A uniform lamina in the form of a sector of a circle, of radius \(a\), is bounded by radii that enclose an angle \(2\beta\). Prove that the mass centre is distant \((2a \sin \beta)/3\beta\) from the centre of the circle. \par A wedge-shaped portion of angle \(2\beta\) is cut from a uniform solid sphere of radius \(r\) by two planes which pass through a diameter. Show that the distance of the mass centre of the wedge from this diameter is \[ \frac{3\pi a \sin \beta}{16\beta}. \] (Note: the paper has \(a\) in the formula, but \(r\) is the radius of the sphere. Assuming \(r\) is correct.) \[ \frac{3\pi r \sin \beta}{16\beta}. \]
Prove that, if \[ \begin{vmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \end{vmatrix} = 0, \] there exist three numbers \(\xi, \eta, \zeta\), not all zero, and such that \begin{align*} \alpha_1 \xi + \beta_1 \eta + \gamma_1 \zeta &= 0, \\ \alpha_2 \xi + \beta_2 \eta + \gamma_2 \zeta &= 0, \\ \alpha_3 \xi + \beta_3 \eta + \gamma_3 \zeta &= 0. \end{align*} Hence (or otherwise) show that if the nine coefficients \(\alpha_i, \beta_i, \gamma_k\) are all positive and are such that \[ \gamma_1 > \alpha_1 + \beta_1, \quad \beta_2 > \gamma_2 + \alpha_2, \quad \alpha_3 > \beta_3 + \gamma_3, \] then the above determinant cannot vanish.
Prove that in any triangle \[ \tan \frac{1}{2} (B-C) = \frac{b-c}{b+c} \cot \frac{1}{2} A. \] X and Y are two stationary observers, of whom Y is one mile due east of X. An aeroplane, travelling with uniform horizontal velocity, is observed simultaneously by X and Y. Its bearings are found to be 50\(^\circ\) east of north, and 72\(^\circ\) west of north, respectively, by the two observers. Fifteen seconds later, the bearings of the plane are found to be 75\(^\circ\) east of south and 63\(^\circ\) west of south by X and Y, respectively. Find the speed of the plane to the nearest mile per hour.
Prove that, if two ranges \((P, Q, \dots)\), \((P', Q', \dots)\) on different lines \(l, l'\) are homographic, then the locus of the point of intersection of lines \((PQ', P'Q)\) is a straight line. \par If the lines \(l, l'\) are the axes of \(x\) and \(y\) and if the condition that points \(P(x,0)\), \(P'(0,y)\) correspond is \[ axy + bx + cy + d = 0, \] prove that the equation of the locus is \[ bx + cy + d = 0. \]
A tetrahedron \(ABCD\) is made of six equal uniform smoothly-jointed rods, each of weight \(W\). It is hung freely from the mid-point of \(AB\), so that \(AB\) and \(CD\) are horizontal. Find the thrust in \(CD\).
The complex variables \(u+iv\) and \(x+iy\) (where \(u,v,x\) and \(y\) are real) are connected by the relation \[ (u+iv)^2 = x+iy. \] Express \(x\) and \(y\) in terms of \(u\) and \(v\); and show that, if the function \(f(x,y)\) becomes on substitution \(\phi(u,v)\), then \begin{align*} x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} &= \frac{1}{2} \left( u \frac{\partial \phi}{\partial u} + v \frac{\partial \phi}{\partial v} \right), \\ y \frac{\partial f}{\partial x} - x \frac{\partial f}{\partial y} &= \frac{1}{2} \left( v \frac{\partial \phi}{\partial u} - u \frac{\partial \phi}{\partial v} \right), \\ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} &= \frac{1}{4} \left( \frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial v^2} \right) / (u^2+v^2). \end{align*}
Prove that \[ (3 \cos \phi - \sec \phi)^2 \geq 12 \] for all real values of \(\phi\). \par Use this result to show that \[ \frac{\sin\theta - \cos\theta}{3 \cos \theta + 3 \cos \phi - \sec \phi} \] lies between \(1 - \sqrt{(5/3)}\) and \(1 + \sqrt{(5/3)}\) for all real values of \(\theta\) and \(\phi\).
If one triangle can be inscribed in a conic \(S_1\) and circumscribed about a conic \(S_2\), prove that there are infinitely many such triangles and that there is a conic \(S_3\) with respect to which they are all self-conjugate.
A smooth wire is bent into the form of a circle of radius \(a\) and is fixed with its plane vertical. A bead of mass \(M\) slides on the wire, and is attached to one end of a light string of length greater than \(2a\); the string passes over a smooth peg fixed at the highest point of the circle and carries a mass \(m\), hanging freely, at its other end. If \(m<2M\), show that positions of equilibrium exist in which the portion of the string between \(M\) and the peg is inclined to the vertical. \par Prove that in such positions the equilibrium is unstable.
A bag contains the ten numbers \(0, 1, 2, \dots, 9\). Three numbers are drawn from the bag simultaneously and at random. Verify that the probability of the three numbers so selected adding up to 13 is equal to 1/12. What is the probability that they add up to 14? \par Find the corresponding probabilities when the three numbers, instead of being drawn simultaneously, are drawn successively and each replaced in the bag before the next draw.