Prove that a plane section of a right circular cone is a conic and find its foci. Prove that the latera recta of all plane sections of a right circular cone, which touch a given sphere whose centre is the vertex of the cone, are equal.
What is meant by the physical independence of forces? Explain briefly the nature of the evidence on which it is based.
Find a construction for the line of quickest descent from a straight line to a circle in the same vertical plane. Apply it to finding the time of quickest descent from a vertical straight line to a circle whose centre is at a distance of 10 feet from it and whose radius is 3 feet.
A particle is projected with velocity \(u\) from the foot of an inclined plane, the vertical plane containing the line of greatest slope, in a direction making an angle \(\alpha\) with the plane; find the range, the inclination of the plane to the horizontal being \(\theta\). If the particle strikes the plane at an angle \(\beta\) with a velocity \(v\), prove that \(\cot\beta=\cot\alpha-2\tan\theta\) and \(v=u\sin\alpha\text{cosec}\beta\).
Discuss the simple harmonic motion of a particle, investigating the velocity at any point of its path, and the time of describing any given portion of the path. An elastic string of natural length \(a\) and modulus \(\lambda\) is stretched between two points on a smooth horizontal table. A particle of mass \(m\) is fastened to its middle point and displaced in the direction of the string; find the time of a small oscillation.
Prove Leibnitz's Theorem for the \(n\)th differential coefficient of a product. If \(y=\sin(p\sin^{-1}x)\), shew that \[ (1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+1)x\frac{d^{n+1}y}{dx^{n+1}} - (n^2-p^2)\frac{d^ny}{dx^n} = 0. \]
Investigate a method of determining the maximum and minimum values of a function of one independent variable. Find the maximum and minimum values of the expression \[ (x+1)^2(9x^2+2x-49). \]
Evaluate: \[ \text{(i) } \int \frac{dx}{5-2x-3x^2}, \quad \text{(ii) } \int \frac{3\cos x+4\sin x}{2\cos x-3\sin x}dx, \quad \text{(iii) } \int_0^1 \frac{dx}{1+2x\cos\alpha+x^2}. \]
Obtain expressions for the area of a curve when its equation is given by (i) \(r=f(\theta)\) and (ii) \(p=f(r)\). Find the area of a loop of the curve \[ p^2(4a^2-3r^2)=r^4. \]
The theory of poles and polars, developed by projective methods.