Given \(F\{s^2(z-x), s^3(z-y)\} = 0\), where \(s=x+y+z\), prove that \[ (s-x)\frac{\partial z}{\partial x} + (s-y)\frac{\partial z}{\partial y} = s-z. \]
\(PQ\) is a chord of the ellipse \(x^2/a^2+y^2/b^2=1\) normal at \(P\). Find the maximum and minimum values of \(PQ\), and shew that if \(e>1/\sqrt{2}\), the minimum value of \(PQ\) is \(3\sqrt{3}a^2b^2/(a^2+b^2)^{3/2}\) and discuss the cases when \(e<1/\sqrt{2}\).
If \[ \frac{a \sin^2 x + b \sin^2 y}{b \cos^2 x + c \cos^2 y} = \frac{b \sin^2 x + c \sin^2 y}{c \cos^2 x + a \cos^2 y} = \frac{c \sin^2 x + a \sin^2 y}{a \cos^2 x + b \cos^2 y}, \] shew that \[ a^3+b^3+c^3-3abc = 0. \]
A chain of length 20 feet and weight 10 lbs. is stretched nearly straight between two points at different levels. Assuming that vertically below the middle point of the chord the chain is approximately parallel to the chord and that the tension there is 100 lbs. weight, prove that the sag measured vertically from the middle point of the chord is approximately 3 inches.
A circular cylinder rolls on a horizontal plane with uniform angular velocity; within it rolls a smaller cylinder also with uniform angular velocity. If the smaller rolls round the circumference of the larger in the time taken by the latter to make one revolution, compare their angular velocities.
A smooth sphere suspended by a string is struck directly by an equal sphere moving downwards, the line of impact being inclined to the vertical. Prove that if there is perfect restitution no kinetic energy is destroyed by the impact.
Find the \(n\)th differential coefficients with respect to \(x\) of \[ x\log x, \quad \sin^3 x, \quad 1/(x^2+1). \]
Determine the stationary values of the function \(e^{ax}\sin bx\), where \(a\) and \(b\) are positive, and illustrate the results by a figure.
The relation between the variables being \(f(x,y)=0\), find \(\dfrac{d^2y}{dx^2}\) in terms of the partial differential coefficients of \(f(x,y)\) with respect to \(x\) and \(y\).
The perpendicular from the origin on the tangent to a curve being denoted by \(p\), and the angle this perpendicular makes with a fixed line by \(\phi\), find an expression for the projection of the radius vector on the tangent. Obtain the relation between \(p\) and \(\phi\) for the curve given by \[ x = a\cos t, \quad y = a\sin^3 t. \]