Problems

\(OA, AB\) are two inextensible strings each of length 5 ft. \(O\) is attached to a fixed point and masses \(m_1, m_2\) are attached at \(A\) and \(B\). The whole swings round so that both strings lie in a vertical plane which rotates about the vertical through \(O\) with uniform angular velocity \(\omega\). Shew that if \(A\) and \(B\) remain at distances 3 ft. and 7 ft. respectively from the axis of rotation the ratio \(m_1/m_2\) is equal to 49/15, and find the necessary value of \(\omega\).

Give a short account, without proofs, of the methods of (1) inversion, (2) orthogonal projection, (3) conical projection, (4) reciprocation, in plane geometry. Compare and contrast the geometrical figures and properties which are unchanged by one or more of these methods of transformation, but not by others; [e.g.\ a point is transformed into a point by (1), (2), (3) but not by (4); conjugate diameters are transformed into conjugate diameters by (2) but not by the others].

Two uniform beams \(AB, AC\) of the same length are smoothly hinged together at \(A\) and placed standing in a vertical plane with the ends \(B, C\) on a rough horizontal plane. Shew that, if the weight of one is double that of the other, the least value of the coefficient of friction necessary for equilibrium is \(\frac{2}{3} \tan \frac{1}{2} (BAC)\). If the coefficient of friction has this value at which end is the friction limiting?

Prove that, if \(nu_n = u_{n-2} + u_{n-3} + \dots + u_2 + u_1\) for all integral values of \(n\) greater than 2, and \(u_1 = 1\), \(u_2 = \frac{1}{2}\), then \[ u_n = \frac{1}{2!} - \frac{1}{3!} + \dots + (-1)^n/n! . \]

\(ABC\) is a triangle, \(P\) any point on the circumscribing circle. Shew that the feet of the perpendiculars from \(P\) on the sides of the triangle lie on a straight line, the pedal line of \(P\). Shew also that if the perpendicular from \(A\) on the pedal line of \(P\) intersects the circle again in \(Q\), then \(PQ\) is parallel to \(BC\).

Prove that, if \(P, Q\) are two polynomials in a variable \(x\) with no common factor, it is possible to find two other polynomials \(A, B\) such that \(AP + BQ = 1\) identically. Prove further that if \(A_1, B_1\) are a pair of polynomials satisfying this identity, every solution is of the form \(A = A_1 + CQ, B = B_1 - CP\), where \(C\) is a polynomial, and deduce that \(A, B\) can be so chosen that the degree of \(A\) is less than that of \(Q\), and the degree of \(B\) less than that of \(P\). State and prove the corresponding theorems relating to positive or negative integers \(p, q\) which are prime to one another.

Three equal uniform rods \(AB, BC, CD\) are smoothly hinged together at \(B\) and \(C\) and rest on a smooth horizontal table with the ends \(A, D\) smoothly pivoted to the table, so that the angles \(ABC, BCD\) are each \(120^\circ\). The middle points of the rods \(AB, BC\) are joined by a string which is kept taut by a couple \(G\) applied to the rod \(CD\). Prove that the tension in the string is \(2G/AB\).

Prove that the sum of the reciprocals of all positive integers which can be written (in the ordinary scale of notation) without the use of the digit 0 is less than \[ 9 + 9^2/10 + 9^3/10^2 + \dots. \]

Having given \begin{align*} ax + by &= 1, \\ a'x + b'y &= 1, \\ ab &= a'b', \\ a + b + a' + b' &= c, \end{align*} and shew that in general \[ x + y = cxy. \]

State and prove Demoivre's Theorem. Give an account of some of its more important applications.