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.
A container in the form of a right circular cone with semi-vertical angle \(\alpha\) is held with its axis vertical and vertex downwards. Water is supplied to the container at a constant volume-rate \(Q\), and it escapes through a leak at the vertex at a rate \(ky\), where \(y\) is the depth of water in the cone, and \(k\) is a constant. Show that $$\pi \tan^2 \alpha \, y^2 \frac{dy}{dt} = Q - ky,$$ and find how long it takes for the water level to rise from zero to \(Q/2k\).
Show that, if $$f(x) = \sum_{n=1}^{\infty} a_n \sin nx, \qquad (1)$$ then $$a_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin nx \, dx. \qquad (2)$$ Assuming that the function $$f(x) = x(\pi - x) \qquad (0 \leq x \leq \pi)$$ can be expressed as an infinite series $$\sum_{n=1}^{\infty} a_n \sin nx,$$ and that the coefficients are still given by the formula (2), show that in this case $$a_{2m} = 0, \quad a_{2m+1} = \frac{8}{\pi(2m+1)^3},$$ and hence sum the series $$1 - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \cdots$$
Let \(f(x)\) be a continuous decreasing function of \(x\) for \(x > 0\), and \(m\) and \(n\) be positive integers with \(m < n\). Show that $$\int_m^{n+1} f(x) \, dx < \sum_{r=m}^n f(r) < \int_{m-1}^n f(x) \, dx,$$ and hence that $$1.19 < \sum_{r=1}^{\infty} \frac{1}{r^3} < 1.22.$$