Problems

Small errors \(\delta a, \delta b, \delta c\) are made in measuring the sides of a triangle; prove that the consequent error in reckoning the radius of the circumcircle is \[ \frac{1}{2}\cot A \cot B \cot C\left(\frac{\delta a}{\cos A} + \frac{\delta b}{\cos B} + \frac{\delta c}{\cos C}\right). \]

Prove that the radius of curvature of a curve, which is given by the equations \[ x=\phi(t), y=\psi(t), \] is \[ \frac{\left\{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right\}^{\frac{3}{2}}}{\left\{\frac{d^2y}{dt^2}\frac{dx}{dt} - \frac{d^2x}{dt^2}\frac{dy}{dt}\right\}}, \] and find the radii of curvature at the origin of the two branches of the curve given by the equations \[ y=t-t^3, \quad x=1-t^2. \]

Trace the curve \[ y^2(a+x) = x^2(3a-x), \] and show that the area of the loop and the area included between the curve and the asymptote are both equal to \(3\sqrt{3}a^2\).

Show that the series \[ 1 - \frac{1}{1+x} + \frac{1}{2} - \frac{1}{2+x} + \frac{1}{3} - \frac{1}{3+x} + \dots \] is convergent, provided that \(x\) is not a negative integer.

Two equal heavy cylinders of radius \(a\) are placed in contact in a smooth fixed cylinder of radius \(b\) (\(>2a\)); a third equal cylinder is placed gently on top of them, the axes of all the cylinders being horizontal. Show that the two lower cylinders will not separate if \[ b < a(1+2/\sqrt{7}). \]

A string \(ABC\) (\(AB=BC=a\)) is stretched out straight on a smooth table with masses \(m\) tied at \(A, B, C\). Impulses each equal to \(I\) perpendicular to the string are applied at \(A\) and \(C\). Find the tensions just before \(A\) and \(C\) collide.

A particle projected with speed \(u\) strikes at right angles a plane through the point of projection inclined at \(\theta\) to the horizon; find the time of flight and the vertical height of the point struck above the point of projection.

A variable tangent \(t\) to a fixed conic meets two fixed tangents in \(A\) and \(B\), and meets any other fixed line \(l\) in \(P\). The harmonic conjugate of \(P\) with respect to \(A, B\) is \(M\). Shew that the locus of \(M\) is a conic which passes through the points in which the fixed tangents meet \(l\).

Four light rods \(AB, BC, CD, DA\) are freely jointed together; \(AB=BC\) and \(CD=DA\). The rod \(AB\) is fixed horizontally and masses \(P, Q\) are suspended from \(C, D\) respectively. Shew that in equilibrium the angles \(DAB, ABC\) will be both acute or both obtuse, and that if \(\alpha, \beta\) are the angles which \(AD, CD\) make with the vertical \[ \frac{\sin 2\beta}{\sin 2\alpha} = 1 + \frac{Q}{P}. \]

Prove that the circumcircles of the four triangles formed by four coplanar lines meet in a point \(O\), and by inverting with respect to this point, or otherwise, shew that the four circumcentres lie on a circle through \(O\).