Problems

Show that \((ay-bx)^2-(bz-cy)(cz-az)\) is the product of two linear factors which are real if \(c^2 > 4ab\). If \(x+y+z+w=0\), prove that \[ wx(w+x)^2+yz(w-x)^2+wy(w+y)^2+zx(w-y)^2+wz(w+z)^2+xy(w-z)^2+4xyzw=0. \]

Prove that the arithmetic mean of \(n\) positive quantities is greater than their geometric mean. If \(s=a_1+a_2+...+a_n\) where all the quantities are positive, prove that \[ \frac{s}{s-a_1} + \frac{s}{s-a_2} + \dots + \frac{s}{s-a_n} \ge \frac{n^2}{n-1}. \]

Sum the series

1. [(i)] \(\displaystyle\sum_{r=1}^n (r+2)r!\).
2. [(ii)] \(\displaystyle\frac{1}{2^3.3!} + \frac{1.3}{2^4.4!} + \frac{1.3.5}{2^5.5!} + \dots\) to infinity.

Prove that if \(p_n/q_n\) is the \(n\)th convergent of \(\displaystyle\frac{a_1}{b_1+}\frac{a_2}{b_2+}\frac{a_3}{b_3+}\dots\), then \[ p_n = b_np_{n-1}+a_np_{n-2}. \] Find the value of \[ \frac{1}{1+}\frac{x}{1-x+}\frac{x}{2-x+}\dots\frac{x}{n+1-x}. \]

Find the \(n\)th differential coefficients of

1. [(i)] \((x+2)/(x^2-2x-3)\),
2. [(ii)] \(x^2\sin^3x\).

If \(y=(\sin^{-1}x)^2\), prove that \[ \lim_{x\to0} \left(\frac{d^{n+2}y}{dx^{n+2}} / \frac{d^ny}{dx^n}\right) = n^2. \]

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 \(3\pi a^2\).