Problems

Find the indefinite integrals

1. [(i)] \(\int x^{2n+1} e^{-x^2} dx\), where \(n = 0, 1, 2, \ldots\)
2. [(ii)] \(\int \frac{(x^2 + a^2)^{\frac{1}{2}}}{x^2} dx\).

A loudspeaker-horn has the form of the surface of revolution obtained by rotating the portion \(0 \leq x \leq a\) of the curve \(y = \frac{b}{3}x^{\frac{1}{2}}\) about the line \(y = 0\). Calculate the area of metal sheet used in its construction.

1. [(i)] Using the substitution \(x = e^t\), or otherwise, solve the differential equation $$x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0.$$
2. [(ii)] Solve $$(x^2 - 1) \frac{dy}{dx} + 2(x + 3)y = 8x(x + 1)^6.$$

\(\psi\) is a given function of the three variables \(x\), \(y\), \(f\). Show that, if the equation \(\psi = 0\) is used to define a function \(f(x, y)\) implicitly, then $$\frac{\partial f}{\partial x} = -\frac{\partial \psi/\partial x}{\partial \psi/\partial f}$$ with a similar expression for \(\partial f/\partial y\). Show also that, if \(\psi(x, y, f)\) actually takes the special form \(\psi = \phi(g, h)\) where \(g = y\), and \(h = f - \log x\), and \(\phi\) is any function of its two arguments, then \(f(x, y)\) as just defined satisfies the differential equation $$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = 1.$$ [All derivatives required throughout the question may be assumed to exist.]

(i) In the equation \[\frac{k_1}{x-a_1} + \frac{k_2}{x-a_2} + \ldots + \frac{k_n}{x-a_n} = 0\] the numbers \(k_i\) are positive and the \(a_i\) are distinct real numbers. Prove that the roots of the equation are all real. (ii) Find necessary and sufficient conditions for the equation \[2x^5 - 5px^2 + 3q = 0,\] where \(p\) and \(q\) are positive, to have (a) one, (b) three real roots.

Two regular polygons of \(n_1\) and \(n_2\) sides are inscribed in two concentric circles of radii \(r_1\) and \(r_2\) respectively. Prove that the sum of the squares on all the lines joining the vertices of one to the vertices of the other is \[n_1n_2(r_1^2 + r_2^2).\]

The function \(\log x\), where \(x\) is real and positive, is defined by the formula \[\log x = \int_1^x \frac{dt}{t}.\] From the definition prove that

1. [(i)] \(\log xy = \log x + \log y\);
2. [(ii)] \(\frac{\log x}{x^k} \to 0\) for all positive \(k\) as \(x \to \infty\).

A conic \(S\) is inscribed in a triangle \(ABC\), its point of contact with \(BC\) being \(D\). \(O\) is a general point of \(S\), and the lines \(OA\), \(OB\), \(OC\) meet \(S\) again in \(A'\), \(B'\), \(C'\) respectively. Prove that the tangent at \(O\) passes through the intersection of the lines \(B'C'\), \(A'D\).

Prove Desargues' theorem that, if the lines joining corresponding vertices of two coplanar triangles are concurrent, the intersections of corresponding sides are collinear. The sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_1\), \(A_1A_3\) of a quadrangle \(A_1A_2A_3A_4\) pass successively through the vertices \(b_1b_2\), \(b_2b_3\), \(b_3b_4\), \(b_4b_1\), \(b_1b_3\) of a quadrilateral \(b_1b_2b_3b_4\), so that \(A_1A_3\) passes through \(b_1b_3\).

Prove Pascal's theorem that the three intersections of pairs of opposite sides of a hexagon inscribed in a conic are collinear. \(A\), \(B\), \(C\), \(D\), \(E\) are five points of a conic \(S\). Show how to construct (i) the tangent at \(A\), (ii) the second intersection with \(S\) of a given line through \(A\). Hence or otherwise, sketch a method of constructing the polar line of a general point \(P\) with respect to \(S\).