Problems

Let \(a\) and \(c\) be given real numbers such that \(0 < a < c\); find the least value of \(x\) for which \[a \sec \theta + x \cos \sec \theta \geq c\] for all \(\theta\) between \(0\) and \(\frac{1}{2}\pi\).

Find \[\int \frac{dx}{x^4 + x^3}; \quad \int \frac{dx}{3 \sin x + \sin^2 x}.\] Evaluate \[\int_0^{2a} \frac{x dx}{\sqrt{(2ax - x^2)}}.\]

1. [(i)] Prove that \[\int_0^a f(x) dx = \int_0^a f(a-x) dx.\] Hence, or otherwise, evaluate the integral \[\int_0^\pi \frac{x \sin x}{1 + \cos^2 x} dx.\]
2. [(ii)] Prove that \[\frac{1}{2} < \int_0^1 \frac{dx}{\sqrt{(4-x^2+x^3)}} < \frac{\pi}{6}.\]

Solve the following equations:

1. [(i)] \(x^2(y^2 - 1) + xy + 1 = 0\),\\ \(y^2 + yz + z^2 = 3\),\\ \(x^2(z^2 - 1) + xz + 1 = 0\).
2. [(ii)] \(\sin 3x = \cos 4x\).

1. [(i)] If all the roots of the equation \(x^3 + px^2 + qx + r^3 = 0\) are positive, show that \(p \leq 3r\) and \(q \geq 3r^2\).
2. [(ii)] The numbers \(a\), \(b\), \(c\) are positive, and $$d = (b + c - a)(c + a - b)(a + b - c).$$ By considering \(d^2\), or otherwise, show that \(d \leq abc\).

If \((1 + x)^n = a_0 + a_1 x + \ldots + a_n x^n\), find

1. [(i)] \(\sum_{r=0}^{r=n} \frac{(-1)^r a_r}{r + 1}\),
2. [(ii)] \(\sum_{r=0}^{r=n-k} a_r a_{r+k}\).

If further, \(n\) is of the form \(4m + 2\), find $$a_1 - a_3 + a_5 - \ldots + a_{n-1}.$$

If $$f(a, b) = \sum_{n=1}^{\infty} \frac{1}{n^2 + an + b},$$ show that $$f(a + \beta, a\beta) = \frac{1}{\beta - a} \left( \frac{1}{a + 1} + \frac{1}{a + 2} + \ldots + \frac{1}{\beta} \right)$$ whenever \(\beta - a\) is a positive integer and \(a\) is not a negative integer. Evaluate \(f(-\frac{1}{2}, 0)\).

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 $$\prod_{r=1}^{r=n-1} \sin \frac{r\pi}{n} = \frac{n}{2^{n-1}}.$$

1. [(i)] Evaluate $$\int_1^{\infty} \frac{dx}{x\sqrt{1 + x^2}}.$$
2. [(ii)] If \(x\) is small, find (to the first order in \(x\)) the error in the approximation $$(1 + x)^{1/x} \simeq e.$$

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?