Find the sum of the terms after the \(n\)th in the expansion of \((1+x)/(1-x)^2\) in ascending powers of \(x\). Prove that the ratio of this sum to the sum of the corresponding terms in the expansion of \((1-x)^{-2}\) can be made equal to any given number \(\lambda\), which is greater than \(2n\), by suitable choice of \(x\). Explain clearly why the restriction upon \(\lambda\) is necessary.
In any triangle \(ABC\) prove that the sum of the squares of the distances of the centre of the inscribed circle from the vertices is \(bc+ca+ab - 6abc/(a+b+c)\). Investigate the corresponding result for the sum of the squares of the distances of the centre of an escribed circle from the vertices.
Prove that the least value of \(a\cos\theta + b\sin\theta\) is the negative square root of \(a^2+b^2\). Prove also that the least value of \[ x^2 + 2x(a\cos\theta+b\sin\theta) + c\cos 2\theta + d\sin 2\theta \] is \[ -\frac{1}{2}(a^2+b^2) - \{c^2+d^2+\frac{1}{4}(a^2+b^2)^2+c(b^2-a^2)-2abd\}^{1/2}. \]
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.