\(Q, R\) are two points on a rectangular hyperbola subtending a right angle at a point \(P\) of the curve. Prove that \(QR\) is parallel to the normal at \(P\).
If \(l_1 = 0, l_2 = 0\) are the equations of two lines, and if \(S = 0\) is the equation of a conic, interpret the equation \(S + \lambda l_1 l_2 = 0\), where \(\lambda\) is a parameter. Hence, or otherwise, show that if a circle meets an ellipse in four points, the joins of these points in pairs are equally inclined to the axis of the ellipse. Circles are drawn to meet the ellipse \(x^2/a^2 + y^2/b^2 = 1\) in four points lying in pairs on two lines through the fixed point \((x_1, y_1)\). Show that the circles form a co-axial system, and find the equation of the radical axis.
Let \(A(\theta)\) and \(B(\theta)\) denote the matrices $$\begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix}, \quad \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$ respectively.
\(S\) is the set of real numbers. Operations, denoted by \(\oplus\) and \(\otimes\), are defined on \(S\) by \begin{align} a \oplus b &= a + b + 1,\\ a \otimes b &= ab + a + b, \end{align} where the operations of addition and multiplication on the right are the usual ones. Show that if \(\oplus\) and \(\otimes\) are taken to define an addition and multiplication on \(S\), then \(S\) is a field. Which element of \(S\) has no multiplicative inverse in this field, and what is the multiplicative inverse of a general element \(x\) of \(S\)?
Verify that \(y = \cos x \cosh x\) satisfies the relation $$\frac{d^2y}{dx^2} = -4y.$$ Hence or otherwise show that $$y = 1 + \sum_{n=1}^{\infty} (-4)^n \frac{x^{4n}}{(4n)!}$$ \([\cosh x = \frac{1}{2}(e^x + e^{-x})]\)
(i) Evaluate $$\int_0^1 \frac{dx}{1+x^3}.$$ (ii) If \(x\) is a function of \(t\) such that $$\frac{dx}{dt} = \sqrt{\frac{x}{1-x}}$$ and \(x = 0\) when \(t = 0\) find the value of \(t\) for which \(x = 1\).
\(z = f(r)\) is a function which decreases steadily from \(h\) to \(0\) as \(r\) increases from \(0\) to \(a\). The inverse function is \(r = g(z)\). Show that $$\int_0^h [g(z)]^2 dz = 2 \int_0^a rf(r) dr$$ (i) by changing the variable in the first integral and integrating by parts; and (ii) by evaluating the volume of the solid of revolution bounded by \(z = f(r)\) and the disc \(z = 0\), \(r < a\) in two different ways. \([r = \sqrt{x^2 + y^2}; x, y, z\) are Cartesian coordinates.]
\(M(\lambda)\) is a function of the real variable \(\lambda\) defined as the greatest value of \(y = x - \lambda x^2 + \lambda x^3\) in the range \(|x| \leq 1\) of the real variable \(x\). Find the least value of \(M(\lambda)\). (ii) If \(y = x - \lambda x^2 + \lambda x^3\), for what values of \(\lambda\) is zero the least value of \(y\) in the range \(x > 0\)?
Determine the number of real positive solutions of the equation \(\log x = ax^b\) for all values of \(a\), \(b\) with \(a\) real and \(b\) real and positive.
\(z = x + iy\) and \(w = u + iv\) are complex numbers related by \(w = z^2\) and represented by points \((x, y)\) and \((u, v)\) in the \(z\) and \(w\) planes. Show that the curves in the \(w\) plane corresponding to the lines \(x = 1\) and \(y = 2\) in the \(z\) plane intersect at right angles. Comment on the fact that the curves intersect at two points.