Four suits of cards, each suit consisting of thirteen cards numbered from 1 to 13, are dealt to four persons. Find the chance that each person's cards contain all the numbers from 1 to 13.
Eliminate \(x, y, z\) from the equations \[ ax^2+by^2+cz^2 = ax+by+cz = yz+zx+xy=0 \] and reduce the result to a symmetrical form.
Prove that, if \(u_n = (\alpha+\beta)u_{n-1} - \alpha\beta u_{n-2}\) and \(u_2=\alpha\beta u_1\), then \[ \frac{u_n}{u_1} = \frac{\alpha\beta}{\beta-\alpha} \{\alpha^{n-2} - \alpha^{-1}\beta^{n-1} - \beta^{n-2} + \beta^{-1}\alpha^{n-1} \}. \]
If \(\theta=t^n e^{-(x^2+y^2)/4t}\), find what value of \(n\) will make \[ \frac{\partial^2\theta}{\partial x^2} + \frac{\partial^2\theta}{\partial y^2} = \frac{\partial\theta}{\partial t}. \]
By finding the fourth differential coefficient of \((\sin^2 x)/x^2\), or otherwise, shew that as \(x\) tends to zero the limit of \[ \frac{15}{x^5} - \frac{2x^4-18x^2+15}{x^6}\cos 2x + \frac{8x^2-24}{x^5}\sin 2x \] is \(\frac{4}{15}\).
Prove that the centre of the circle inscribed in the triangle formed by the external common tangents to the escribed circles of any triangle \(ABC\) is the point of intersection of the perpendiculars from the centres of the escribed circles upon the corresponding sides, and that the radius of this circle is equal to \(R(1+\cos A + \cos B + \cos C)\), where \(R\) is the radius of the circle circumscribing the triangle \(ABC\).
Any number of forces \(P_1, P_2, \dots, P_n\) in the same plane are in equilibrium. The direction of each is given, and also the magnitude of \(P_1\) and the ratios of the magnitudes of \(P_3, P_4, \dots, P_n\). Shew how to construct the magnitude of each force.
Three equal spheres rest in contact on a rough horizontal plane. An equal sphere of the same material is placed so as to rest symmetrically on them. Shew that, if the coefficient of friction \(\mu\) is \(>\sqrt{3}-\sqrt{2}\) and all surfaces are equally rough, equilibrium will be maintained.
Two equal particles \(A, B\) attached to the ends of a light string of length \(a\) are placed on a smooth horizontal table with the string \(AB\) perpendicular to the edge of the table and \(B\) hanging just over the edge. The system is released from rest in this position. Prove that when first the string is horizontal the distance of \(B\) from the vertical through the edge of the table is \(\frac{1}{4}a(\pi-2)\), and find the tension in the string.
A heavy particle is supported in equilibrium by two equal elastic strings with their other ends attached to two points in a horizontal plane and each inclined at an angle of \(60^\circ\) to the vertical. The modulus of elasticity is such that when the particle is suspended from any portion of the string its extension is equal to its natural length. The particle is displaced vertically a small distance and then released. Prove that the period of its small oscillations is \(2\pi\sqrt{2l/5g}\), where \(l\) is the stretched length of either string in equilibrium.