A boy standing at the corner \(B\) of a rectangular pool \(ABCD\) with \(AB = 2\)m, \(AD = 4\)m has a boat in the corner \(A\) at the end of a string of length 2m. He walks slowly towards \(C\) along \(BC\) keeping the string taut. Locate the boy on \(BC\) when the boat is 1m from \(BC\). $$[\int \operatorname{cosec}\theta d\theta = \ln \tan \frac{1}{2}\theta.]$$
Find a solution of \(d^2y/dx^2 = y\) for which \(y = 0\) when \(x = l\), and \(y = a\) when \(x = 0\). Assuming a particular integral of the form \(x(A\cosh x + B\sinh x)\), or otherwise, solve $$\frac{d^2y}{dx^2} = y + 2\cosh(l-x),$$ given that \(y = 0\) when \(x = l\), and \(y = a\) when \(x = 0\).
A family of parabolas is given by the equation $$(x-at)^2 = 4a(y-at^2), \quad (1)$$ where \(a\) is a positive constant and \(t\) is a real-valued parameter. Show that the number of members of the family passing through a given point \((x_0, y_0)\) is 0, 1 or 2 according as \(x_0^2\) is greater than, equal to, or less than \(5ay_0\). Show that, for each fixed value of \(t\), the function \(y\) of the variable \(x\) defined by the equation (1) satisfies the differential equation $$5x^2\left(\frac{dy}{dx}\right)^2 - 4ax\left(\frac{dy}{dx}\right) + x^2 - ay = 0. \quad (2)$$ Deduce a solution of (2) which cannot be obtained by giving any fixed value to \(t\) in (1). How many solutions of (2) are there for which \(dy/dx\) is everywhere continuous and \(y = 0\) when \(x = 0\)?
Write down the expansions of \(e^x\) and \((1-x)^{-1}\) as power series in \(x\). Show that, for \(0 < a < \frac{1}{2}\), $$\int_0^a \frac{e^x-1}{x}dx < a + \frac{1}{4}a^2(1-\frac{1}{8}a)^{-1}.$$ Show also that $$1.80 < \int_0^1 \frac{e^x-1}{x}dx < 1.83.$$
Explain graphically why, if \(x_1\) and \(x_2\) are each approximations to the same root of the equation \(f(x) = 0\), the expression $$\frac{x_1f(x_2) - x_2f(x_1)}{f(x_2) - f(x_1)}$$ may be expected to be a better approximation to the root. Show that $$f(x) \equiv x^3 + 3x^2 - 5x - 1 = 0$$ has a root between 1 and 1.5, and hence find an approximation to the root by the above formula, taking \(x_1 = 1\) and \(x_2 = 1.5\). Find a better approximation, and comment on its accuracy.
The function \(f(z)\) possesses a derivative \(f'(z)\) for all real values of \(z\), and is such that $$f(x + y) = f(x)f(y)$$ for all real values of the independent variables \(x\) and \(y\). By differentiating the relation with respect to \(x\) and \(y\) in turn, show that $$\frac{f'(x)}{f(x)} = \frac{f'(y)}{f(y)},$$ and hence determine the form of \(f\). Determine similarly the form of the function \(g\) that satisfies $$g(x+y) = \frac{g(x) + g(y)}{1 + g(x)g(y)}.$$
The components \(f_i(t)\) (\(i = 1, 2, \ldots, n\)) of the \(n\)-dimensional vector \(\mathbf{F}\) are functions of time \(t\), and not all of them are constant. Show that the vectors \(\mathbf{F}\) and \(\dot{\mathbf{F}}\) (where \(\dot{\mathbf{F}}\) is the vector with components \(df_i/dt\)) are orthogonal for all \(t\) if and only if \(\mathbf{F}\) has constant length. Is it possible for \(\mathbf{F}\) and \(\dot{\mathbf{F}}\) to be orthogonal for all \(t\) if \(\mathbf{F}\) has constant length? Another vector \(\mathbf{G}\), with components \(g_i(t)\) (\(i = 1, 2, \ldots, n\)), is parallel to \(\mathbf{F}\) for all \(t\). \(\mathbf{G}\) has constant length, and \(\mathbf{F}\) has length proportional to \(e^{kt}\), where \(k\) is a constant. Show that \(\dot{\mathbf{F}}\) and \(k\mathbf{G}+\dot{\mathbf{G}}\) are parallel for all \(t\). [If the vectors \(\mathbf{A}, \mathbf{B}\) have components \(a_i, b_i\) (\(i = 1, 2, \ldots, n\)) respectively, \(\mathbf{A}\) and \(\mathbf{B}\) are said to be orthogonal if \(\sum_{i=1}^n a_i b_i = 0\), and are said to be parallel if there is a scalar \(\lambda\) such that \(a_i = \lambda b_i\) for \(i = 1, 2, \ldots, n\). The length of \(\mathbf{A}\) is \(\sqrt{a_1^2 + a_2^2 + \ldots + a_n^2}\).]
A mapping of the \((X, Y)\) plane onto the \((x, y)\) plane is given by $$x = \sin X \cosh Y,$$ $$y = \cos X \sinh Y.$$ Find and sketch the curves in the \((x, y)\) plane which correspond under this mapping to the lines \(X = \text{const.}\) and \(Y = \text{const.}\) To which curves in the \((X, Y)\) plane do the lines \(x = 0, y = 0\) and \(x = y\) correspond?
A running track is in the form of a convex circuit. The width of the track is \(d\). By how much does the length of the outer edge of the track exceed that of the inner edge? (You should explain carefully how you arrive at your answer.)
Write a program in any standard language (or draw a flow diagram for such a program) which will print out a list of candidates in order of merit, together with their marks on each paper and their overall mark. You may assume that candidates are identified by code numbers and not by names.