If \(x = c + \frac{1}{4}\cos^8\theta\), \(y = (1-x)\cot\theta\), where \(c\) is a positive constant and \(\theta\) is a variable parameter, find a relation of the form \(y^2 = f(x, c)\). Sketch the graphs of this relation for the cases (i) \(c > 1\), (ii) \(\frac{1}{4} < c < 1\). What value of \(c\) makes the graph a circle?
Show that the function \(y = \sin^2(m\sin^{-1}x)\) satisfies the differential equation \[(1-x^2)y'' = xy' + 2m^2(1-2y).\] Show that, at \(x = 0\), \[y^{(n+2)} = (n^2 - 4m^2)y^{(n)} \quad (n \geq 1)\] and derive the MacLaurin series for \(y\).
Let \(f\) be the function of two real variables defined by \[f(x, y) = x^2 + xy + y^4.\] Find the range of \(f\) when the domain of \(f\) is:
Give a definition of an integral as the limit of a sum. By considering \[\sum_{n=0}^{N-1} (aq^n)^p(aq^{n+1} - aq^n),\] where \(p\) is a positive integer, \(q^N = b/a\) and \(b > a > 0\), show that \[\int_a^b x^p dx = \frac{b^{p+1} - a^{p+1}}{p+1}.\]
By considering \(\int_0^1 [1 + (\alpha-1)x]^n dx\), or otherwise, show that \[\int_0^1 x^k(1-x)^{n-k} dx = \frac{k!(n-k)!}{(n+1)!}.\] Deduce the value of \[\int_0^{\pi/2} \sin^{2n+1}\theta \cos^{2n+1}\theta d\theta.\]
(i) Find a first-order differential equation satisfied by each member of the family \(F\) of curves \[y = c\exp(x^2) \quad (-\infty < c < \infty).\] Write down the differential equation satisfied by any curve which is orthogonal to every member of \(F\) and hence find the set of orthogonal trajectories to \(F\). (ii) Verify that \(y = x^2/4a\) is a solution of the differential equation \[y = x\frac{dy}{dx} - a\left(\frac{dy}{dx}\right)^2.\] Explain why each tangent to the curve \(y = x^2/4a\) is also a solution of the differential equation.
(i) Use de Moivre's theorem to express \(\cos 6\theta\) and \(\sin 6\theta\) in terms of powers of \(\cos\theta\) and \(\sin\theta\). (ii) Let the roots of the equation \(z^4 - 1 = i\sqrt{3}\) be \(z_r\) (\(r = 1, 2, 3, 4\)), where \(z_r\) lies in the \(r\)th quadrant of the complex plane. Show that \[(z_1 + z_3) = -(z_3 + z_4) = 2^{-\frac{1}{2}}(1 + i\sqrt{3}).\]
Prove that, if \(x\) and \(y\) are real numbers, and \(\max(x, y)\) denotes the greater of \(x\) and \(y\) when \(x \neq y\), and their common value when \(x = y\), then \[\max(x, y) = \frac{1}{2}(x + y) + \frac{1}{2}|x - y|.\] Explain what is meant by saying that a real-valued function \(f\), defined on an interval of the real line, is continuous at a point of the interval. Suppose that \(f\) and \(g\) are defined in the same interval \(I\), and that \(h(x) = \max[f(x), g(x)]\) for all \(x\) in \(I\); prove that, if \(f\) and \(g\) are both continuous at a point \(x_0\) in \(I\), then \(h\) is also continuous at \(x_0\). Give an example to show that if \(f\) and \(g\) are both differentiable at \(x_0\), then \(h\) is not necessarily differentiable at \(x_0\); and another example to show that \(h\) can be differentiable at \(x_0\).
Two sequences \((x_0, x_1, x_2, \ldots)\) and \((y_0, y_1, y_2, \ldots)\) of positive integers are defined inductively by taking \(x_0 = 2\), \(y_0 = 1\), and counting rational and irrational parts in the equations \[x_n + y_n\sqrt{3} = (x_{n-1} + y_{n-1}\sqrt{3})^2 \quad (n = 1, 2, 3, \ldots).\] Prove that \[x_n^2 - 3y_n^2 = 1 \quad (n = 1, 2, 3, \ldots),\] and that when \(n \to \infty\), the sequences \(x_n/y_n\) and \(3y_n/x_n\) tend to the limits \(\sqrt{3}\) from above and below respectively. By carrying this process far enough, obtain two rational numbers enclosing \(\sqrt{3}\) and differing from one another by less than \(5 \times 10^{-9}\).
The triangle \(ABC\) is inscribed in a circle \(K\) of radius \(R\), and its angles are all acute. If small changes \(\delta a\), \(\delta b\), \(\delta c\) are made in the sides \(a\), \(b\), \(c\) of the triangle in such a way that it remains inscribed in \(K\), prove that \[\frac{\delta a}{\cos A} + \frac{\delta b}{\cos B} + \frac{\delta c}{\cos C} = 0\] approximately. Discuss what happens when \(C\) is a right angle. Show also that, if \(S\) is the area of the triangle, then the small change \(\delta S\) in \(S\) under the same conditions is given approximately by the equation \[\frac{\delta S}{S} = \frac{\delta a}{a} + \frac{\delta b}{b} + \frac{\delta c}{c}.\] [The formula \(a = 2R\sin A\) may be assumed.]