Two small rings of weights \(w\) and \(kw\) can slide along a rough wire in the form of a circle of radius \(a\) in a vertical plane, and are joined by a light string of length \(2a \sin\alpha\). The system is in equilibrium with the string straight, horizontal and above the centre of the circle. If \(k>1\), show that \(\tan^{-1} \frac{T}{kw} \ge \alpha-\lambda\) and \(\tan^{-1}\frac{T}{w} \le \alpha+\lambda\), where \(T\) is the tension of the string and \(\lambda\) is the angle of friction. What conditions must be satisfied by \(k, \alpha\) and \(\lambda\) if equilibrium is possible?
If \(P(x), Q(x)\) are polynomials in \(x\) with a highest common factor \(H(x)\), shew that polynomials \(A(x), B(x)\) can be found such that \(AP+BQ=H\) identically. Hence shew that \(F(x)/G(x)\), where the degree of \(F\) is less than that of \(G\), may be expressed as the sum of real partial fractions. Express in partial fractions \[ \frac{x}{(x+2)(x-1)^n}, \] where \(n\) is a positive integer.
The inscribed circle of the pedal triangle \(DEF\) of a triangle \(ABC\) touches the sides \(EF, FD, DE\) at \(A', B', C'\), respectively. Calculate the angles and sides of the triangle \(A'B'C'\) in terms of those of \(ABC\), (i) when \(ABC\) is acute-angled, (ii) when \(ABC\) has an obtuse angle at \(A\), expressing your results as symmetrically as you can.
If \(A, B, C, D\) are four points on a conic, prove that the points of intersection of \(AB, CD\), of \(AC, BD\) and of \(AD, BC\) form a triangle self-conjugate with respect to the conic. A variable rectangular hyperbola circumscribes a fixed triangle \(ABC\) whose pedal triangle is \(DEF\), and \(P\) is one of its intersections with a fixed straight line \(l\). If \(PD\) meets the hyperbola again in \(Q\), prove that the locus of \(Q\) is a straight line concurrent with \(EF\) and \(l\).
Two uniform discs can rotate in the same vertical plane about their centres. The centre of one disc, which is of radius \(a\), is at a height \(3a\) vertically above the centre of the other disc, which is of radius \(2a\), the edges of the discs being in contact. A particle of weight \(w\) is attached to the upper disc at its highest point, and a particle of weight \(W\) is attached to the lower disc at its lowest point. Assuming that the discs can rotate freely without slipping, prove that this position of equilibrium is stable if \(2w < W\) and unstable if \(2w > W\). Discuss the stability when \(2w=W\).
Prove that the equation whose roots are the cubes of the roots \(x_1, x_2, \dots, x_n\) of the equation \(a_0+a_1x+\dots+a_nx^n=0\) (\(a_n \ne 0\)) is \[ X^3 + xY^3 + x^2Z^3 - 3xXYZ = 0, \] where \begin{align*} X &= a_0 + a_3x + a_6x^2 + \dots, \\ Y &= a_1 + a_4x + a_7x^2 + \dots, \\ Z &= a_2 + a_5x + a_8x^2 + \dots, \end{align*} the sums continuing as long as the suffixes of the \(a\)'s do not exceed \(n\). Prove also that, if \(a_0 \ne 0\), \[ \prod_{r=1}^n (x_r + 1 + x_r^{-1}) = \frac{(-1)^n}{a_0 a_n} (A^2+B^2+C^2-BC-CA-AB), \] where \(A, B, C\) are the values of \(X, Y, Z\) for \(x=1\).
Find for what values of \(k\) the equation \[ \sin x \sin 3x = k \] has real solutions in \(x\). Find all real solutions, if any, in the following cases:
Prove that the tangent to a parabola at a point \(P\) bisects the angle between the focal distance \(SP\) and the perpendicular from \(P\) to the directrix. \(P, P'\) are two points on a parabola. \(SP\) meets the diameter through \(P'\) at \(Q\), and \(SP'\) meets the diameter through \(P\) at \(Q'\). Prove that a circle can be drawn to touch the lines \(PQ, QP', P'Q', Q'P\).
On a certain day when the speed and direction of the wind remain steady, it is found that when an aircraft travels at a constant air-speed with its nose pointing due North its track over the ground is 15\(^\circ\) East of North, and that when it travels with the same air-speed with its nose pointing 60\(^\circ\) East of North its track over the ground is 80\(^\circ\) East of North. By making a scale drawing, find the direction of the wind.
(i) The points \(z_r = x_r + i y_r\) (\(r=1,2,3\)) in an Argand diagram are the vertices \(A_1, A_2, A_3\) of a triangle. Find the complex coordinates of the vertices of a triangle \(B_1B_2B_3\) similar to \(A_1A_2A_3\) but of twice the linear dimensions, with \(B_1\) at the point \(z_0\) and with the sides making angles of 30\(^\circ\) (measured in the positive sense) with the corresponding sides of \(A_1A_2A_3\). (ii) Find expressions for the roots of the equation \(z^5 = (1-z)^5\). Indicate the position of these roots on an Argand diagram.