Prove that the circle which has with the parabola \(y^2-4ax=0\) the common chords \(x+4y-5a=0, x-4y+7a=0\), passes also through the poles of these chords with respect to the parabola.
Prove that, in the rectangular hyperbola \(2xy=c^2\), the normals at the extremities of the chords \(x+2y-c=0, 4x-2y+3c=0\) meet in a point, and find the point.
Find the first significant term in the expansion in ascending powers of \(\theta\) of \[ \frac{2\theta - 28\sin\theta+\sin 2\theta}{9+6\cos\theta}. \]
The radii of two parallel plane sections of a sphere are \(a,b\), and the distance between them is \(c\). Prove that the included volume is \(\frac{1}{6}\pi c(c^2+3a^2+3b^2)\), and the included surface \(\pi\{(a^2+b^2+c^2)^2-4a^2b^2\}^{\frac{1}{2}}\). What is the condition that the centre of the sphere should lie within the volume considered?
Prove that if \(\cos 2A + \cos 2B + \cos 2C + 4\cos A \cos B \cos C + 1 = 0\) then \(A \pm B \pm C\) must be an odd multiple of 180\(^\circ\). Prove that \begin{align*} \Sigma \sin^2 A \sin(B+C-A) &- 2\sin A \sin B \sin C \\ &= \sin(B+C-A)\sin(C+A-B)\sin(A+B-C). \end{align*}
Shew how to solve a triangle \(ABC\) having given \(B-C, b-c\) and the perpendicular distance of \(A\) from \(BC\), and adapt the solution to logarithmic computation.
Prove that, if \(x<1\) \[ \frac{1-x^2}{1-2x\cos\theta+x^2} = 1+2x\cos\theta+2x^2\cos 2\theta+2x^3\cos 3\theta+\dots. \] Obtain the first 3 terms of the expansion of \(\cos n\theta\) in ascending powers of \(\cos\theta\), when \(n\) is even.
Find the real quadratic factors of \(x^{2n} - 2x^n\cos n\alpha + 1\). Prove that, if \(n\) is an odd integer \[ \sin^2\theta \sin^2(\theta+\frac{\pi}{n})\dots\sin^2(\theta+\frac{n-1}{n}\pi) + \cos^2\theta\cos^2(\theta+\frac{\pi}{n})\dots\cos^2(\theta+\frac{n-1}{n}\pi) = 2^{2-2n}. \]
Shew that the effect of a couple is independent of its position in the plane in which it acts. \(ABCD\) is a skew quadrilateral. Prove that forces completely represented by the lines \(AB, BC, CD, DA\) are equivalent to a couple in a plane parallel to \(AC\) and \(BD\) and determine its moment.
A drawer of depth \(b\) (from back to front) is jammed by pulling at a handle at a distance \(c\) from the centre of the front. Prove that the coefficient of friction must be at least \(b/2c\).