Three spheres, each of radius 3 inches, rest in mutual contact on a horizontal table, and a fourth sphere, of radius 2 inches, rests upon them. Find (i) the height above the table of the highest point of the smaller sphere, and (ii) the inclination to the horizontal of a plane which touches the smaller sphere and two of the larger ones.
Find values of \(a, b, c, d\) such that the curve \(y=ax^3+bx^2+cx+d\) touches the lines \(3x-y-6=0, 3x+3y+2=0\) at their points of intersection with the axes of \(x\) and \(y\) respectively. Prove that the curve touches the axis of \(x\), and that the curvature at the point of contact is 2.
Prove that a function \(f(x)\) has a minimum for \(x=a\), if \(f'(a)=0\) and \(f''(a)>0\). A thin closed rectangular box is to have one edge \(n\) times the length of another edge, and the volume is to be \(V\). Prove that the least surface \(S\) is given by \(nS^3=54(n+1)^2V^2\).
Sketch the curve \(a^2y^2 = x^2(a^2-x^2)\). Find the area of a loop of the curve, and prove that the volume generated by revolution of a loop about the \(y\)-axis is \(\pi a^3/4\).
Given the circumcentre, the nine-point circle and the difference of two angles of a triangle, construct the triangle.
The tangents at the points \(P, Q\) of \(x^2/a^2+y^2/b^2=1\) meet on the confocal \[ x^2/(a^2+\lambda) + y^2/(b^2+\lambda)=1. \] \(R\) is the other extremity of the diameter of the first conic through \(Q\). Prove that the tangents at \(P\) and \(R\) meet on the confocal \[ \frac{x^2}{a^2(b^2+\lambda)} + \frac{y^2}{b^2(a^2+\lambda)} = \frac{1}{\lambda}. \]
A conic is inscribed in a triangle. Prove that the straight lines drawn from the vertices of the triangle to the points of contact of the opposite sides meet in a point \(P\) and shew that if the centre of the conic moves along a straight line the locus of \(P\) is a conic through the vertices of the triangle.
If \(1, \alpha, \alpha^2, \alpha^3, \alpha^4\) are the fifth roots of unity, prove that \[ \alpha\tan^{-1}\alpha + \alpha^2\tan^{-1}\alpha^2 + \alpha^3\tan^{-1}\alpha^3 + \alpha^4\tan^{-1}\alpha^4 \] \[ = \pi\cos\frac{3\pi}{5} + \sin\frac{3\pi}{5}\log\left(\tan\frac{\pi}{20}\right) + \sin\frac{\pi}{5}\log\left(\tan\frac{3\pi}{20}\right). \]
\(P(x), Q(x)\) are given polynomials of which the latter can be expressed as the product of real linear factors. Into what Partial Fractions can the function \(P(x)/Q(x)\) be decomposed? Prove your result and shew that such decomposition is possible in one way only.
\(x_1, x_2, y_1, y_2, z_1, z_2\) are given. Shew that the numbers \begin{align*} X &= \lambda x_1 + \mu x_2 \\ Y &= \lambda y_1 + \mu y_2 \\ Z &= \lambda z_1 + \mu z_2 \end{align*} satisfy for all values of \(\lambda, \mu\) a relation of the form \[ aX+bY+cZ = 0, \] where all of \(a, b, c\) are not zero. State and prove the converse proposition.