Define angular velocity. A point is describing a circle with uniform velocity; prove that the angular velocity of the line joining it to any point on the circumference of the circle is constant. One end \(A\) of the connecting rod of a piston is moving in a circle with velocity \(u\) while the other end \(B\) is moving in a straight line which passes through \(O\) the centre of the circle with velocity \(v\). Prove that \(v=u\sin BAO \sec OBA\), and find the angular velocity of \(AB\).
A string passing over a smooth fixed pulley carries a mass \(2m\) at one end and another smooth pulley of mass \(m\) at the other end. A string having masses \(m\) and \(2m\) at its ends passes over the second pulley. If motion of the system takes place under gravity, find the accelerations of the various parts of the system and the acceleration with which the centre of gravity of the whole system descends.
A point is moving in a circle with velocity \(v\). Prove that \(v^2/r\) is its acceleration towards the centre. A heavy particle is suspended from a point by a string of length \(a\). Prove that if it is projected when at its lowest point with a horizontal velocity greater than \(\sqrt{5ag}\) it will describe a circle in the vertical plane. Also describe the motion of the particle when its velocity of projection (i) lies between \(\sqrt{5ag}\) and \(\sqrt{2ag}\), (ii) is less than \(\sqrt{2ag}\).
Given the inscribed and circumscribed circles of a triangle in position, prove that the orthocentre lies on a fixed circle.
If \(A,B,C,D\) are four points on the same straight line, and circles are drawn through \(AB, BC, CD, DA\) to pass through a common point \(O\), prove that the product of the radii of the first and third circles is equal to the product of the radii of the second and fourth.
\(T\) is a point on a tangent at a point \(P\) of an ellipse so that a perpendicular from \(T\) on the focal distance \(SP\) is of constant length. Shew that the locus of \(T\) is a similar and similarly situated concentric ellipse.
If \[ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0 \quad \text{and} \quad x^2+y^2+z^2=0, \] prove that \[ \left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)(x+y+z)^3+9=0. \]
Find the value of \[ \begin{vmatrix} a^2-bc & b^2-\omega^2ca & c^2-\omega ab \\ c^2-\omega^2ab & \omega a^2-bc & b^2-ca \\ b^2-\omega ac & c^2-ab & \omega a^2-bc \end{vmatrix}, \] where \(\omega\) is an imaginary cube root of unity.
Eliminate \(\theta\) from the equations \[ b\cos(\alpha-3\theta)=2a\cos^3\theta, \quad b\sin(\alpha-3\theta)=2a\sin^3\theta. \]
The lengths of the perpendiculars from the angular points of a triangle on the straight line joining the orthocentre and the centre of the inscribed circle are \(p,q,r\). Prove that \(p\sin A, q\sin B, r\sin C\) are proportional to \[ \sec B-\sec C, \quad \sec C-\sec A, \quad \sec A-\sec B, \quad \text{respectively}. \]