A uniform rod \(AB\), of mass \(m\) and length \(2l\), rests on a smooth horizontal table, to which it is freely pivoted at \(A\). A particle, of mass \(m\), moving horizontally with velocity \(V\), strikes the rod normally at \(B\) and adheres to it. Find the kinetic energy \(T_1\) of the system after impact. Find also the impulsive reaction at \(A\). If the rod \(AB\) had been completely free to move on the table, find what the kinetic energy \(T_2\) would have been, and show that \(T_2/T_1 = 16/15\).
Define the modulus \(|z|\) and the conjugate \(\bar{z}\) of a complex number \(z\). Show that \(z\bar{z}=|z|^2\) and that \(z\) is of unit modulus if and only if \(|z|=1\). Show that if \(\alpha, \beta\) are fixed distinct complex numbers and if \(\dfrac{z-\alpha}{z-\beta}\) is constant then the point representing \(z\) in the Argand diagram lies on a circle or on a straight line according to the value of the constant. The complex numbers \(z\) and \(w\) are related by the equation \[ \frac{2z-i}{z-2i} + 2\frac{2w-i}{w-2i} = 0. \] Show that \(|z|=1\) if and only if \(|w|=1\).
Prove that the geometric mean of a finite set of positive numbers is not greater than their arithmetic mean. When does equality occur? Find the volume of the greatest box whose sides of length \(x, y\) and \(z\) satisfy \[ 36x^2+9y^2+4z^2=36. \]
The three numbers \(X, Y\) and \(Z\) are related to the three numbers \(x, y\) and \(z\) by the two equations \[ \frac{X}{x+3y-z} = \frac{Y}{3x+4y-2z} = \frac{Z}{-x-2y+2z}. \] Find one set of constants \(\alpha, \beta, \gamma\) and \(\lambda\) so that each of these three ratios is equal to \[ \lambda \left(\frac{\alpha X + \beta Y + \gamma Z}{\alpha x + \beta y + \gamma z}\right) \] for all \(x, y\) and \(z\).
The numbers \(a_1, b_1, a_2, b_2, \dots\) and the numbers \(c_1, c_2, c_3, \dots\) are all positive and \[ 0 < h < \frac{a_1}{b_1} < \frac{a_2}{b_2} < \dots < \frac{a_n}{b_n} < \dots < H. \] Show that \[ h < \left( \frac{a_1^m c_1 + a_2^m c_2 + \dots + a_n^m c_n}{b_1^m c_1 + b_2^m c_2 + \dots + b_n^m c_n} \right)^{\frac{1}{m}} < H \quad (m=1, 2, 3, \dots). \] Show also that \[ h < \left( \frac{\frac{1}{b_1^m c_1} + \frac{1}{b_2^m c_2} + \dots + \frac{1}{b_n^m c_n}}{\frac{1}{a_1^m c_1} + \frac{1}{a_2^m c_2} + \dots + \frac{1}{a_n^m c_n}} \right)^{\frac{1}{m}} < H \quad (m=1, 2, 3, \dots). \]
A sequence of functions \(f_n(x)\), \(n=0, 1, 2, \dots\), is defined by \[ \begin{cases} f_0(x) = 1 \\ f_{n+1}(x) = (1+x)^{f_n(x)} \end{cases} \] Assuming that \(f_n(x)\) can be expanded in powers of \(x\), show that \[ f_n(x) = 1+x+x^2+\frac{3}{2}x^3+\dots \text{ for } n\ge3. \] Show that \(f_{n+1}(x)=f_n(x)+x^{n+1}+\text{higher powers of } x\). Deduce that the coefficient of \(x^m\) in the expansion of \(f_n(x)\) in powers of \(x\) is independent of \(n\) for \(n \ge m\).
Sketch the graph of a function \(f(x)\) that satisfies the conditions (i) \(f(0)=0\), (ii) \(f'(0)<0\), (iii) \(f''(x)>0\) for \(x>0\), (iv) \(f(x)\) tends to a limit as \(x\to\infty\). Also sketch the graph of a function \(g(x)\) that satisfies the conditions (i) \(g(0)=0\), (ii) \(g'(0)<0\), (iii) \(g''(0)>0\), (iv) \(\dfrac{g(x)}{x} \to 1\) as \(x\to\infty\).
Show that, if \(u=x^2\), \[ x\frac{d^2f(x)}{dx^2} - \frac{df(x)}{dx} = 4x^3 \frac{d^2f(x)}{du^2}. \] Find a function satisfying the equation \[ x\frac{d^2f(x)}{dx^2} - \frac{df(x)}{dx} - 4x^3 f(x) = 0 \] and containing two arbitrary constants.
Sketch the curve \[ y^2(1+x^2) = (1-x^2)^2, \] and find the area of its loop.
Evaluate \[ \int_0^{2\pi} \frac{\sin^2\theta d\theta}{2-\cos\theta}, \quad \int_{1}^2 \sqrt{\frac{x-1}{x+1}} \frac{dx}{x}. \]