A particle is projected from the origin with velocity \(u\) in a direction making an angle \(\alpha\) with the horizontal in a medium that resists the motion by a force \(kv\) per unit mass, where \(v\) is the velocity of the particle. Write down the Cartesian equations of motion of the particle, and hence show that the trajectory is given by $$y = \frac{g}{k^2}\log\left[1-\frac{kx}{u\cos\alpha}\right] + \frac{x}{u\cos\alpha}\left[u\sin\alpha + \frac{g}{k}\right].$$ Show further that the maximum height is attained after a time $$\frac{1}{k}\log\left[1+\frac{ku\sin\alpha}{g}\right].$$ Verify that on passing to the limit \(k \to 0\), these results reduce to those obtained by omitting the resistance in the original equations.
A gramophone turntable with radius \(a\) and moment of inertia \(I\) is rotating freely with angular velocity \(\omega_1\) about a vertical frictionless spindle. An insect of mass \(m\) (to be regarded as a particle) flies with velocity \(u\) along a horizontal line tangential to the rim of the turntable, and alights on it, coming to rest relative to it. Find the new angular velocity, \(\omega_2\). The insect then walks with constant velocity \(v\) to the centre of the turntable. Prove that the turntable makes $$\frac{a\omega_2}{2\pi v}(y+\gamma^{-1})\tan^{-1}\gamma$$ revolutions during the walk, \(\gamma\) being defined by \(\gamma^2 = ma^2/I\).
Define the modulus \(|z|\) of the complex number \(z\) and show that \(|z_1 + z_2| \leq |z_1| + |z_2|\). Show that, if $$\sum_{k=1}^{n} |z_k| = \left| \sum_{k=1}^{n} z_k \right|,$$ then there is a complex number \(z_0\) such that, for \(1 \leq k \leq n\), \(z_k/z_0\) is real and non-negative.
Prove that, if \(a \neq 1\) or 2, then the equations \begin{align} ax + y + z &= 1,\\ x + ay + z &= b,\\ 3x + 3y + 2z &= c \end{align} have a solution whatever the values of \(b\) and \(c\). Find particular values of \(b\) and \(c\) such that the equations have a solution whatever value is given to \(a\).
\(X\), \(Y\) are fixed points of a circle and the tangent at a variable point \(A\) of the circle meets the tangents at \(X\), \(Y\) in \(B\), \(C\) respectively. Find the positions of \(A\) for which the area \(BCYX\) has a minimum. Show that, if a quadrilateral encloses a circle of unit radius, then its area is at least 4.
\(R\) is the radius of the circumcircle of the triangle \(ABC\). Show that the distance between the orthocentre and the circumcentre is $$R\sqrt{(1 - 8\cos A \cos B \cos C)}$$ and find its maximum when \(R\) is given and \(A\) is a given obtuse angle.
Show that \(\cos(2n + 1)\psi\) may be expressed as a sum of odd powers of \(\cos\psi\) and that the coefficients of \(\cos\psi\) and \(\cos^3\psi\) in this expression are \((-1)^n(2n + 1)\) and \((-1)^{n+1}\frac{2}{3}n(n + 1)(2n + 1)\). By considering the roots of a suitable equation, show that $$\sum_{k=0}^{n-1} \sec^2 \left( \frac{2k + 1}{2n + 1} \frac{\pi}{2} \right) = \frac{2}{3}n(n + 1).$$
Let \(f(x) = x^4 - x^3 - x^2 - x + 1\). Show that \(f(x) = 0\) has two real roots. By considering \(f(x + 1)\) and \(f(x + \frac{1}{2})\), or otherwise, prove that \(f(x) + 2 > 0\) for all real \(x\).
Sketch the curve \(3y^2x^2 - 7y^2 + 1 = 0.\) Show that the line \(y = mx\) meets the curve in three distinct points if, and only if, \(|m| > 2\sqrt{3-4\sqrt{5}+7-4}.\)
Prove Leibniz's theorem on the differentiation of the product of two functions. By considering $$\frac{d^{2n}}{dx^{2n}}(1 - x^2)^{2n} \quad \text{for } x = 1,$$ or otherwise, prove that $$\sum_{k=0}^{s} (-1)^k \binom{2n}{k} \binom{4n - 2k}{2n} = 2^{2n}.$$ [If \(s\), \(t\) are non-negative integers and \(s \leq t\), then $$\binom{t}{s} = \frac{t!}{s!(t-s)!},$$ where \(0!\) is taken to be 1.]