Problems

Three equal smooth balls \(A, B, C\) are placed in order on a smooth floor with their centres in a line perpendicular to a smooth wall which is perfectly elastic, the centre of the ball \(A\) being at a distance from the wall which is small in comparison with the distance of the centre of \(B\). If the ball \(C\) is projected towards the wall, prove that \(A\) comes to rest temporarily after two collisions with \(B\), independently of the coefficient of elasticity between a pair of balls. Prove further that, if the coefficient of elasticity is nearly unity, \(A\) comes permanently to rest after its fourth collision with \(B\).

Solve the equations:

1. [(i)] \(\sqrt{x^2+12y} + \sqrt{y^2+12x} = 33, \quad x+y=23\);
2. [(ii)] \(x^2+2yz=-11, \quad y^2+2zx=-2, \quad z^2+2xy=13\).

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.