Two particles \(A, B\), of the same weight, are joined by a light inextensible string, and placed on a rough horizontal table with the string taut. The coefficient of friction between each particle and the table is \(\mu\). A horizontal force is applied to \(B\) in a direction making an acute angle \(\theta\) with \(AB\) produced, and the magnitude of the force is gradually increased until equilibrium is disturbed. If the initial displacement of \(B\) is in a direction making an angle \(\phi\) with \(AB\) produced, prove that \(\phi=90^\circ\) when \(\theta\ge 45^\circ\), and that \(\phi=2\theta\) when \(\theta < 45^\circ\).
Give an account of the application of determinants to the solution of linear algebraic equations. Solve completely the equations \begin{align*} x+y+z &= a+b+c, \\ a^2x+b^2y+c^2z &= a^3+b^3+c^3, \\ a^3x+b^3y+c^3z &= a^4+b^4+c^4, \end{align*} distinguishing between the cases (i) \(bc+ca+ab \neq 0\), (ii) \(bc+ca+ab=0\), it being assumed that no two of the numbers \(a, b, c\) are equal.
Prove that, if the fraction \(p/q\) is in its lowest terms, there are exactly \(q\) different values of the expression \((\cos \theta + i \sin \theta)^{p/q}\). Prove that the equation whose roots are \(\tan (4r+1)\dfrac{\pi}{20}\), \((r=0,1,2,3,4)\), is \[x^5 - 5x^4 - 10x^3 + 10x^2 + 5x - 1 = 0.\]
Show that the polars of a fixed point \(P\) with respect to the conics through four given points \(A, B, C, D\) are concurrent. For what particular positions of \(P\) do these polars all coincide? Dualize the above result, and hence or otherwise show that the mid-points of pairs of opposite vertices of a complete quadrilateral are collinear.
A smooth wire bent into the form of a circle of radius \(a\) is fixed in a vertical plane. One end of a light elastic string of modulus \(\lambda\) and natural length \(a\) is attached to the highest point of the wire, and the other end to a bead of weight \(W\) that can slide along the wire. Show that, when the bead rests at the lowest point of the wire, the equilibrium is stable if \(\lambda < 2W\), and unstable if \(\lambda > 2W\). Investigate the stability if \(\lambda = 2W\).
Prove that, if \(f(u,v)\) is a homogeneous polynomial in \(u\) and \(v\) of degree \((n-1)\), \[ \frac{f(\sin x, \cos x)}{\sin(x-\alpha_1)\dots\sin(x-\alpha_n)} = \sum_{r=1}^n \frac{f(\sin\alpha_r, \cos\alpha_r)}{\sin(\alpha_r-\alpha_1)\dots\sin(\alpha_r-\alpha_n)} \frac{1}{\sin(x-\alpha_r)}, \] (where the factor \(\sin(\alpha_r - \alpha_r)\) is omitted from the denominator of the \(r\)th term). Prove that, if \(n\) is odd, \[ \frac{1}{\sin(x-\alpha_1)\dots\sin(x-\alpha_n)} = \sum_{r=1}^n \frac{A_r}{\sin(x-\alpha_r)}, \] and find the coefficients \(A_r\). Show what modification is required if \(n\) is even.
Explain what you understand by a convergent series. Investigate for what ranges of values of \(x\) the following series are convergent:
Show that the locus of a point \(P\) in space whose distances from three fixed points \(A, B, C\) are in given ratios is a circle whose centre lies in the plane \(ABC\) and whose plane is perpendicular to the plane \(ABC\).
Illustrate the use of the principle of virtual work by solving the following problem. A smooth cone of semi-vertical angle \(\beta\) is fixed with its axis vertical and its vertex upwards. A uniform inextensible string of length \(2\pi a\) and weight \(W\) is placed over the cone and rests in equilibrium in a horizontal circle. Show that the tension in the string is \(\dfrac{W}{2\pi\tan\beta}\) and that the reaction between the string and the cone is \(\dfrac{W}{2\pi a \sin\beta}\) per unit length of the string.
Starting from any definition of the logarithmic function \(\log x\) that you please, give an account of its leading properties. Include a proof that, as \(x \to \infty\), \(\dfrac{\log x}{x^k} \to 0\) for any positive constant \(k\).