Problems

Prove that the radius of curvature of the cardioid \(r=a(1+\cos\theta)\) at the point whose vectorial angle is \(\theta\), subtends an angle \(\tan^{-1}\left(2\tan\frac{\theta}{2}\right)\) at the origin.

Show that the fraction \(x/(x+1)(2x+1)\) can be expanded as a power series \[ a_1x+a_2x^2+\dots+a_nx^n+\dots, \] giving the values of \(x\) for which the expansion is valid and the value of \(a_n\). Find as single fractions the sums of the series

1. [(i)] \(2a_1x+3a_2x^2+\dots+(n+1)a_nx^n+\dots\),
2. [(ii)] \(a_1^2x+a_2^2x^2+\dots+a_n^2x^n+\dots\).

and

Trace carefully the curves

1. [(i)] \(y = \frac{x(x-1)}{2x-1}\),
2. [(ii)] \(y^2 = \frac{x(x-1)}{2x-1}\),

and show how inspection of the former would indicate that the latter consists of two distinct branches.

A point \(P\) is situated on the side \(BC\) of a triangle \(ABC\). The lengths \(PA, PB, PC\) are \(p+x, p+y, p+z\) ft. respectively, where \(x, y, z\) are small in comparison with \(p\): and the angle \(APB\) exceeds a right angle by \(\theta\) minutes where \(\theta\) is small. Find expressions for the lengths \(AB, AC\) which are linear in \(x, y, z\) and \(\theta\), and prove that the angle \(A\) of the triangle exceeds a right angle by \[ \frac{5400(y+z-2x)}{\pi p} \text{ minutes.} \]

Determine for what ranges of \(x\) the function \((\log x)/x\) (i) increases and (ii) decreases as \(x\) increases. Hence, or otherwise, prove the following theorems wherein \(n\) is a given positive number and only positive values of \(x\) are considered:

1. [(I)] The equation \(n^x=x\) has two roots, one root, or none according to the value of \(n\).
2. [(II)] The inequality \(n^x>x^n\) is a consequence of either \(x>n\) or \(x

State the critical values of \(n\).

Prove that through any point \(P\) on a hyperbola a circle can be described which cuts the hyperbola again at the angular points of an equilateral triangle. If the asymptotes are inclined at an angle \(\omega\), and are taken to be axes of coordinates, show that the locus of the centre of the circle as \(P\) varies is a hyperbola having an equation of the form \[ (2x \cos\omega - y)(2y \cos\omega - x) = c^2, \] where \(c\) is a constant. Discuss the case when the original hyperbola is rectangular.

Show that through any point in space one line can be drawn to meet each of two other lines which do not intersect each other. Show how to draw through three given non-intersecting lines three planes which have a common line and one of which passes through an arbitrary point. Prove that if all the lines are drawn which intersect three given lines the ranges of points in which they intersect those lines are such that the cross ratio of four points on one of the given lines is equal to the cross ratio of the corresponding four on each of the other given lines.

Show how to find the three pairs of lines joining the four points of intersection of two conics \(S=0, S'=0\). Show that the equation of any conic which has double contact with both conics can be written in either of the forms \[ S-kP^2=0, \] or \[ S'-k'Q^2=0, \] where \(P=0\) and \(Q=0\) are two lines which form a harmonic pencil with one of the pairs of lines joining the four points of intersection of \(S=0\) and \(S'=0\). Find the equations of the parabolas which have double contact with both \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \quad \text{and} \quad \frac{x^2}{b^2}+\frac{y^2}{a^2}=1. \]

Show that, if a point moves along any curve under the action of a force always at right angles to the direction of motion, the point moves with constant speed. A particle is attached to the end of a string which is partly wound round a post whose section is a regular polygon of \(r\) sides each of length \(a\). Initially a length \(l\), equal to an integral multiple of \(a\), is unwound and in a straight line with one of the sides. The particle is then projected at right angles to the string with velocity \(v\) so that the string winds in a horizontal plane round the post. Show that the time taken to wind up is \[ \frac{\pi l(l+a)}{rav}. \]

An aqueduct of cross section 2 sq. ft. delivers water with a velocity of 2 ft. per sec. at the top of a water wheel of 12 ft. diameter. The water leaves the wheel at a point 3 ft. below the centre. Calculate the horse-power developed. The compartments which catch the water being 40 in number, find the minimum capacity of each in order that all the water may be caught when the rim velocity of the wheel does not exceed that with which the water is delivered to it; calculate also the maximum turning couple which the wheel can exert when working at full power; examine the effect of friction at the axle, and discuss what happens if the load is greater than the couple calculated above.