If \(a_1\) and \(a_2\) are positive numbers and if \(p\) is a positive integer, shew that \[ 2^p(a_1^p+a_2^p) \ge 2(a_1+a_2)^p. \] Shew also that if \(a_1, a_2, \dots, a_n\) are \(n\) positive numbers, their arithmetic mean is not less than their geometric mean, i.e. that \[ (a_1+a_2+\dots+a_n) \ge n(a_1a_2\dots a_n)^{1/n}. \]
Prove that the polars of a point with respect to a system of confocal conics envelop a parabola, and find the directrix of the parabola.
Two tangents to a parabola inclined at an angle \(\alpha\) are taken as Cartesian axes \(Ox, Oy\). Shew that the equation to the parabola can be written \[ (x/a)^{\frac{1}{2}} + (y/b)^{\frac{1}{2}} = 1, \] and that the focus \((h,k)\) of the parabola satisfies the equations \[ h^2+2hk\cos\alpha+k^2=ah=bk. \]
If \[ f_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}), \] prove that \[ x\frac{d^2f_n(x)}{dx^2} + (1-x)\frac{df_n(x)}{dx} + n f_n(x) = 0. \]
Evaluate \[ \int_0^\infty \frac{dx}{(x+1)\sqrt{(5x^2+12x+8)}}. \]
A circle of radius \(r\) is rotated through 180\(^\circ\) about an axis which lies in the plane of the circle and is at a perpendicular distance \(a(>r)\) from the centre of the circle. Find the mass-centre of a uniform solid occupying the volume generated by the circle, and explain why it does not coincide with the mass-centre of a uniform semi-circular wire occupying the path traced out by the centre of the circle.
A smooth rectangular plank of mass \(M\) fits accurately in a smooth horizontal groove along which it is free to slide. The plank has a deep rectangular groove with vertical sides cut in its upper face, the edges of the groove being parallel to the edges of the plank, and in this groove a ball of mass \(m\) fits closely and runs freely. With the plank at rest the ball is projected with velocity \(V\) along the groove. The coefficient of restitution between the ball and the ends of the groove is \(e\). Find an expression for the velocities of the plank and the ball just after the \(n\)th impact. Shew that the velocities of the plank and the ball both tend to the same limit.
The horizontal and inclined faces of a wedge of mass \(M\) meet at an angle \(\alpha\) in a line \(AB\). The smooth horizontal face has a smooth groove cut in it perpendicular to \(AB\), and the wedge rests on a smooth table, the groove running on a smooth rail fixed to the table so that any motion of the wedge is perpendicular to \(AB\). A particle of mass \(m\) slides down the rough inclined face of the wedge. Shew that, if the system starts from rest, the time taken by the particle to describe a given distance down the wedge bears to the time taken when the wedge is fixed the ratio \[ \left\{1 - \frac{m \cos\alpha(\cos\alpha+\mu\sin\alpha)}{M+m}\right\}^{\frac{1}{2}}:1, \] where \(\mu\) is the coefficient of friction between particle and wedge.
A particle placed close to the vertex of a smooth cycloid whose axis is vertical and vertex upward is allowed to run down the curve under gravity. Shew that it leaves the curve when it is moving in a direction making an angle of 45\(^\circ\) with the horizontal.
A uniform rod \(AB\) of mass \(M\) and length \(l\) hangs vertically down from a smooth hinge \(A\). When the rod is at rest an impulse is applied in a horizontal direction to a point \(C\) of the rod. Determine the position of \(C\) in order that there may be no impulsive reaction at \(A\), and then determine the magnitude of the impulse required in order that the rod may just reach the upward vertical position.