Problems

Find the equation of the tangent at a point on the curve \(f(x,y)=0\). If the tangent at \(P\) on \(y^3=3ax^2-x^3\) meets the curve again at \(Q\), prove that \[ \tan QOx + 2\tan POx = 0, \] \(O\) being the origin. Also show that if the tangent at \(P\) is a normal at \(Q\), then \(P\) lies on \[ 4y(3a-x)=(2a-x)(16a-5x). \]

Find the asymptotes of the curve \[ 2x(y-3)^2 = 3y(x-1)^2 \] and trace the curve.

Prove that if \(\phi\) is the angle the radius vector of a plane curve makes with the tangent \[ \frac{dr}{ds} = \cos\phi, \quad r\frac{d\theta}{ds} = \sin\phi, \quad \frac{d^2r}{ds^2} = \frac{\sin^2\phi}{r} - \frac{\sin\phi}{\rho} \] where \(\rho\) is the radius of curvature. If the tangent at \(P\) to this curve is produced to \(P'\) at a distance from \(P\) equal to \(OP\), where \(O\) is the origin, prove that the angle \(\phi'\) between \(OP'\) and the tangent to the locus of \(P'\) is \(\tan^{-1}\frac{\rho r^2}{2r^3-\rho r^2}\), where \(\rho\) is the radius of curvature of the given curve at \(P\) and \(r'=OP'\).

Integrate

1. [(i)] \(\displaystyle\int \frac{\sqrt{x-1}}{x\sqrt{x+1}}\,dx\),
2. [(ii)] \(\displaystyle\int_0^1 \frac{dx}{(1+x)(2+x)\sqrt{x(1-x)}}\),
3. [(iii)] \(\displaystyle\int \frac{\sin x\,dx}{4\cos x+3\sin x}\),
4. [(iv)] \(\displaystyle\int \frac{\sqrt{a^2+b^2\cos^2x}}{\cos x}\,dx\).

Trace \(r=a(2\cos\theta-1)\), find the areas of its loops and show that their sum is \(3\pi a^2\).

(i) Solve the equations \[ x^2 + 2xy^2 + 2y^4 = 1, \quad \frac{1}{x^2} - \frac{2}{xy^2} + \frac{2}{y^4} = 1. \] (ii) If \[ \frac{a}{x-md} + \frac{b}{x-mc} + \frac{c}{x+mb} + \frac{d}{x+ma} = 0, \] and \(a+b+c+d=0\), prove that the only finite value of \(x\) is \(\displaystyle\frac{m(ac+bd)}{a+b}\).

Sum to \(n\) terms the series

1. [(i)] \(\displaystyle\frac{x}{(1+x)(1+ax)} + \frac{ax}{(1+ax)(1+a^2x)} + \frac{a^2x}{(1+a^2x)(1+a^3x)} + \dots\)
2. [(ii)] \(\displaystyle\frac{1}{4!} + \frac{2\cdot 3}{5!} + \frac{3\cdot 3^2}{6!} + \frac{4\cdot 3^3}{7!} + \dots\)

If \(a_r\) is the coefficient of \(x^r\) in the expansion of \((1+x+x^2)^n\) in a series of ascending powers of \(x\), prove that

1. [(i)] \(a_0+a_1+a_2+\dots+a_{2n}=3^n\),
2. [(ii)] \(a_1+2a_2+3a_3+\dots+2na_{2n}=n\cdot3^{n-1}\),
3. [(iii)] \(a_0^2-a_1^2+a_2^2-\dots+a_{2n}^2=a_n\).

If £\(P\) is the present value of an annuity of £\(A\), to continue for \(n\) years, at \(100r\) per cent. per annum compound interest, prove that \[ \frac{Pr}{A} = 1-(1+r)^{-n}. \] If £\(Q\) is the present value of an annuity of £1 on the life of a man, shew that in order to receive £\(R\) at his death the payment to be made immediately and repeated annually is \[ £\frac{R(1-Qr)}{Q(1+r)}. \]

Find the number of combinations of \(m\) unlike things \(r\) at a time. Prove that the number of combinations \(n\) at a time of \(2n\) things, of which \(n\) are alike and the rest all different, is \(2^n\).