Find the following limits: $$\lim_{x \to 0} \frac{2\sin x - \sin 2x}{x^3}, \quad \lim_{x \to 0} x \sin\left(\frac{1}{x}\right), \quad \lim_{x \to 0} \frac{\cos ec x - \cot x}{x}, \quad \lim_{x \to 1} \frac{\cos \frac{1}{2}\pi x}{\log x}.$$
Evaluate the integrals: $$\int_0^1 \frac{\sin^{-1} x}{(1+x)^2} dx, \quad \int_0^a \frac{x dx}{x + \sqrt{a^2 - x^2}}, \quad \int_1^2 \frac{(x-1) dx}{\sqrt{x+1}} \cdot \frac{1}{x}, \quad \int_0^{\pi/4} \frac{dx}{\sec x + \cos x}.$$
Sketch roughly the possible forms of the curve given by the equation $$y(ax^2 + 2bx + c) = a'x^2 + 2b'x + c',$$ where \(a\), \(b\), \(c\), \(a'\), \(b'\), and \(c'\) are real, and \(a\) and \(a'\) are non-zero. Prove that a necessary condition for \(y\) to take every real value at least once as \(x\) takes all real values can be put in the form $$(ca' - ac')^2 \leqslant 4(ab' - ba')(bc' - cb').$$
Consider the curve given by the intrinsic equation \(s = c\sin\psi\) for values of \(\psi\) between \(-\frac{1}{2}\pi\) and \(+\frac{1}{2}\pi\), and taking the tangent and normal at the point from which \(s\) is measured as axes of \(x\) and \(y\), respectively, obtain expressions for \(x\) and \(y\) in terms of \(\psi\) as a parameter. With the same axes find the locus of the centre of curvature in terms of the same parameter. Discuss briefly the nature of the two curves if the restriction on the value of the parameter is disregarded.
A thin uniform lamina is in the form of a sector of a circle, of radius \(a\) and angle \(\frac{2\alpha}{3}\). Show that the mass-centre of the lamina is distant \(\frac{3a\sin\beta}{\beta}\) from the centre of the circle. A circular disc, of radius \(a\) and made of uniform sheet material, is cut along a radius. A quadrant of the circle is folded along a radius so as to lie in contact with the remainder. Find the distance of the mass-centre of the folded sheet from the centre of the circle.
A set of rectangular axes \(Ox\), \(Oy\) is taken in a given plane; a force \(R\) in the plane may be regarded as the resultant of two components \((X, Y)\) respectively parallel to \(Ox\) and \(Oy\). Prove that the moment of \(R\) about \(O\) is equal to the algebraic sum of the moments of \(X\) and \(Y\) about \(O\). A number of forces \((X_i, Y_i)\) act respectively at points \((x_i, y_i)\) of the plane, where \(i = 1, 2, 3, \ldots, n\). Show that, if the system reduces to a single force, the line of action of the force is $$x \sum Y_i - y \sum X_i + \sum (y_iX_i - x_iY_i) = 0.$$ Deduce that, if the algebraic sum of the moments of the forces of the system about each of three points in the plane vanishes, then in general the points must be collinear. Mention any exceptional cases.
A light rod is freely hinged at its lower end to a point on horizontal ground, and rests symmetrically across a uniform circular cylinder, of radius \(a\) and weight \(w\), lying on the ground with its axis horizontal; the rod and the end faces of the cylinder are perpendicular to the axis. The coefficients of friction at the points of contact of the cylinder with the rod and the ground are \(\mu_1\), \(\mu_2\) respectively. The rod makes an acute angle \(2\theta\) with the ground. If a weight \(W\) is hung from a point of the rod distant \(l\) from the hinge, find the conditions that slipping does not occur at either point of contact. Find also the conditions that slipping will never occur however much \(W\) is increased.
A rigid hoop, of radius \(a\), is made of thin smooth wire, and is fixed with its plane vertical. A small bead, of weight \(w\), is free to slide on the hoop and is joined to one end of a light elastic string, of modulus of elasticity \(\lambda\). The other end of the string is fastened to the highest point of the wire, and the unstretched length \(l\) of the string is less than \(2a\). Show that asymmetrical positions of equilibrium exist if \(2lw/\lambda < 2a - l\). If these asymmetrical positions exist, show that the equilibrium is stable.
To a man cycling on level ground with speed \(U\) in a direction due E, the wind appears to blow from a direction \(\alpha\) E. of N. When he cycles due S. with the same speed the corresponding apparent direction is \(\beta\) E. of N. Calculate the components of the velocity of the wind in the W. and S. directions.
A particle is moving in a straight line on a smooth horizontal plane. Its motion is opposed by a force proportional to the cube of the velocity. Show that: