Problems

Find the condition that \(ax+b/x\) can take any real value for real values of \(x\). Express \(\xi = (x-a)(x-b)/(x-c)(x-d)\) in terms of \(y\) where \(y=(x-d)/(x-c)\), and hence or otherwise shew that \(\xi\) can take all real values if \((c-a)(c-b)(d-a)(d-b)\) is negative.

Sum the series:

1. [(i)] \(2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \dots\) to \(n\) terms;
2. [(ii)] \(1^4+2^4+3^4+\dots\) to \(n\) terms.

Prove by induction that \(5s_4+7s_5 = 12s_2s_3\) where \(s_r\) denotes \(\sum_{k=1}^n k^r\).

Find the law of formation of successive convergents to the continued fraction \[ \frac{a_1}{b_1+} \frac{a_2}{b_2+} \frac{a_3}{b_3+} \dots. \] Prove that \(\frac{1}{1-} \frac{1}{4-} \frac{1}{1-} \frac{1}{4-} \dots\) to \(n\) quotients is \(\frac{2n}{n+1}\).

What is meant by the statement that the series \(u_1+u_2+u_3+\dots\) is convergent? Discuss the convergence of the series

1. [(i)] \(\sum_1^\infty \frac{1}{n^{1+a}}\),
2. [(ii)] \(\sum_1^\infty \frac{(-1)^n}{1+n^a}\).

Define the differential coefficient of a function of \(x\). Differentiate (i) \(x^x\), (ii) \(\cos^{-1}\left(\frac{a+b\sin x}{b+a\sin x}\right)\). If \(y^3+3x^2y+1=0\), prove that \((x^2+y^2)\frac{d^2y}{dx^2} + 2(x^2-y^4)\frac{dy}{dx} = 0\).

If \(y=(x+\sqrt{x^2+1})^n\), prove that \[ (x^2+1)\frac{d^2y}{dx^2}+x\frac{dy}{dx}-n^2y=0. \] Expand \(y\) in ascending powers of \(x\) and shew that the coefficient of \(x^{n+r}\) is zero, where \(r\) and \(n\) are positive integers.

Find the equation of the tangent at any point of the curve given by \[ x=f(t), \quad y=\phi(t). \] If \(lx+my+n=0\) is a tangent to the curve \(x=b/(t-1)^3, y=l/(t-1)^2\), prove that % Note: OCR'd y equation has l, seems like typo for 1 \[ 4l(l+n)^2+36lmn+27m^2n+4mn^2=0. \]

Prove the formulae for the radius of curvature of a plane curve \[ \frac{1}{\rho} = \frac{\frac{d^2y}{dx^2}}{\left(1+\left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}}}, \quad \rho = r\frac{dr}{dp} = \frac{r^3}{r^2-p\frac{d^2r}{d\theta^2}} \cdot \] % Note: The second formula for rho is unusual and likely contains OCR errors. Standard formula is rho = r dr/dp. The second part is non-standard. Prove that the radius of curvature at a point \((r,\theta)\) of the curve \(r^n=a^n\cos n\theta\) subtends an angle \(\tan^{-1}\left(\frac{1}{n}\tan n\theta\right)\) at the pole.

Evaluate:

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

Find the length and area of the loop of \(3x^2 = y(1-y)^2\).