A heavy beam inclined at an angle \(\alpha\) to the horizontal rests with one end against a vertical wall and the other on the ground, the coefficient of friction in each case being \(\tan\lambda\). The beam is in a vertical plane perpendicular to the plane of the wall. Shew that if the beam is kept from slipping down by a horizontal string of tension \(T\) attached to the lower end, or by a vertical string of tension \(T'\) attached to the upper end, then \[ T'=T\tan(\alpha+\lambda). \]
When a ship is steaming due North the line of smoke makes an angle \(\alpha\) to the East of South; on the ship turning due East the line of smoke makes an angle \(\beta\) to the South of West; on its turning due South the smoke makes an angle \(\gamma\) to the East of North. Shew that \[ \cot\beta(\cot\alpha-\cot\gamma) = \cot\alpha+\cot\gamma-2. \]
Solve the equations:
Find the sum of the squares of the first \(n\) odd numbers. \par Prove that the sum of the squares of all numbers less than 60 and prime to 60 is 19120.
Prove that \[ 1+\sec 20^\circ = \cot 30^\circ \cot 40^\circ \] and solve the equation \[ 1+\sec\theta = \tan 3\theta \cot 2\theta. \]
On opposite sides of a base \(BC\) are described two triangles \(ABC, BCD\), such that \(\angle ABC=30^\circ, \angle ACB=80^\circ, \angle DBC = \angle DCB = 50^\circ\). Shew that, if \(AD\) is drawn, \(\angle DAC=30^\circ\), and find the other angles of the figure.
When are two ranges said to be homographic? Shew that two homographic ranges on the same straight line have two common points, real, coincident, or imaginary. \par In what case is one of the common points at infinity?
Prove that the tangents from a point \(O\) to a conic subtend angles at a focus which are equal or supplementary. \par Shew that the same proposition is true if instead of a focus is taken the foot of the perpendicular from \(O\) on the major axis.
Prove that two of the tangents of the parabola \(y^2=ax\) are identical with two of the tangents of \(a^2x^2=c^2(4y-3c)\), and find their equations. Prove also that the two curves touch one another.
Prove that the two circles \[ (x-\alpha)^2+(y-\beta)^2 = \lambda(x^2+y^2), \quad (\alpha+\mu\beta)(x^2+y^2) = (\alpha^2+\beta^2)(x+\mu y) \] cut orthogonally, and interpret the result geometrically.