Problems

A roof whose slope is inclined at \(30^\circ\) to the horizontal runs into another roof whose slope is inclined at \(45^\circ\), their horizontal ridge lines being inclined to one another at an angle of \(60^\circ\). Find the inclination to the horizontal of the line of intersection of the two slopes.

Points \(P\) and \(Q\) are taken upon two opposite sides \(AB\), \(CD\) of a square \(ABCD\). Shew that, if the diagonals of the trapezium \(APQD\) meet in \(X\) and those of \(BPQC\) meet in \(Y\), \(XY\) will pass through the centre of the square.

A smooth sphere is suspended from a fixed point by a string of length equal to its radius. To the same point a second string is attached which after passing over the sphere supports a weight equal to that of the sphere. Shew that the string which supports the sphere makes an angle \(\sin^{-1}(\frac{1}{4})\) with the vertical.

From a point \(P\) outside a circle two lines \(PAB, PDC\) are drawn, cutting the circle at \(A, B, C, D\). Prove that \(PA.PB\) is equal to \(PC.PD\). \par If \(AC, DB\) meet in \(Q\), show that the circles \(PBD, PAC, QAB, QCD\) all pass through a point which lies on \(PQ\).

Prove that the curve in which a right circular cone is cut by a plane possesses the following properties of a conic:

1. [(i)] the focus-directrix property, \(SP=e.PM\),
2. [(ii)] the focal property, \(SP+S'P = \text{constant}\),
3. [(iii)] the property that \(PN^2\) is proportional to \(AN.NA'\), where \(PN\) is the perpendicular from \(P\) upon the major axis \(AA'\).

Prove that the locus of the vertices of right circular cones, which have a given ellipse as a plane section, is a hyperbola whose foci lie at the ends of the major axis of the ellipse. \par A number of tetrahedra have the same base \(ABC\), and the point where the inscribed sphere touches the face \(ABC\) is the same for all. Prove that the point where the sphere escribed to \(ABC\) touches the face \(ABC\) is also the same for all the tetrahedra.

A sphere is divided by two parallel planes into three portions of equal volume; find to three places of decimals the ratio of the thickness of the middle portion to the diameter of the sphere.

Two parabolas have foci \(S_1\), \(S_2\), and the directrix of each passes through the focus of the other. Prove that each parabola touches the line which bisects \(S_1S_2\) at right angles, and that the two other common tangents are perpendicular to one another.

It is required to place forces in the sides of a given plane quadrilateral so that they shall have a resultant which is given in all respects. If forces \(P, Q, R, S\) be one solution, and \(P', Q', R', S'\) be a second solution, shew how to find the ratios that \(P-P'\), \(Q-Q'\), \(R-R'\), \(S-S'\) bear to each other.

The tangent and normal at a point \(P\) of a parabola whose focus is \(S\) meet the axis of the parabola in \(T\) and \(G\) respectively: prove that \(ST=SP=SG\). \par Prove also that, as \(P\) moves along the curve, \(GP^2 \propto GS\).

The prime factors of a number \(N\) are known, viz. \[ N = P_1^{a_1} P_2^{a_2} P_3^{a_3} \dots P_r^{a_r}, \] where \(P_1, P_2, P_3 \dots P_r\) are different prime numbers. Counting \(N\) itself and unity as divisors of \(N\), shew that the number of divisors of \(N\) is \[ (1+a_1)(1+a_2)(1+a_3)\dots(1+a_r), \] and find a formula for the sum of these divisors. Find also the number of ways in which \(N\) can be resolved into two factors prime to one another, and a formula for the sum of such factors. \par Prove that 60 is the smallest number which has 12 divisors; and generally that the smallest number which has \(k\) divisors is one of the numbers of the form \[ 2^{a_1} 3^{a_2} 5^{a_3} 7^{a_4} 11^{a_5} \dots, \] where \(a_1, a_2, a_3, \dots\) are numbers such that \(a_1 \ge a_2 \ge a_3 \ge a_4 \dots\), and \[ (1+a_1)(1+a_2)(1+a_3)\dots = k. \]