From a stretch of level country, the ground rises at a steady slope of 1 in 30. A railway cutting runs directly into the hill, ending in the vertical face of a tunnel when the maximum depth of excavation is 60 ft. The bottom of the cutting is 35 ft. wide and the sides slope outwards at 45\(^\circ\): estimate the total volume to be excavated.
Equal weights are suspended from the joints of a chain composed of five straight light smoothly jointed links. The extreme links are fastened to two points \(P, Q\) in the same horizontal line by smooth joints. The projection of each link on the horizontal is equal to \(a\), the distance \(PQ\) being \(5a\). The depth of the lowest link, which is horizontal, below \(PQ\) is \(6a\). Find the inclination of each link to the horizontal.
Given two circles (the centre of each of which lies inside the other), show how to draw a rhombus \(ABCD\) with two opposite angular points \(A, C\) on one circle, and \(B, D\) on the other circle. Prove that all such rhombuses have equal sides.
State and prove Menelaus' theorem on transversals. In the triangle \(ABC\), \(AB=AC\) and \(DEF\) is a transversal cutting the sides \(BC, CA, AB\), in \(D, E, F\) respectively. Prove that, without regard to sign, \[ AF \cdot BD \cdot DE = AE \cdot CD \cdot DF. \]
Define an involution of pairs of points on a straight line, and prove that it is determined by two pairs. Shew that a system of conics through four points cut an arbitrary straight line in an involution. Prove that two conics of the system can be found to touch the line, and construct the points of contact, when real.
Two ships are at opposite ends of a diameter of a circle 10 miles in radius. One sails at 2 miles per hour along the diameter, and the other at 4 miles per hour along the circumference. Find an equation determining the time at which they are first at a minimum distance apart, and solve this equation by the use of tables to within five minutes of time.
An aeroplane rests on the ground and is supported in front by a pair of wheels of radius \(a\) and behind by a tail skid which touches the ground at a distance \(l\) from the line joining the points of contact of the wheels. It is found that, if the aeroplane is tilted up through an angle \(\theta\), the vertical force required to support the tail skid is half the original pressure on the ground. Prove that the aeroplane will tip on its nose, if tilted through a total angle greater than \[ \cot^{-1}\left(\frac{1}{2}(\cot\theta - a/l)\right). \]
State and prove Pascal's theorem concerning any hexagon inscribed in a conic. \(OM, ON\) are fixed straight lines through a point \(O\) of a hyperbola. Through a variable point \(P\) of the hyperbola \(PM\) is drawn parallel to one asymptote to meet \(OM\) in \(M\), and \(PN\) parallel to the other asymptote to meet \(ON\) in \(N\). Prove that \(MN\) passes through a fixed point.
Prove that the sum of the odd coefficients in the binomial expansion is equal to the sum of the even coefficients, and each is equal to \(2^{n-1}\), where \(n\) (a positive integer) is the index of the expansion.
Develope ab initio the principal properties of Determinants. Include in particular the proof of the theorem that, if \[ D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}, \quad \Delta = \begin{vmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \end{vmatrix}, \text{ then} \] \[ D\Delta = \begin{vmatrix} a_1\alpha_1+b_1\beta_1+c_1\gamma_1 & a_1\alpha_2+b_1\beta_2+c_1\gamma_2 & a_1\alpha_3+b_1\beta_3+c_1\gamma_3 \\ a_2\alpha_1+b_2\beta_1+c_2\gamma_1 & a_2\alpha_2+b_2\beta_2+c_2\gamma_2 & a_2\alpha_3+b_2\beta_3+c_2\gamma_3 \\ a_3\alpha_1+b_3\beta_1+c_3\gamma_1 & a_3\alpha_2+b_3\beta_2+c_3\gamma_2 & a_3\alpha_3+b_3\beta_3+c_3\gamma_3 \end{vmatrix}. \]