Shew how to draw a perpendicular to a plane from a point outside it. Prove that if two straight lines neither intersect nor are parallel, there is one straight line perpendicular to both of them.
Prove that the two tangents drawn from a point to a parabola subtend equal angles at the focus. The normal at a point \(P\) in a parabola meets the curve again at \(Q\), and the tangents at \(P\) and \(Q\) intersect at \(T\). If \(S\) is the focus and \(K\) is the middle point of \(TQ\), prove that \(TSK\) is a right angle.
Prove that the latus-rectum of the conic, in which a given right-circular cone is cut by a plane, is proportional to the perpendicular to the plane from the vertex of the cone.
Shew that the equation \(axy+bx+cy+d=0\) may be written in the form \[ \frac{x-p}{x-q} = \lambda \frac{y-p}{y-q}, \] where \(\lambda\) has two values of which the product is \(-1\).
Find the general term in the expansion in powers of \(x\) of the expression \[ \frac{1-2x-x^2}{(1-x^2)(1+x+x^2)}. \]
Sum the series \(n^2+2(n-1)^2+3(n-2)^2+\dots\) where \(n\) is a positive integer; and find the \(n\)th term of the recurring series \(1+3x+10x^2+36x^3+\dots\).
Prove that any convergent to \(a_1+\frac{1}{a_2+}\frac{1}{a_3+}\dots\) is nearer to the continued fraction than any convergent preceding it. Shew that the \(n\)th convergent to the continued fraction \(1-\frac{1}{2-}\frac{1}{2-}\dots\) is equal to \(\frac{n}{n-1}\).
Five equal light rods are jointed together to form a regular pentagon \(ABCDE\) and two light rods \(BE\) and \(BD\) make the framework rigid. A weight of 10 lb. is attached to \(A\) and the framework is suspended from \(C\); find the stresses in all the rods of the system.
Explain what is meant by the angle of friction. Find the direction in which the least force necessary must be applied to move a body up a rough plane inclined at an angle \(\alpha\) to the horizon, the angle of friction being \(\lambda\). The end \(A\) of a uniform rod \(AB\) rests on a rough horizontal plane, and the end \(B\) is connected by a string to a point \(C\) above it. When \(A\) is as far as possible from \(C\) for equilibrium, \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) with the vertical. Prove that the angle of friction between the rod and the plane is \(\cot^{-1}(\cot\beta-2\cot\alpha)\).
Shew that two couples in the same plane balance each other if their moments are equal and opposite. If \(r\) and \(R\) are the radii of the inscribed and circumscribed circles of a triangle \(ABC\) and \(D, E, F\) are the points at which the sides are touched by the inscribed circle, shew that forces represented by \(r.BC, r.CA\) and \(r.AB\) are balanced by forces represented by \(2R.FE, 2R.DF\) and \(2R.ED\).