Solve the equations:
Expand in ascending powers of \(x\) the fraction \[ \frac{2x + (9+3x^2)^{1/2}}{3-x} \] as far as the fifth power of \(x\), and shew that, for small values of \(x\) it leads to a good approximation for \(e^x\). Deduce that \(e^{1/4} = 1.2840\dots\).
Prove that, in any triangle, \(a \cot A = b \operatorname{cosec} C - a \cot C\). If \(a=19.1, b=15.1, \tan C = 3\), find \(\tan A\) and \(\tan B\).
A figure of four triangles and three squares is constructed by describing squares P, Q, R externally on the three sides of a triangle and joining their adjacent corners. P contains 5 sq. cm., Q 10 sq. cm. and the areas of the triangles are together half of that of the squares. Prove that the square R contains either 5 or 13 sq. cm.
Three parallel chords of a circle, AL, BM, CN are drawn. Shew that the perpendiculars from L on BC, M on CA, N on AB intersect on the circle.
Find rational expressions for the focal distances of a point \(x,y\) on the hyperbola \(2xy=c^2\).
Prove that the chord of the ellipse \(x^2/a^2+y^2/b^2=1\) which is bisected at right angles by \(lx+my=1\) has for its equation \[ (a^2-b^2)(x/l - y/m) = a^4/l^2 + b^4/m^2. \]
Prove that, if \(y^3+3x^2+cx^3=0\), \(y^5 y'' + 2x^2 = 0\).
Perform the integrations \[ \int \frac{dx}{(x+1)^3(x-1)}; \quad \int \frac{dx}{\{(x+1)^3(x-1)\}^{1/2}}; \quad \int \sec x dx. \]
Shew that, in addition to the nine-point circle of a triangle, there are four circles which touch the escribed circles of the triangle and that these are the inverses of the sides of the triangle and of the nine-point circle with regard to the circle which cuts the escribed circles orthogonally.