x+r & px \\ x+r & px & qx \end{vmatrix} = 0. \]

Shew that the geometric mean of \(n\) positive numbers is not greater than their arithmetic mean. If \(x, y, z\) are positive numbers such that \(x+y+z=1\), find the greatest value of \(x^ay^bz^c\).

If \(0 < x < 1\), shew that \(n^2x^n \to 0\), as \(n \to \infty\). Find the limit as \(n \to \infty\) of \[ \frac{x^{2n}}{n+x^{2n}}; \] distinguish between the two cases.

\(z, w, a\) are complex numbers and \(a\) lies inside the unit circle in the Argand diagram and \[ w = \frac{z-a}{1-\bar{a}z} \] (where \(a\bar{a}=|a|^2\)). Shew that \(|z|<1\) implies \(|w|<1\) and conversely.

Find the values of \(x\) which give maxima and minima of \[ \sin x + \frac{1}{3}\sin 3x + \frac{1}{5}\sin 5x. \] Distinguish between the maxima and minima.

Prove that the coefficient of \(x^n\) in the expansion of \[ \frac{a}{ax^2-2bx+c} \] in ascending powers of \(x\), where \(ac>b^2\), is \[ \left(\frac{a}{c}\right)^{\frac{n+1}{2}} \frac{\sin(n+1)\theta}{\sin\theta}, \] where \(\sqrt{(ac)}\cos\theta=b\).

1. Obtain a reduction formula for \[ \int (\sec x)^n \,dx. \]
2. Shew that, if \(m\) and \(n\) are positive integers and \(n \ge 2\), then \[ \int_0^\infty \frac{x^m}{(1+x)^{m+n}} \,dx = \frac{m!(n-2)!}{(m+n-1)!}. \]

Evaluate

1. \(\displaystyle \int_0^a \frac{dx}{x+\sqrt{(a^2-x^2)}}\),
2. \(\displaystyle \int_0^{2\pi} \frac{\sin^2\theta \,d\theta}{a-b\cos\theta}\), where \(a>b>0\),
3. \(\displaystyle \int_1^\infty \frac{dx}{(1+x)\sqrt{x}}\).

Define the area of the surface of a body formed by the revolution of a curve about a straight line in its plane. A circular arc revolves about its chord; prove that the area of the surface generated is \(4\pi a^2(\sin\alpha-\alpha\cos\alpha)\), where \(a\) is the radius and \(2\alpha (<\pi)\) is the angular measure of the arc.