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 \[ \sec B-\sec C, \quad \sec C-\sec A, \quad \sec A-\sec B, \quad \text{respectively}. \]
Prove that the line joining the extremities of two variable radii of two given concentric circles which are equally inclined to a given direction is normal to a fixed conic.
Two equal smooth cylinders, each of radius \(a\), rest in parallel positions on a horizontal plane. On them rests an equilateral triangular prism in a symmetrical position touching the cylinders with two of its faces and with the third face horizontal. The cylinders are prevented from moving outwards by means of two stops in the plane. Prove that the height of each stop must be at least \[ a\left\{1-\frac{2\kappa+1}{2(\kappa^2+\kappa+1)^{\frac{1}{2}}}\right\}, \] where \(\kappa\) is the ratio of the weight of a cylinder to that of the prism.