Problems

If \(a\) and \(b\) are real numbers, show that the equation \[ x^4 + ax^3 + (b-2)x^2 + ax + 1 = 0 \] has four real roots if and only if one of the following two sets of conditions is satisfied:

1. \(b \le 0, a^2 < 16b^2\);
2. \(b \ge 8, 4(b-4) < a^2 < \frac{1}{4}b^2\).

The parameters of three points \(P_1, P_2, P_3\) on the conic \[ x:y:z = \theta^2:\theta:1 \] are the roots of the equation \[ a_0\theta^3+3a_1\theta^2+3a_2\theta+a_3=0. \] The line joining \(P_1\) to the pole of \(P_2P_3\) meets the conic in \(P_1'\), and \(P_2', P_3'\) are similarly defined. Prove that the three pairs of points \(P_1, P_1'\); \(P_2, P_2'\); \(P_3, P_3'\) belong to an involution on the conic, the parameters of whose double points are the roots of \[ (a_0a_2-a_1^2)\theta^2 + (a_0a_3-a_1a_2)\theta + (a_1a_3-a_2^2) = 0. \]

A pack contains an even number of cards \(s\). Two piles A and B of \(p\) cards each (\(0 \le p \le \frac{1}{2}s\)) are dealt out and an arbitrary number of the remaining cards is then dealt into a third pile C. If the order within any pile is disregarded, what is the total number of different possible deals? Compare the number so found with the number of different deals possible when two piles of \(p\) cards each are formed from similar packs of \(s\) cards each. Hence, or otherwise, deduce the relation \[ \sum_{p=0}^{\lfloor s/2 \rfloor} 2^{s-2p} {}_sC_p {}_{s-p}C_p = {}_{2s}C_s. \]

The quantity \(x\) (\(0 < x < 1\)) is determined by the equation \[ \cot(\lambda\sqrt{1-x}) = -\sqrt{\frac{x}{1-x}}, \] \(\lambda\) being positive. Discuss how the number of possible values of \(x\) depends on \(\lambda\) and show that if \(\lambda\) is such that only one value of \(x\) satisfies the equation, and if that value is small compared to unity, then approximately \[ \lambda^2 = \frac{\pi^2}{4} + \pi\sqrt{x} + \left(1+\frac{\pi^2}{4}\right)x. \]

Assuming that a function \(f(x)\) satisfies the relation \[ f''(x) = \frac{n(n-1)}{x^2}f - f', \] and taking \[ g(x) = x^n \frac{d}{dx} \left( \frac{f(x)}{x^n} \right), \] find an expression for \(g''(x)\) in terms of \(g\). Hence, show how to calculate the sequence of functions \(F_n(x)\) in which \(F_0(0)=0\), \(F_0'(0)=1\), and \[ F_n''(x) - \frac{n(n+1)}{x^2}F_n(x) + F_n(x) = 0, \] and evaluate \(F_2(x)\) explicitly.

A wedge is cut from a uniform solid circular cylinder by a plane which makes an angle \(\alpha\) with the base of the cylinder and which touches at a point \(O\) the circular boundary of the base. Prove that the mass-centre of the wedge is at a distance \(\frac{3}{8}(4+\tan^2\alpha)a\) from \(O\) where \(a\) is the radius of the cylinder, and explain why this distance does not tend to \(a\) as \(\alpha \to 0\).

Forces \((X_r, Y_r)\), \(r=1,2,\dots,n\), act on a rigid body at the points \((x_r, y_r)\) referred to rectangular Cartesian axes. Prove that the forces are equivalent to a force \((X, Y)\) acting at the origin together with a couple of moment \(M\). If \(X^2+Y^2 \ne 0\) find the equation of the line of action of the resultant of these forces, and also find the equation of the line of action of the resultant if the line of action of each force is rotated (from \(Ox\) to \(Oy\)) through the angle \(\alpha\), the points of application of the forces being unchanged. Show that the two lines of action intersect in the point \[ \left( \frac{MY+M'X}{X^2+Y^2}, \frac{MX-M'Y}{X^2+Y^2} \right), \] where \[ M' = \sum_{r=1}^n (x_rX_r + y_rY_r). \]

A particle of mass \(m\) is projected vertically upwards under gravity with initial velocity \(V\tan\alpha\), and the resistance of the air is assumed to be of magnitude \(mg(v/V)^2\) when the velocity of the particle is \(v\). Show that the particle returns to the point of projection with velocity \(V\sin\alpha\). If \(\tan\alpha\) is small, prove that the height attained by the particle is less than it would be if there were no air resistance by \(\frac{1}{3}V^2g^{-1}\tan^4\alpha\), approximately.

The velocity of the mass-centre of two particles of masses \(m_1, m_2\) moving in a plane is \(V\) and their relative velocity is \(V'\); prove that the total kinetic energy of the two particles is \(\frac{1}{2}MV^2 + \frac{1}{2}M'V'^2\), where \(M=m_1+m_2\) and \(MM'=m_1m_2\). If a shell of mass \(M\) travelling with velocity \(V\) is broken into two fragments of masses \(m_1, m_2\) by an explosion which increases the kinetic energy by an amount \(E\), find the smallest possible relative velocity of the fragments and find the ratio \(m_1/m_2\) in this case.

Two particles of masses \(m, m'\) are attached to the middle point \(A\) and to the end point \(A'\) of a light inextensible string \(OAA'\) of length \(2l\). The end \(O\) is fixed and the system executes a small oscillation under gravity in a vertical plane through \(O\). If \(x, x'\) are the horizontal distances of the particles from the vertical line through \(O\) at time \(t\), and \(n^2=g/l\), prove that \begin{align*} m\frac{d^2x}{dt^2} + (m+2m')n^2x - m'n^2x' &= 0, \\ \frac{d^2x'}{dt^2} + n^2x' - n^2x &= 0, \end{align*} and hence show that if \(m=3m'\), a motion is possible in which \(x+x'=0\).