P and Q are two points on a semi-circle whose diameter is AB; AP and BQ meet in M, AQ and BP meet in N. Prove that MN is perpendicular to AB, and that the circle on MN as diameter cuts the semi-circle orthogonally.
Find the locus of a point P which moves in a plane containing three distinct fixed points \(A_1\), \(A_2\), \(A_3\) subject to the restriction that \(PA_1^2 + PA_2^2 + PA_3^2\) is constant. Show that there is a value of this constant for which the locus reduces to a single point, and identify this point. If \(B_1\), \(B_2\), \(B_3\) are three other fixed points of the plane, prove that the locus of a point P which moves in the plane so that \(PA_1^2 + PA_2^2 + PA_3^2 = PB_1^2 + PB_2^2 + PB_3^2\) is in general a straight line. Discuss possible exceptional cases.
A, B, C, D are four points on a parabola. The lines through B and D parallel to the axis of the parabola meet CD and AB, respectively, in E and F. Prove that AC is parallel to EF.
\(A_1\), \(A_2\), \(A_3\), \(A_4\) are four points of the rectangular hyperbola whose general point is \((d, c/t)\). If the normals at these points are concurrent, prove that each of the points is the orthocentre of the triangle formed by the other three. Show also that the conic through \(A_1\), \(A_2\), \(A_3\), \(A_4\) and the centre of the hyperbola is a second rectangular hyperbola whose axes are parallel to the asymptotes of the first.
By writing \(n^{1/n} = 1 + x_n\) and using the fact that \((1 + x)^n \geq \frac{1}{2}n(n - 1)x^2\) if \(n \geq 2\) and \(x \geq 0\), prove that \(n^{1/n}\) tends to 1 as \(n \to \infty\). Sketch the graph of \(y = x^{1/x}\) for \(x > 0\).
Let \(C_1\) be the plane curve whose polar equation is \(r\theta = 1\), \(\theta \geq \pi\) and let \(C_2\) be the curve whose equation is \(r(\theta + \pi) = 1\), \(\theta \geq \pi\). Show that these curves do not cross. Find the spiral-shaped area enclosed by \(C_1\), \(C_2\) and the line segment \(\theta = \pi\), \(1/2\pi \leq r \leq 1/\pi\). Find the area of the snail-shaped region bounded by the arc \(\pi \leq \theta \leq 3\pi\) of \(C_1\) and the line segment \(\theta = \pi\), \(1/3\pi \leq r \leq 1/\pi\).
(i) Explain why the transformation from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\) given by \(r = (x^2 + y^2)^{\frac{1}{2}}\), \(\theta = \tan^{-1}(y/x)\) needs further comment. (ii) Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(ad - bc \neq 0\), and let \(T(x) = \frac{ax + b}{cx + d}\). Assuming that \(T(x) \neq x\), find a necessary and sufficient condition in terms of \(a\), \(b\), \(c\) and \(d\) for the identity \(T(T(x)) \equiv x\) to hold. Show that if \(T\) satisfies this condition and \(c \neq 0\) then there are two distinct solutions (possibly complex) of \(T(x) = x\).
For any integer \(n\), define \(I_n = \int_0^{\pi/2} \frac{\cos nx - 1}{\sin x} dx\). By considering \(I_n - I_{n-2}\), or otherwise, evaluate \(I_3\) and \(I_4\).
The following four functions are defined for all real \(x\): (i) \(\log(2e^x)\); (ii) \(e^x\); (iii) \(|x|\); (iv) \((x^2 + 1)^{\frac{1}{2}}\). Show that the first function can be represented as a polynomial in \(x\). Prove that the other three functions cannot be so represented.
Let \(z_1\), \(z_2\) be complex numbers such that \(z_1 + z_2\) and \(z_1 z_2\) are both real. Show that either \(z_1\) and \(z_2\) are both real or \(|z_1| = |z_2|\). Let \(\Phi\) be the set of all complex numbers \(z\) such that \(iz + (1/iz)\) is real, together with 0. Describe the set \(\Phi\), and find all those complex numbers \(w\) such that \(wz\) is in \(\Phi\) whenever \(z\) is in \(\Phi\).