Sketch the graph of the function given by \[f(x) = \frac{x-a}{x(x-2)},\] where \(a\) is a constant, in each of the following cases:
Let \[y = x^{\alpha}(1-x)^{1-\alpha},\] where \(0 < x < 1\), and where \(\alpha\) is fixed. Show that, if \(0 < \alpha < 1\), then the greatest value taken by \(y\) is \(M(\alpha)\), where \[M(\alpha) = \alpha^{\alpha}(1-\alpha)^{1-\alpha}.\] What happens if \(\alpha < 0\)? Show further by considering \(\log_e M(\alpha)\), or otherwise, that, whatever the value of \(\alpha\) in the range \(0 < \alpha < 1\), \(M(\alpha)\) is at least \(\frac{1}{4}\).
Throughout this question \(y = f(x)\) denotes a continuous curve such that \(d^2y/dx^2 > 0\) for all \(x\). Illustrate the geometrical meaning of the condition on \(d^2y/dx^2\) by means of a sketch or sketches, stating what can be said about the values of \(x\) (if any) where \(f\) increases, and where it decreases. (i) How many distinct solutions of the equation \(f(x) = 0\) can there be? Justify your answer, and give examples of all the possibilities. (ii) Prove that the curve \(y = f(x)\) lies entirely on one side of any tangent to itself. (iii) Can \(f(x)\) tend to a (finite) limit as \(x \to \infty\)? Can there be numbers \(A\) and \(B\) such that \(A \leq f(x) \leq B\) for all \(x\)? In each case, if the answer is 'yes', give an example of a function with the relevant property; if the answer is 'no', indicate briefly why this is so (a detailed proof for this is not required).
(i) Evaluate \[\int_{1/a}^{a} \frac{x^2}{1+x^2} dx,\] where \(a > 1\). (ii) Find a substitution that transforms \[\int_{1/a}^{a} \frac{1}{1+x^2} dx \text{ to } \int_{1/a}^{a} \frac{x}{1+x^2} dx.\] By considering the sum of these two integrals in the case \(a = 2\), or otherwise, evaluate \[\int_{\frac{1}{2}}^{2} \frac{1}{1+x^2} dx.\] Would you expect \[\int_{1/a}^{\beta} \frac{1}{1+x^2} dx \text{ and } \int_{1/a}^{\beta} \frac{x}{1+x^2} dx\] to be equal when \(\beta > a > 1\)? Justify your answer.
Sketch the curve given parametrically by the equations \[x = a \cos^3 \theta, \quad y = a \sin^3 \theta, \quad \text{for } 0 \leq \theta \leq 2\pi.\] Find the volume of the solid of revolution obtained by rotating this curve about the \(x\)-axis.
An assembly hall has a semi-circular dais of radius \(a\), set with its bounding diameter against a straight wall which extends a distance greater than \(a\pi\) on either side of the dais. There is an electric power point in the wall, where it meets one end of the curved side of the dais; the power point is at floor level (here and elsewhere in this question, the word 'floor' means the floor of the hall and not that of the dais). A standard lamp with a flex of length \(a\pi\) is plugged into the power point; the lamp and its flex are to be on the floor, and there are no obstructions nearby except the dais and the wall. Consider the case in which the flex is at full stretch and the straight part of it has length \(a\theta\) \((0 < \theta < \pi)\). Find, to first order in \(\delta\theta\), the distance through which the lamp has to be moved in order to increase \(\theta\) by a small amount \(\delta\theta\); find also the approximate area swept out by the flex in this operation. Hence determine the total area of the region on which the lamp can be placed, and the length of the boundary of this region.
Let \(ABCDE\) be a regular pentagon and let \(AC\) and \(BE\) intersect at \(H\). Prove that \(AB = CH = EH\) and that \(AB\) is tangent to the circle \(BHC\).
Prove that the three altitudes (i.e. perpendiculars from the vertices to the opposite sides) of a triangle \(ABC\) meet at a single point \(H\). What is the equation of the locus of \(H\) as \(A\), \(B\) and \(C\) vary on the curve \(y = 1/x\)?
Show that if \(\alpha\) is a repeated root of the equation \[a_n x^n + \ldots + a_1 x + a_0 = 0,\] then \(\alpha\) is also a root of the equation \[n a_n x^{n-1} + \ldots + 2 a_2 x + a_1 = 0.\] Hence, or otherwise, solve the equation \[24x^4 - 20x^3 - 6x^2 + 9x - 2 = 0,\] given that three of its four roots are identical.
The polynomial \(f(x) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \ldots + a_n\) has integer coefficients. Prove that any root of \(f(x) = 0\) which is rational is an integer. What can be said about the rational roots of \(2x^n + b_1 x^{n-1} + \ldots + b_n = 0\), where the \(b_i\) are integers? Prove that the equation \(2x^5 - 9x^3 + 1 = 0\) has no rational roots. How many real roots does it have?