Problems

A uniform rod of weight \(W\) and length \(l\) is suspended from a fixed point by two light elastic strings attached to its ends. If the strings have the same modulus of elasticity, \(W\), and are of natural lengths \(l\) and \(l/2\), prove that their lengths in the position of equilibrium are \(l/x\) and \(l/y\), where \[ y=1+x, \] and \[ \frac{1}{x^2} + \frac{1}{(1+x)^2} = \frac{1}{2}\left\{1 + \frac{1}{(1-x)^2}\right\}. \]

State and prove Menelaus' Theorem. Prove that the centres of similitude of three circles (in the same plane) taken in pairs lie at the six vertices of a complete quadrilateral. \(A, B, C\) and \(D\) are the centres of four spheres \((A), (B), (C)\) and \((D)\). \(P\) is a centre of similitude of \((A)\) and \((B)\): \(Q\) is a centre of similitude of \((C)\) and \((D)\): and \(R\) is a centre of similitude of \((A)\) and \((C)\). Prove that the plane \(PQR\) contains one of the centres of similitude of each of the remaining three pairs of spheres.

Two circles intersect in \(A, B\). Through \(A\) a straight line \(CAD\) is drawn cutting the circles in \(C, D\) respectively. Shew that any chords \(CC'\) and \(DD'\) of the circles will intersect on the circle \(BC'D'\).

Find equations giving the foci of the conic whose tangential equation is \[ Al^2 + 2Hlm + Bm^2 + 2Gln + 2Fmn + Cn^2 = 0. \] Shew that the general tangential equation of conics having four-point contact at the origin with a fixed conic touching \(y=0\) there is \[ Al^2 + 2Gln + 2Fmn + (C+\lambda)n^2 = 0; \] and that the locus of their foci is \[ (x^2+y^2)(Gy-Fx) = Axy. \]

One end \(A\) of a uniform rod \(AB\) of weight \(W\) and length \(l\) is smoothly hinged at a fixed point, while \(B\) is tied to a light string which passes over a small smooth pulley a distance \(d\) vertically above \(A\) and carries a weight \(W/4\). If \(l

By reciprocation or otherwise prove the following: \(O\) is a given point of a given hyperbola. Show that, if lines parallel to the asymptotes are drawn through any point of the curve, the parabolas with focus \(O\) which touch two such lines all touch a certain fixed line, and that the feet of the perpendiculars from \(O\) on the two such lines are collinear with a certain fixed point. Show also that, if any triangle is inscribed in the hyperbola, the feet of the perpendiculars from \(O\) on its sides are concyclic with this fixed point.

\(ABC\) is a triangle, \(P\) any point on the internal bisector of the angle \(BAC\); \(BP, CP\) are produced to meet the external bisector of the angle \(BAC\) in \(Q, R\) respectively. Shew that \(PA, BR\) and \(CQ\) meet in a point.

Shew by partial integration that \[ f(a+b) = f(a) + bf'(a) + \dots + \frac{b^n}{n!}f^{(n)}(a) + \int_0^b \frac{(b-t)^n}{n!} f^{n+1}(a+t)dt. \] Expand \((\sin^{-1}x)^2\) in ascending powers of \(x\).

Two uniform ladders \(AB, AC\), of the same length and of the same weight, \(W\), are smoothly jointed at \(A\) and stand with \(B\) and \(C\) in contact with a rough horizontal plane. If a man of weight \(W\) can stand on any rung of the ladders, prove that the coefficient of friction must not be less than \(\frac{3}{2}\tan\frac{1}{2}(BAC)\). Also find the minimum coefficient of friction necessary if in all cases two men (of the same weight \(W\)) can stand one on each ladder.

Express the coordinates of points of the cubic curve \(y^2=x^2(1+x)\) in terms of a parameter \(t\) by putting \(y=tx\); and show that the parameters \(t_1, t_2, t_3\), of three collinear points satisfy the relation \[ t_2 t_3 + t_3 t_1 + t_1 t_2 + 1=0. \] The tangent at a point \(P\) meets the curve again in a point \(Q\): express the parameter of the point \(Q\) in terms of that of \(P\) and hence show that two straight lines \(QP_1, QP_2\) can be drawn from any point \(Q\) of the curve to touch it other than the tangent at \(Q\). Show also that as \(Q\) varies, the chord \(P_1P_2\) touches the conic \(y^2+8(x+1)(x+2)=0\).