Problems

A navigator wishes to determine the position \(D\) of his ship; he observes three landmarks \(A\), \(B\), \(C\) at his eye level, and measures the angles \(ADB\) and \(BDC\); he then reads off the distances \(AB\) and \(BC\) and the angle \(ABC\) from the chart. Obtain a formula that will enable him in general to calculate the angle \(DAB\) and hence to determine his position. Explain why the method breaks down when \(ABCD\) is a cyclic quadrilateral.

Obtain the general solutions of the trigonometrical equations:

1. [(i)] \(\sin^{-1}(2x) - \sin^{-1}(\sqrt{3}x) = \sin^{-1} x\);
2. [(ii)] \(\sin(\pi \cos x) = \cos(\pi \sin x)\).

Suppose that the functions \(f(x)\) and \(g(x)\) can each be differentiated \(n\) times. Prove that one can write \[\frac{d^n}{dx^n}\{g[f(x)]\} = g'[f(x)]u_1(x) + g''[f(x)]u_2(x) + \cdots + g^{(n)}[f(x)]u_n(x),\] where the functions \(u_k(x)\) depend on \(f(x)\), and on \(n\), but not on \(g(x)\). Show that \(u_k(x)\) is the coefficient of \(s^k\) in the expansion of \[e^{-sf(x)} \frac{d^n}{dx^n} [e^{sf(x)}]\] as a power series in \(s\). Hence, or otherwise, prove that \[u_k(x) = \frac{1}{k!} \sum_{r=0}^k (-1)^{k-r} \binom{k}{r} [f'(x)]^{k-r} \frac{d^n}{dx^n} [f(x)]^r,\] where \(\binom{k}{r}\) denotes the coefficient of \(t^r\) in the binomial expansion of \((1+t)^k\).

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 \(f(-\frac{1}{2}, 0)\).