Find the equation of the perpendicular bisector of the line joining the points \((x_1, y_1)\), \((x_2, y_2)\). A fixed circle has centre \(C\) and radius \(2a\). \(A\) is a fixed point inside the circle and \(P\) is a variable point on the circumference. Prove that the perpendicular bisector of \(AP\) touches the foci are at \(C\) and \(A\), and whose major axis is of length \(2a\).
Two planes \begin{align} x - 3y + 2z &= 2, \\ 2x - y - z &= 9, \end{align} meet in the line \(l\). Find the equations of (i) the plane through the origin which contains \(l\), (ii) the plane through the origin which is perpendicular to \(l\). Find also the coordinates of the reflection of the origin in \(l\).
The surfaces of two spheres have more than one real common point. Prove that they intersect in a circle. A triangle \(BCD\) is given in a plane \(\alpha\). Prove that there are just two possible positions, reflections of each other in \(\alpha\), for a point \(A\), which is such that the angles \(BAC\), \(CAD\), \(DAB\) are right angles, if and only if the triangle \(BCD\) is acute-angled. Find the distance of \(A\) from \(\alpha\) in the case where \(BCD\) is an equilateral triangle with sides of unit length.
If \(y = \sin(x \sin^{-1} x)\), prove \[(1-x^2) y'' - xy' + x^2 y = 0,\] where \(y'\) and \(y''\) represent the first and second derivatives of \(y\). Prove that the Maclaurin series of \(y\) is \[x\left[x + \frac{(1^2-x^2)x^3}{3!} + \frac{(3^2-x^2)(1^2-x^2)x^5}{5!} + \ldots\right].\]
(i) Prove \[\int_0^a \frac{x^2 dx}{x^2 + (x-a)^2} = \int_0^a \frac{(x-a)^2 dx}{x^2 + (x-a)^2} = \frac{1}{2}a.\] (ii) Evaluate \[\int x^3 \tan^{-1} x dx.\] (iii) Given \(a > b > 0\), evaluate \[\int_0^\pi \frac{\cos x dx}{a^2 + b^2 - 2ab \cos x}.\]
Two variables \(x\) and \(y\) are to be determined as functions of time \(t\). It is found that the rate of change of \(x\) is equal to the sum of \(k_1\) times the instantaneous value of \(x\) and \((-k_2)\) times the instantaneous value of \(y\). The rate of change of \(y\) is similarly equal to \((-k_3)\) times the value of \(x\) plus \(k_4\) times the value of \(y\). Here \(k_1\), \(k_2\), \(k_3\) and \(k_4\) are positive constants. Obtain a second-order differential equation for \(x(t)\) and show that if \(k_1 k_4 > k_2 k_3\) the solution is of the form \[x = A e^{\alpha t} + B e^{\beta t},\] where \(A\) and \(B\) are arbitrary constants and \(\alpha\) and \(\beta\) are positive.
(i) Given \(\arg(z + a) = \frac{1}{4}\pi\) and \(\arg(z - a) = \frac{3}{4}\pi\), where \(a\) is a given real positive number, find the complex number \(z\). [\(\arg z = 0\) means that \(z\) is a real positive multiple of \(\cos \theta + i \sin \theta\).] (ii) Given \[|z + c| + |z - c| < 2d,\] where \(c\) is complex and \(d > |c|\), and also \[\pi < \arg z < 2\pi,\] describe geometrically the region of the complex plane in which \(z\) must lie.
Show that \(x \tan x = 1\) has an infinite number of real roots, and that if \(n\) is a large integer there is a root near \(n\pi\). Show that a better approximation is \(n\pi + (1/n\pi)\), and find a better one still.
Show that \(y = \sin x \tan x - 2 \log \sec x\) increases steadily as \(x\) increases from \(0\) to \(\frac{1}{2}\pi\). Show also that \(y\) has no inflexion in this range. Sketch the curve \(y(x)\) in \[0 \leq x < \frac{1}{2}\pi.\]