Show how, by graphical means, a general indication of the position of the real roots of the equation \(x\cos x = 1\) may be determined, and obtain an approximate value for that one of them lying nearest to \(3\pi/2\).
Prove that for an algebraic equation \(f(x)=0\), there can at most be only one real root in a range of values of \(x\) not containing any real root of the derived equation \(f'(x)=0\). Consider the equation \(3x^5-25x^3+60x+k=0\) for different real values of \(k\), and prove that it cannot have more than three real roots, and that it will have more than one real root only if \(16 \le |k| \le 38\).
If for the segment of a sphere intercepted by a plane, \(\lambda\) denotes the ratio of the area of the curved part of the surface of the segment to that of the whole sphere, and \(\mu\) denotes the ratio of the volume of the segment to that of the whole sphere, prove that \(\mu=\lambda^2(3-2\lambda)\). If \(\lambda'\) denote the ratio of the area of the total surface of the segment to that of the sphere, show that \((\lambda'-\mu)^2=4(\lambda'-1)(\mu-\lambda')\).
Derive the polar equation of a plane curve whose tangent is inclined at a constant angle \(\alpha\) to the radius vector from \(O\). Prove that the length \(d\) of a chord subtending an angle \(\beta\) at \(O\) is given by \[ d = r(1-2e^{\beta\cot\alpha}\cos\beta+e^{2\beta\cot\alpha})^{\frac{1}{2}}, \] where \(r\) is the radius vector to the end of the chord nearer to \(O\). Prove also that the line joining a point of the curve to its centre of curvature subtends a right angle at \(O\).
Three unequal uniform rods \(AB, BC\) and \(CD\), of lengths \(a, b\) and \(c\) respectively, are smoothly jointed together at \(B\) and \(C\); the ends \(A\) and \(D\) slide smoothly along a horizontal rail. Find the positions of equilibrium, and illustrate by sketches the various cases.
A regular hexagonal framework \(ABCDEF\) is formed from six equal uniform rods, each of weight \(W\), smoothly jointed together; it is kept in shape by three light rods \(BE, BF\) and \(CE\). Find the thrust or tension in each of these three rods if the framework is suspended from \(A\).
A motor-car stands on level ground with its back wheels, which are of radius \(a\), in contact with a fixed obstacle of rectangular cross-section and height \(\frac{1}{2}a\). The coefficient of friction between the wheels and the ground and between the wheels and the obstacle is \(\mu\). These back wheels together carry a vertical load \(V\), including their own weight. A torque of gradually increasing moment \(M\) is applied to the back axle. Find how, and for what value of \(M\), equilibrium is broken, distinguishing the cases that can arise. Neglect the friction in the bearings.
\(ABC\) is a triangular lamina. Forces of magnitude \(k \cdot AB\) and \(k \cdot BC\) act outwards along the perpendicular bisectors of the two edges \(AB\) and \(BC\) respectively. Show that their resultant is a force of magnitude \(k \cdot AC\) acting inwards along the perpendicular bisector of the third edge \(AC\). State and prove a more general theorem about the resultant of forces acting along the perpendicular bisectors of the edges \(AB, BC, \dots, MN\) of a lamina in the form of a polygon \(ABC\dots MN\). Find the magnitude and line of action of the resultant of forces of magnitude \(k \cdot AB, k \cdot BC, \dots, k \cdot MN\) acting at the mid-points of the edges \(AB, BC, \dots, MN\) respectively of a lamina \(ABC\dots MN\) along lines (in the plane of the lamina) making the same angle \(\alpha\) with the corresponding edges.
A light inextensible string \(ABC\) is laid upon a smooth horizontal table with \(AB\) and \(BC\) straight and \(\angle ABC\) equal to \(135^\circ\). Equal particles are attached at \(A\) and \(B\), and the end \(C\) is then jerked into motion in the direction \(BC\) with velocity \(V\). Calculate the speed with which \(A\) is jerked into motion.
A particle is attached to a point \(P\) of a light uniform elastic string \(AB\). The ends of the string are fixed to points in a vertical line, and the particle oscillates along this line. Show that, provided that the two parts of the string remain taut, the motion is simple harmonic. Show also that the period is the same whatever the distance between the points to which \(A\) and \(B\) are attached and whichever end of the string is uppermost.