Prove that the normals to a parabola at the points \(Q\), \(R\) intersect on the curve if and only if \(QR\) passes through a certain fixed point. Suppose that this condition is satisfied, and let the normals at \(Q\), \(R\) meet at \(P\). If \(P'\) is the intersection of the parabola with the line parallel to the axis passing through the common point of \(QR\) and the directrix, show that \(PP'\) passes through the focus.
Interpret the equation \(S + \lambda T^2 = 0\), where \(S = 0\) and \(T = 0\) are the equations of a conic and one of its tangents, and \(\lambda\) is a constant. Hence or otherwise find the equations of the circles of curvature of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the ends of the major axis. Show that these circles touch each other if \(a^2 = 2b^2\), and find the condition that each should be touched by the circle with the minor axis as diameter.
Two regular tetrahedra are formed from among the vertices of a cube of edge length \(a\). Find the volume of the portion of the cube external to both tetrahedra.
Show that if \(\mathbf{p}\), \(\mathbf{q}\), \(\mathbf{u}\) are non-zero vectors, with \(\mathbf{u}\) not a scalar multiple of \(\mathbf{p} - \mathbf{q}\), and if \(\lambda\), \(\mu\) are positive scalars with \(\lambda + \mu\), then the four points with position vectors \(\mathbf{p}\), \(\mathbf{q}\), \(\lambda\mathbf{p} + \mu\mathbf{u}\) are vertices of a trapezium. By considering the two triangles into which the trapezium is divided by a diagonal, or otherwise, show that the position vector of the centroid of the trapezium is \(\frac{(2\lambda + \mu)\mathbf{p} + (\lambda + 2\mu)\mathbf{q}}{3(\lambda + \mu)}\) \(ABCD\) is a trapezium, with \(AB\) parallel to \(DC\). \(H\), \(I\), \(J\) are points on \(CD\) and \(K\), \(M\), \(N\) are such that \(AH\), \(KC\) are parallel to \(BD\), and \(BL\), \(MD\) are parallel to \(AC\). Prove that \(HK\), \(LM\) meet in the centroid of the trapezium.
\(OABC\) is a tetrahedron, and \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) are the position vectors of \(A\), \(B\), \(C\) with respect to the origin \(O\). If \((\mathbf{b}, \mathbf{c}) = (\mathbf{c}, \mathbf{a}) = (\mathbf{a}, \mathbf{b})\), prove that each edge of the tetrahedron is perpendicular to the opposite edge. Show that, in this case, the join of \(O\) to the point whose position vector is \([(\mathbf{a}, \mathbf{a}) - (\mathbf{b}, \mathbf{c})]^{-1}\mathbf{a} + [(\mathbf{b}, \mathbf{b}) - (\mathbf{c}, \mathbf{a})]^{-1}\mathbf{b} + [(\mathbf{c}, \mathbf{c}) - (\mathbf{a}, \mathbf{b})]^{-1}\mathbf{c}\) is normal to the plane \(ABC\), and that the altitudes of the tetrahedron meet in a point \(K\). Find the position vector of \(K\).
Show that, for \(0 < \lambda < 1\), the least positive root of the equation $$\sin x = \lambda x \qquad (1)$$ is a decreasing function of \(\lambda\). How many real positive roots of (1) are there when $$\lambda = \frac{2}{(4n+1)\pi},$$ with \(n\) an integer?
Let \(y_0(x) = x\), \(y_n(x) = 1 - \cos y_{n-1}(x)\) (\(n \geq 1\)). For fixed \(n\), find the limit of \(x^{-2^n}y_n(x)\) as \(x\) tends to zero.
Sketch the curves described by the following equations:
Show that, if \(y = \tanh^{-1} x\), then $$(1-x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} = 0.$$ Hence determine the value of the \(n\)th derivative of \(y\) at \(x = 0\). [You may use the theorem of Leibnitz.]
For positive \(Q\), evaluate the integrals $$I(Q) = \int_0^{\pi/2} \frac{\sin^3 \theta \, d\theta}{1 + Q^2 \cos^2 \theta}, \quad J(Q) = \int_0^{\pi/2} \frac{\cos^2 \theta \sin \theta \, d\theta}{1 + Q^2 \cos^2 \theta},$$ and show that \(I(Q) > J(Q)\) when \(Q\) is sufficiently large.