Problems

A series of pairs of quantities \(p_1, q_1; p_2, q_2; \dots\) are formed according to the law \[ p_n = p_{n-1}+q_{n-1}, \quad q_n=p_{n-1}. \] Find the limiting value of \(p_n/q_n\) as \(n \to \infty\).

Shew that two quadratic expressions \(ax^2+2bx+c\) and \(a'x^2+2b'x+c'\) can generally be expressed in the forms \(p(x-\alpha)^2+q(x-\beta)^2\) and \(r(x-\alpha)^2+s(x-\beta)^2\) respectively. Find a function \(\frac{ax^2+2bx+c}{a'x^2+2b'x+c'}\) which has turning values 3 and 4 when \(x=2\) and \(-2\) respectively, and has the value 6 when \(x=0\).

Determine graphically or otherwise for what values of \(\lambda\) the equation \(2x^3-15x^2+24x-\lambda=0\) has three real roots.

Find from the definition the differential coefficients of

1. [(1)] \(x^n\) where \(n\) is a rational number.
2. [(2)] \(\sin^{-1}x\).

Explain why there are two values in the latter case. Shew that when \(x=0\) the \(n\)th differential coefficient of \((\text{vers}^{-1}x)^2\) is \[ 2(n-1)!/(3\cdot5\cdot7 \dots 2n-1). \]

Find the maximum and minimum values of \[ y=(x-1)^2(x-2)^3(x-3) \] and draw a rough graph of the curve.

Prove that the radius of curvature of a curve is given by the formula \(\rho = r \frac{dr}{dp}\), and that the radius of curvature at the corresponding point of the inverse curve is \(k^2\rho/(2p\rho-r^2)\), where \(k^2\) is the constant of inversion. Deduce that a circle of radius \(a\) whose centre is at a distance \(c\) from the centre of inversion inverts into a circle of radius \(k^2a/(a^2-c^2)\).

Integrate

1. [(1)] \(\int \frac{dx}{x^4-1}\),
2. [(2)] \(\int_0^{\pi} \frac{dx}{a^2\cos^2x+b^2\sin^2x}\),
3. [(3)] \(\int \frac{dx}{(x+1)\sqrt{x^2-1}}\).

Find a formula of reduction for \[ \int x^n \cos ax dx. \]

Trace the curve \(r=a(2\cos\theta-1)\), find the areas of its loops and shew that their sum is \(3\pi a^2\).

Find the necessary and sufficient conditions of equilibrium of a system of coplanar forces. Forces \(P, Q, R\) act along the lines \(x=0, y=0\), and \(x\cos\theta+y\sin\theta=p\). Find the magnitude of the resultant and the equation of its line of action. (The axes of coordinates are rectangular.)

Shew how to obtain the resultant of a system of parallel forces, and establish the existence of their `centre.' Parallel forces of 1, 2, 3 lbs. weight act at the corners \(A, C, E\) of a regular hexagon \(ABCDEF\), and forces 4, 5, 6 lbs. weight act at the corners \(B, D, F\) in a direction parallel to the first three but in the opposite sense. Find the point of application of their resultant.