Prove that the equation of the tangent at \(\theta\) to the curve given by \(x=a\sin^2\theta\), \(y=a\cot\theta\), is \(x+y\sin^2\theta\sin 2\theta+a\cos^2\theta\cos 2\theta=a\). Find the point \(\phi\) in which the tangent meets the curve again, and find also the points of inflexion on the curve.
Prove that, if \(y^3=2x-3y\), \[ (x^2+1)y''+xy'=\frac{1}{9}y. \]
Prove that, if \(O, N, H\) are the circumcentre, nine-point centre and orthocentre of a triangle \(ABC\), and \(R\) is its circumradius, \begin{align*} OH^2 &= R^2(1-8\cos A \cos B \cos C), \\ AN^2 &= \frac{1}{4}R^2(1+8\cos A \sin B \sin C), \end{align*} and \[ AN^2+BN^2+CN^2+ON^2=3R^2. \]
Prove that \[ \begin{vmatrix} \sin^2 x & \sin^2(a-x) & \sin^2(a+x) \\ \sin^2 y & \sin^2(a-y) & \sin^2(a+y) \\ \sin^2 z & \sin^2(a-z) & \sin^2(a+z) \end{vmatrix} = 4\sin^2 a \cos a \sin(y-z)\sin(z-x)\sin(x-y). \]
Resolve the expression \(x^{2n}-2x^n\cos n\theta+1\) into \(n\) real quadratic factors, and deduce the factors of \(\cos n\phi - \cos n\theta\) regarded as a function of \(\cos\phi\).
Draw the graphs of \(\text{cosech } x\) and \(\sinh \frac{1}{x}\), and determine on which side of the hyperbola \(xy=1\) each curve lies.
Prove that two couples in the same plane are equivalent if their moments are equal. \(ABCD\) is a square. Forces are represented in magnitude, direction and position by the lines \(AB, BC, CD, DA, AC, DB\). Find the magnitude, direction and position of their resultant.
Find necessary and sufficient conditions of equilibrium of a system of coplanar forces. Four rods smoothly jointed at their extremities form a cyclic quadrilateral \(ABCD\), the opposite corners being joined by strings in a state of tension. Prove that the tensions in \(AC\) and \(BD\) are as \(\sin A:\sin B\).
Explain the cone of friction. A triangle formed of equal uniform rods of length \(a\) hangs in a vertical plane on a rough horizontal peg. Prove that the peg may be in contact with any point on a length \(\mu a/\sqrt{3}\) of either side, where \(\mu\) is the coefficient of friction.
A train travels from rest to rest between two stations 5 miles apart. The total mass is 200 tons; there is a road resistance of 12 lb. weight per ton, and the engine exerts a uniform pull of 5 tons weight until the maximum speed of 30 miles per hour is reached. This speed is maintained until, steam being shut off, an additional resistance equal to \(\cdot075\) the weight of the train is applied to bring the train to rest. Find the time between the stations. (Take \(g=32\).)