Problems

Discuss completely the convergence of the logarithmic series for different real values of the variable, and prove that, if \(n>1\), \[ \frac{1}{n} + \frac{1}{2n(n-1)} > \log\left(\frac{n}{n-1}\right) > \frac{2}{2n-1}. \]

Comment on the following statements:

1. [(i)] If \(\theta\) is small, \(\sin\theta\) is approximately equal to \(\theta\).
2. [(ii)] The ``sum to infinity'' of a series is the sum of \(n\) terms when \(n\) becomes infinite.
3. [(iii)] An equation of the \(n\)th degree has \(n\) roots.
4. [(iv)] \(8^{\frac{1}{3}} = 2\).

Explain how it is that any meaning is assigned to such an expression as \(8^{\frac{1}{3}}\).

The internal bisectors of the angles of the triangle \(ABC\) (with sides \(a,b,c\) and area \(\Delta\)) meet the opposite sides in \(D, E, F\). Shew that the area of the triangle \(DEF\) is \[ 2\Delta abc / (b+c)(c+a)(a+b). \] Hence, or otherwise, shew that this triangle can never occupy more than a quarter of the original triangle, and that this occurs only when the original triangle is equilateral.

In walking a mile up the line of greatest slope of an inclined plane a man finds that he has risen 300 feet. He then turns to the left and walks on in a straight line for a second mile, rising a further 250 feet. If the line of greatest slope runs due North, find (i) the true bearing of the direction of his second mile, and (ii) the true bearing of his starting point as viewed from the point he finally reaches.

Find the equation determining the values of \(x\) for which \(\dfrac{\sin mx}{\sin x}\) is stationary. Hence, or otherwise, shew that, if \(m\) is an integer, \(\dfrac{\sin^2 mx}{\sin^2 x}\) never exceeds \(m^2\).

Two curves through the point \((x,y)\) are said to have contact of the \(n\)th order there if they have the same values of \(y', y'', \dots y^{(n)}\) at the point. Shew that, in general, a circle can be found to have contact of the second order with a given curve \(y=f(x)\) at a given point \((x,y)\), but not of higher order. Find the centre and radius of such a circle. Hence, or otherwise, find the evolute of the curve \(y^2=x^3\).

Determine the following:

1. [(i)] \(\dfrac{d}{dx} \{(\log x)^{\log x}\}\),
2. [(ii)] \(\dfrac{d^n}{dx^n} (x^2 \cos^2 x)\),
3. [(iii)] \(\displaystyle\int \frac{dx}{(x-1)\sqrt{(x^2+x-2)}}\).

Find a reduction formula for \(\int \tan^n x dx\) and so determine \(\int \tan^5 x dx\).

Give a rough sketch of the curve \[ 3x^2 = y(y-1)^2, \] and determine the greatest breadth of the loop, its perimeter and its area.

Find all the real solutions of the equations:

1. [(i)] \(x(x^2+y^2)=6y, \quad y(x^2-y^2)=x\).
2. [(ii)] \(\sqrt{(x^2+y^2)} + \sqrt{(x^2-y^2)}=2y, \quad x^4-y^4=a^4\).

If \[ (x+a_1)(x+a_2)(x+a_3)\dots\dots(x+a_n) = x^n + p_1 x^{n-1} + p_2 x^{n-2} + \dots\dots + p_n, \] prove that \[ a_1^3+a_2^3+a_3^3+\dots\dots+a_n^3 = p_1^3 - 3p_1p_2 + 3p_3. \] Prove also that the sum of the products \(r\) at a time of the \(n-1\) quantities \(a_2, a_3, \dots a_n\) is equal to \[ p_r - a_1 p_{r-1} + a_1^2 p_{r-2} - \dots\dots + (-a_1)^r. \]