The roots of the equation \(x^3 + px - q = 0\) are \(\alpha, \beta, \gamma\), and \(s_n = (\alpha^n + \beta^n + \gamma^n)/n\). Express \(s_5, s_7, s_9, s_{11}\) in terms of \(s_2\) and \(s_3\).
Prove that, if the equation \[ a_0 x^n + a_1 x^{n-1} + \dots + a_n = 0 \] is satisfied for more than \(n\) distinct values of \(x\), then \(a_0, a_1, \dots, a_n\) are all zero. A function \(f(x)\) is said to be rational, if it can be expressed in the form \(P(x)/Q(x)\), where \(P(x), Q(x)\) are polynomials in \(x\). A function \(f(x)\) is said to be periodic, with period \(k\), if \(f(x+k)=f(x)\) for all values of \(x\) for which \(f(x)\) is defined. Prove that a periodic function cannot be a rational function.
Solution:
Express \[ \frac{ax^2+2bx+c}{(x-\alpha)^2(x-\beta)^2} \] in partial fractions, when all the coefficients are general and not subject to any conditions. Find the condition that \[ \int \frac{ax^2+2bx+c}{(Ax^2+2Bx+C)^2} \,dx \] should be a rational function (as defined in Question A 2), when the two quadratic expressions have no common factor and \(B^2 > AC\).
(i) Define an involution of points on a straight line, and prove that a necessary and sufficient condition that the three pairs of points \((A, A')\), \((B, B')\), \((C, C')\) should be in involution is \[ BC' \cdot CA' \cdot AB' + B'C \cdot C'A \cdot A'B = 0. \] (ii) Explain precisely what is meant by an involution of points on a conic. (iii) It is required to find a triangle inscribed in a conic with its sides passing through three given points; shew that in general there are two such triangles, and find necessary and sufficient conditions that there should be an infinite number of such triangles. (iv) \(A, B\) are two given points on a conic and it is required to find two other points \(P, Q\) on the conic such that the circle on \(PQ\) as diameter passes through \(A\) and \(B\); shew that in general there is only one solution and consider specially the cases when the conic is a circle or a rectangular hyperbola.
Prove that with a suitable choice of homogeneous coordinates \((x,y,z)\) the locus equation of any conic \(s\) can be taken in the form \[ yz+zx+xy = 0. \] Any point \(P_r\) on this conic has coordinates \((\frac{1}{\alpha_r}, \frac{1}{\beta_r}, \frac{1}{\gamma_r})\), where \(\alpha_r : \beta_r : \gamma_r\) are parameters and \(\alpha_r+\beta_r+\gamma_r=0\); find the equations of (i) the chord \(P_1P_2\), (ii) the tangent at \(P_1\). The tangents from the point \(P(1/\alpha, 1/\beta, 1/\gamma)\) to the conic \[ s' = a^2x^2+b^2y^2+c^2z^2-2bcyz-2cazx-2abxy=0 \] meet the conic \(s\) again at the points \(Q, R\); prove that the equation of the chord \(QR\) is \[ a\beta\gamma x + b\gamma\alpha y + c\alpha\beta z = 0, \] and deduce that \(QR\) touches \(s'\) and that the triangle \(PQR\) is self-polar with respect to a third fixed conic. State the poristic property of the two conics \(s\) and \(s'\) which is established by these results.
A uniform cube of edge \(2b\) rests in equilibrium on the top of a fixed rough cylinder of radius \(a\) whose axis is horizontal. By considering the potential energy when it is rolled over through an angle \(\theta\), shew that the equilibrium is stable, if \(b\) is less than \(a\). Shew also that, if this is the case, the cube can be rolled into another position of equilibrium, which is unstable. Discuss the stability of the first position when \(b\) is greater than \(a\) and also when \(b\) is equal to \(a\).
\(AFBCED\) is a light horizontal beam 12 ft. long, bearing equal weights \(W\) at \(A,B,C,D\) and supported at \(F\) and \(E\). The lengths of \(AB, BC, CD\) are each 4 ft., of \(AF\) 2 ft. and of \(ED\) 3 ft. Find graphically the pressures on \(F\) and \(E\), and obtain a diagram showing the distribution of the bending moment along the beam.
A particle moves in a straight line with retardation \(a^2v^3 + b^2v^5\). The initial velocity is \(a/b\). Shew that the time taken to reduce the velocity to \(\frac{a}{b\sqrt{3}}\) is \[ \frac{b^2}{a^3}(1-\log_e 2), \] and that the space moved in this time is \[ \frac{b}{a^2}(\sqrt{3}-1-\frac{\pi}{12}). \]
A bead of mass \(m\) slides on a smooth straight wire which is made to rotate about a point of itself in a horizontal plane with uniform angular velocity \(\omega\). The bead is attached by an elastic string of natural length \(a\) to the centre of rotation and the tension in the string when it is stretched to length \((a+x)\) is \(\lambda x/a + \mu(x/a)^2\), where \(\lambda\) and \(\mu\) are positive constants. Shew that there is one position in which the bead can remain in relative rest on the wire, and that this position is one of stable equilibrium relative to the wire. Shew that the period of small oscillations about this position is \(2\pi/n\), where \[ n^2 = \frac{1}{am}\{(\lambda - ma\omega^2)^2 + 4\mu ma\omega^2\}^{\frac{1}{2}}. \]