A pendulum consists of a rigid uniform wire of negligible thickness in the form of a circle of radius \(a\) and centre \(C\), and mass \(M\). Two particles of mass \(m\) are attached rigidly to the circumference of the circle at two points \(A\) and \(B\). If \(O\) is a point of the circumference on the diameter perpendicular to \(AB\), show that the pendulum swinging about \(O\) in the plane of the circle has a period for small oscillations which is independent of the length \(AB\) and equal to that of a simple pendulum of length \(2a\).
Solve the system of equations: \begin{align*} x+y+z &= a, \\ x+\omega y + \omega^2 z &= b, \\ x+\omega^2 y + \omega z &= c, \end{align*} where \(\omega\) is a complex cube root of unity, expressing \(x, y\) and \(z\) as simply as you can in terms of \(a, b, c\) and \(\omega\). \newline Assuming \(a, b\) and \(c\) to be real, show that \(x\) is real, and that \(y\) and \(z\) are conjugate complex numbers. Prove also that \[ x^2 + |y|^2 + |z|^2 = x^2+2yz = \frac{1}{3}(a^2+b^2+c^2), \] where \(|y|\) and \(|z|\) are the moduli of \(y\) and \(z\). Find the area of the triangle formed by the points representing the numbers \(x, y, z\) on the Argand diagram.
Show that, if \(p(r)\) is the probability of throwing a total \(r\) with three dice, then \(p(r) = p(21-r)\). Prove the formula \[ p(r) = \frac{(r-2)(r-1)}{432} \quad (3 \le r \le 8), \] and obtain formulae valid for other ranges of \(r\). Show that the chance that the total will lie between 9 and 12 (inclusive) is about \(0 \cdot 48\).
Solution: The probability of throwing \((a,b,c)\) is the same as the probability of throwing \((7-a,7-b,7-c)\) (imagine counting the face down dots). Therefore \(p(r) = p(21-r)\). All possible ways of writing \(x_1+x_2+x_3\) with \(1 \leq x_i\) are valid ways to obtain sums from \(3\) to \(8\) (no value can be larger than \(6\)). Therefore there are \(\binom{r-1}{2}\) ways to obtain the number \(r\), and each way has probability \(\frac{1}{6^3}\), so \(\frac{(r-1)(r-2)}{216 \times 2}\) as required. Since \(p(21-r) = p(r)\) we must have that \(p(r) = \frac{(19-r)(20-r)}{432}\) for \(13 \leq r \leq 18\). The number of ways to obtain \(9\) (and hence \(12\)) is \(\binom{9-1}{2} - 3\) (for the ways which involve a \(7\)). The number of ways to obtain \(10\) is \(\binom{10-1}{2} - 3 \cdot 1 - 3 \cdot 2 \cdot 1\) Therefore we must have \[ p(r) = \begin{cases} \frac{(r-2)(r-1)}{432} & \text{if }3 \leq r \leq 8 \\ \frac{25}{216} & \text{if }r=9 \\ \frac{27}{216}=\frac{1}{8} & \text{if }r=10, 11 \\ \frac{25}{216} & \text{if }r=12 \\ \frac{(19-r)(20-r)}{432} & \text{if }13 \leq r \leq 18 \end{cases}\] The probability in question is \(\frac{50+54}{216} = \frac{104}{216} = \frac{13}{27} \approx 0.48\)
Find the least value of the expression \[ y = \frac{1}{n} \sum_{r=0}^{n-1} \left( x - \sin \frac{r\pi}{n} \right)^2 \] for real values of \(x\). \newline If the minimum is \(y_n\), attained for \(x=x_n\), prove that, when \(n \to \infty\), \[ x_n \to \frac{2}{\pi}, \quad y_n \to \frac{1}{2} - \frac{4}{\pi^2}. \]
The asymptotes and a point \(P\) of a hyperbola are given. Describe and justify constructions for
How many conics (real or imaginary) can be found (in general) to pass through \(5-n\) given points and to touch \(n\) given lines, for \(n=0, 1, 2, 3, 4, 5\)? Give reasons for your answers, and indicate any familiar metrical cases that may be obtained by specialising some of the points or lines.
A solid cube casts a shadow on a plane \(\pi\) from a source of light at a point \(O\) so that the outline of the shadow is a hexagon. Show that, if three faces of the cube are illuminated, a conic may be inscribed in the hexagon. \newline In what circumstances is the same result true if only two faces are illuminated?
Prove that \begin{align*} \int_0^\infty \frac{dx}{\cosh x + \cos \theta} &= \frac{\theta}{\sin \theta} \quad (0 < \theta < \pi), \\ \int_0^\pi \frac{d\theta}{\cosh x + \cos \theta} &= \frac{2}{\sinh x} \tan^{-1}(\tanh \frac{1}{2}x) \quad (x > 0), \end{align*} where the inverse tangent is to be taken between \(0\) and \(\frac{1}{2}\pi\). Discuss carefully the determinations of any inverse trigonometric functions you use. \newline Denoting the integrals by \(I(\theta)\) and \(J(x)\), respectively, verify that \[ \int_0^\pi I(\theta) d\theta = \int_0^\infty J(x) dx. \]
In a triangle \(ABC\) the side \(a\) and the angles \(B, C\) (measured in radians) are taken as independent variables. Show that \[ \frac{\partial b}{\partial B} = \frac{c}{\sin A}, \] and interpret this result geometrically. \newline If the angles \(B\) and \(C\) undergo small variations \(\delta B\) and \(\delta C\) (positive or negative) while the vertices \(B\) and \(C\) remain fixed in position, prove that the vertex \(A\) is displaced through a distance \(\delta s\) given approximately by \[ (\sin A \, \delta s)^2 = c^2 \delta B^2 + b^2 \delta C^2 + 2bc \cos A \, \delta B \delta C. \] The position of \(A\) is estimated by taking bearings from \(B\) and \(C\). The position and length of the base line \(BC\) may be taken to be accurately known, but the measurement of each of the angles \(B\) and \(C\) is liable to a small error \(\epsilon\) in either direction. Show that, if \(B+C \le \frac{1}{2}\pi\), the maximum distance between the true and the estimated positions of \(A\) due to these errors is \[ \frac{\epsilon a}{\sin(B+C)} \] to the first order in \(\epsilon\). How must this result be modified if \(B+C > \frac{1}{2}\pi\)?
A uniform chain \(AB\) of length \(l=a+b\) hangs from the end \(B\) with a portion \(AP\) of length \(a\) resting on a smooth plane inclined at an angle \(\alpha\) to the horizontal. Prove that the height of \(B\) above the level of \(A\) is \(h\), where \[ h^2 = l^2 \sin^2 \alpha + b^2 \cos^2 \alpha. \]