In an Argand diagram a quadrilateral (which may be crossed) has its vertices at the points \(ab, aB, AB\) and \(Ab\) taken in that order, where \(a, b, A\) and \(B\) are any non-zero complex numbers. Prove that the origin \(z=0\) cannot be inside the quadrilateral (or, in the case of a crossed quadrilateral, inside either of the triangles formed by the sides of the quadrilateral).
If \(\Delta_n\) denotes the determinant \[ \begin{vmatrix} \lambda & 1 & 0 & 0 & \cdots \\ 1 & \lambda & 1 & 0 & \cdots \\ 0 & 1 & \lambda & 1 & \cdots \\ 0 & 0 & 1 & \lambda & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{vmatrix} \] with \(n\) rows and columns (where elements \(a_{rr}\) in the main diagonal all have the value \(\lambda\), elements \(a_{r,r+1}\) and \(a_{r+1,r}\) all have the value 1, and the rest vanish) prove that \[ \Delta_n = \lambda \Delta_{n-1} - \Delta_{n-2}. \] Deduce that, if \(\lambda=2\cos\theta\), the value of the determinant \(\Delta_n\) is \(\sin(n+1)\theta \operatorname{cosec}\theta\).
Assume that, if a function of \(x\) vanishes for two values of \(x\), its derivative vanishes for an intermediate value of \(x\). If \[ \phi(x) = \int_x^b f(t) dt - \tfrac{1}{2}(b-x)\{f(x)+f(b)\} + \tfrac{1}{12}(b-x)^3 R, \] where the constant \(R\) is so chosen that \(\phi(a)\) vanishes, show that \(\phi'(x)\) vanishes for \(x=\alpha\) and \(x=b\), where \(a< \alpha< b\). Deduce that \[ \int_a^b f(t) dt = \tfrac{1}{2}(b-a)\{f(a)+f(b)\} - \tfrac{1}{12}(b-a)^3 f''(\beta), \] where \(a< \beta < b\). Hence show that the difference between \(\int_a^{a+nh} f(t) dt\) and \[ \tfrac{1}{2}h\{f(a)+f(a+nh)\} + h\{f(a+h)+f(a+2h)+ \dots + f(a+\overline{n-1}h)\} \] is less than \(\frac{1}{12}nh^3M\), where \(M\) is the greatest value of \(|f''(t)|\) in \(a< t< a+nh\).
By the use of Maclaurin's theorem, or otherwise, prove that \[ \sin x \sinh x = \frac{2x^2}{2!} - \frac{2^3x^6}{6!} + \frac{2^5x^{10}}{10!} - \dots. \]
A function \(z=f_m(x)\) is defined as the solution of the differential equation \[ \frac{dz}{dx} = m \frac{z}{x} \] (where \(m\) is constant) such that \(z=1\) when \(x=1\). Without solving the differential equation explicitly prove that \begin{align*} f_m(x) f_n(x) &= f_{m+n}(x), \\ f_m(x) f_m(y) &= f_m(xy). \end{align*} Deduce the values of \(f_m(1)\) and \(f_0(x)\), and prove that \[ f_{-m}(x) = f_m\left(\frac{1}{x}\right) = [f_m(x)]^{-1}. \]
Establish the equations \[ x=c\sinh^{-1}\frac{s}{c}, \quad y=\sqrt{(s^2+c^2)}, \quad T=wy \] for a uniform flexible chain hanging under gravity. If \(O\) is the lowest point of the chain (\(s=0\)) and \(P\) is any other point, verify from these equations that the moment about \(O\) of the weight of the portion \(OP\) is equal to the moment of the tension at \(P\).
A chain of length \(b\) is trailed on level ground behind a uniformly moving cart to which it is attached at height \(h\) above the ground. Prove that the length \(a\) of chain in contact with the ground satisfies the equation \[ a^2 - 2(b+\mu h)a + b^2 - h^2 = 0, \] where \(\mu\) is the coefficient of friction. Find which of the roots of this quadratic equation represent possible values of \(a\).
A particle of mass \(m\) moves in a straight line under a force \(mf(t)\); the motion is opposed by a resistance \(mkv\) where \(k\) is constant and \(v\) is the velocity. The function \(f(t)\) has the constant value \(F\) when \(0< t< T, 2T
Find whether any of the roots of the equation \[ x^5 + 8x^4 + 6x^3 - 42x^2 - 19x - 2 = 0 \] are integers, and solve it completely.
If \(a, b, c\) are three constants, all different, show that the system of equations \begin{align*} x+y+z &= 3(a+b+c), \\ ax+by+cz &= (a+b+c)^2, \\ xyz &= ayz+bzx+cxy \end{align*} has in general only one set of unequal solutions, and find that set.