Trace the curves given by \[ \sin x = 2 \cos y, \] for which \(y = \frac{1}{6}\pi\), and \(y = \frac{1}{3}\pi\) when \(x=0\), respectively. Find by means of your graph the solutions (correct to about one-tenth of a radian) of the equations \[ \sin x = 2 \cos y, \quad 7x=4y, \] for which \[ 0 < x < \pi. \]
Prove that, if \(\dfrac{p}{q}, \dfrac{r}{s}\) are fractions such that \(qr-ps=1\), then the denominator of any fraction whose value lies between \(\dfrac{p}{q}\) and \(\dfrac{r}{s}\) is not less than \(q+s\). Prove that there are two and only two fractions with denominators less than 19 which lie between \(\frac{13}{18}\) and \(\frac{3}{4}\).
A particle of mass \(m\) is connected to a slightly extensible string of modulus \(\lambda\), the other end of which is fixed to a point in a smooth horizontal table. The particle lies on the table and is projected with velocity \(v\) at right angles to the string which is initially just taut. Show that the maximum extension of the string is approximately \(2mv^2/\lambda\).
Discuss fully the graphical determination of the resultant of a system of co-planar forces whose magnitudes and lines of action are given. Illustrate your discussion by finding the resultant of three forces acting in the sides of a triangle and proportional to them. A light beam is loaded at various points and rests on two supports at the same level. If the loads are given, shew how to determine the magnitudes of the reactions on the supports by a graphical construction. Shew also that the funicular polygon constructed in finding the reactions is a diagram of bending moments for the beam.
An equilateral triangle \(ABC\) is drawn on an inclined plane. The heights of \(A\), \(B\), \(C\) above a horizontal plane are as \(1:12:14\). Show that the side \(BC\) makes an angle \(\sin^{-1}(1/7)\) with a horizontal line drawn on the inclined plane.
Two complex variables \(Z=X+iY\) and \(z=x+iy\) are connected by the equation \[ Z = \frac{e^z-1}{e^z+1}. \] Prove that if \(-\frac{1}{2}\pi < y < \frac{1}{2}\pi\), then \[ X^2+Y^2 < 1. \]
A warship is firing at a target 3000 yards away dead on the beam, and is rolling (simple harmonic motion) through an angle of \(3^\circ\) on either side of the vertical in a complete period of 16 secs. A gun is fired during a roll 2 seconds after the ship passes the vertical. The gun was correctly aimed at the moment of firing, but the shell does not leave the barrel till 0.03 sec. later. Show that the shell will miss the centre of the target by about 4 feet.
Prove that the motion of a rigid lamina moving in its own plane is at any instant (in general) equivalent to a rotation about a certain point \(I\). What is the exceptional case? Prove that, if the vectors \(Oa, Ob\) represent the velocities of two points \(A, B\), the triangles \(Oab, IAB\) are directly similar and that their corresponding sides are perpendicular. Given \(Oa, Ob\) find a geometrical construction for the vector \(Oc\) which represents the velocity of a third point \(C\). Shew in particular that \(AC:CB = ac:cb\), and that, if \(ABC\) is a straight line, so is \(abc\). Four rods are freely jointed together so as to form a quadrilateral \(PQRS\). Shew that if \(PQ\) is fixed the angular velocities of \(QR, PS\) are in the ratio \(PT:QT\), where \(T\) is the point of intersection of \(PQ, RS\).
The sides \(BC\), \(CA\), \(AB\) of a triangle \(ABC\) are \(3x+2y=39\), \(2x-y=5\), \(9x-y=33\), respectively. Show that the line joining \(C\) to the origin bisects one of the angles between the lines joining \(A\) and \(B\) to the origin.
Prove that, if \(s_n = a_1+a_2+\dots+a_n\), where \(a_1, a_2, \dots\) are positive, and \[ t_n = a_1 + \frac{a_2}{s_1} + \frac{a_3}{s_2} + \dots + \frac{a_n}{s_{n-1}}, \] then \[ t_n > \log s_n. \] Deduce that the infinite series \(\sum \frac{1}{n}\) is divergent.