Problems

A particle oscillates on a smooth cycloid from rest at a cusp, the axis being vertical and the vertex downwards. Shew that

1. [(i)] the hodograph is a pair of equal circles touching each other.
2. [(ii)] the resultant acceleration of the particle is equal to \(g\).
3. [(iii)] the pressure on the curve is \(2mg \cos\psi\), where \(\psi\) is the inclination of the tangent to the horizontal.
4. [(iv)] the projection of the particle on the axis has a simple harmonic motion.

Eliminate \(x, y, x', y'\) from the following equations: \[ \frac{x}{a} + \frac{y}{b} = 1, \quad \frac{x'}{a'} + \frac{y'}{b'} = 1, \quad x^2+y^2=c^2, \quad x'^2+y'^2=c'^2, \quad xy' - x'y=0. \]

Shew graphically the change in the value of the function \[ (x-a)(x-b)/(x-c)(x-d), \] as \(x\) changes from \(-\infty\) to \(+\infty\), where \(a, b, c, d\) are real numbers such that

1. [(i)] \(a>b>c>d\),
2. [(ii)] \(a>c>b>d\),
3. [(iii)] \(a>c>d>b\).

Find rationalising factors for the expressions

1. \(x^{2/3} + x^{1/3} + 1\),
2. \(ax^{2/3} + bx^{1/3} + c\);

and also find the results of rationalisation.

Find the general term of the recurring series whose scale of relation is \[ u_n - u_{n-1} - 5u_{n-2} - 3u_{n-3} = 0, \] and whose first three terms are 2, 5, 20.

Define a convergent series. State and prove the theorem used in discussing the convergency of such series as \[ \frac{1}{1} - \frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \dots \] and \[ 2 - \frac{5}{4} + \frac{8}{7} - \frac{11}{10} + \frac{14}{13} - \dots. \] Prove that the second series can be made convergent by bracketing the terms in pairs.

Find from the definition the differential coefficient of \(\sin x\), establishing the limiting value required in the proof. By calculating successive differential coefficients, or otherwise, shew that if \(0 < x < \frac{\pi}{2}\), then \[ 2x + x \cos x - 3 \sin x > 0. \]

Having given that \[ x(1-x)\frac{d^2y}{dx^2} - (3-10x)\frac{dy}{dx} - 24y = 0, \] prove that \[ x(1-x)\frac{d^{n+2}y}{dx^{n+2}} + \{n-3-2(n-5)x\}\frac{d^{n+1}y}{dx^{n+1}} - (n-3)(n-8)\frac{d^ny}{dx^n} = 0. \] Hence find, by Maclaurin's Theorem, that value of \(y\) which is zero when \(x=0\), and is such that its fourth differential coefficient is unity when \(x=0\).

Define the curvature of a plane curve, and deduce the expression \[ \pm \frac{d^2y/dx^2}{\{1+(dy/dx)^2\}^{3/2}}, \] for the curvature at a point on the curve \(y=f(x)\). Prove that the centre of curvature at the point \((x, y)\) on the curve \(ay=x^2\) is the point \((-4x^3/a^2, 3y+a/2)\). Find the coordinates of the points where the locus of centres of curvature cuts the original curve, and shew that at these points the curvature of the locus of centres of curvature is \(\sqrt{6}/27a\).

Evaluate \[ \int x \sin^{-1} x \, dx, \quad \int \frac{3x^2+x-1}{(x^2+1)(x+1)^2} \, dx, \quad \int \frac{cx+f}{(ax^2+2bx+c)^3} \, dx. \] Prove that, if \(a\) and \(b\) are positive, then \[ \int_0^\pi \frac{\sin^2 x \, dx}{a^2+b^2-2ab\cos x} = \frac{\pi}{2a^2} \text{ or } \frac{\pi}{2b^2}, \] according as \(a >\) or \(< b\).