A heavy lever (weight \(w\) lb. per foot length) with the fulcrum at one end, is to be used to raise a weight W, which is at a given distance \(a\) feet from that end. Prove that in order that the weight may be lifted with the least effort, the length of the lever should be \[ \sqrt{\frac{2aW}{w}} \text{ feet}. \]
Find the equation of the normal at any point \((at^3, at^2)\) of the curve \(x^2 = ay^3\), and show that it meets the curve again in two points whose parameters \(t_1, t_2\) are connected with \(t\) by the relation \[ \frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t}=0. \] (Note: The parameterization in the question appears to be inconsistent with the curve equation. Transcribed as written.) Prove that if \(t^2 = \frac{2}{9}\) the normal touches the curve.
Prove that the \(n\)th differential coefficient of \(e^{ax}\sin bx\) is \[ (a^2+b^2)^{\frac{n}{2}}e^{ax}\sin\left(bx+n\tan^{-1}\frac{b}{a}\right). \] Prove that the curve \[ y = e^{-x^2} \] has points of inflexion where \(x=\pm\frac{1}{\sqrt{2}}\). Sketch the curve roughly.
Prove that the line joining the vertex of a triangle to the point on the inscribed circle, which is furthest from the base, passes through the point of contact of the escribed circle with the base.
Given a circle of which AB is a diameter, C and D two points on the circumference, find a point P on the circumference such that PC and PD will cut AB in points equidistant from the centre.
Prove that the locus of the extremities of parallel diameters of a system of coaxal circles is a rectangular hyperbola.
Prove that, if \((1+x)^n = p_0+p_1x+\dots+p_nx^n\), where \(n\) is a positive integer, \[ \frac{p_0}{n+1} - \frac{1}{2}\frac{p_1}{n+2} + \dots + (-1)^r \frac{p_r}{n+r+1} + \dots = \frac{2^n(n!)^2}{(2n+1)!}. \]
The bisector of the angle A of a triangle ABC meets the circumcircle in D. Prove that the line joining D to the orthocentre of the triangle makes an angle \(\theta\) with BC, where \[ \sec^2\theta \sin^2(B-C) = 2(1-\cos A+2\cos B \cos C - 4\cos A \cos B \cos C). \]
Prove that, if \(\tan\frac{\theta}{2} = 2\tan\frac{\alpha}{2}\), \[ \frac{1}{2}(\theta-\alpha) = \tan^{-1}\frac{\sin\alpha}{3-\cos\alpha} = \frac{1}{3}\sin\alpha+\frac{1}{2\cdot 3^2}\sin 2\alpha + \frac{1}{3\cdot 3^3}\sin 3\alpha + \dots. \]
From any point on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), chords are drawn through the foci. Prove that the locus of the points of intersection of the tangents at the other extremities of the chords is the ellipse \[ \frac{x^2}{a^2}+\frac{b^2y^2}{(2a^2-b^2)^2} = 1. \]