Solve the equations: \begin{align*} x+y+z+w &= 1, \\ ax+by+cz+dw &= \lambda, \\ a^2x+b^2y+c^2z+d^2w &= \lambda^2, \\ a^3x+b^3y+c^3z+d^3w &= \lambda^3. \end{align*} Prove that, if \(a, b, c, d, \lambda\) be all real and unequal, at least two and not more than three of \(x, y, z, w\) are positive.
If \(\phi(x)\) be a function such that \(\phi(x)+\phi(y) = \phi\left(\frac{x+y}{1-xy}\right)\) for all values of \(x, y\) such that \(xy \neq 1\), shew that \[ (1+x^2)\phi'(x) = \phi'(0). \] If \(\phi'(0)=1\) and \(x>0\), shew that
If \(\frac{p_{r,s}}{q_{r,s}}\) be the value of the continued fraction \[ \frac{1}{a_r +}\frac{1}{b_r +}\frac{1}{a_{r+1} +}\frac{1}{b_{r+1} +}\dots\frac{1}{+ b_{s-1} +}\frac{1}{a_s}, \] shew that \[ p_{1,n} - p_{1,n-1} = a_n \frac{\partial p_{1,n}}{\partial a_n}, \quad p_{\lambda,n} - p_{\lambda+1,n} = a_\lambda \frac{\partial p_{\lambda,n}}{\partial a_\lambda}; \] shew also that \[ \frac{\partial p_{1,n}}{\partial a_n}\frac{\partial p_{1,n}}{\partial a_1} - p_{1,n-1}p_{2,n} = 1, \] and that \[ a_\lambda p_{1,n} = p_{1,\lambda} p_{\lambda,n} - p_{1,\lambda-1} p_{\lambda+1,n}. \]
Two straight lines are given by the equations \[ p = ax+by+c=0, \quad p' = a'x+b'y+c'=0; \] shew that the harmonic conjugate of the line \(p+kp'=0\) with respect to \(p=0, p'=0\) is \(p-kp'=0\). If \(p=0\) be one of the lines bisecting the angles between the line \(p'=0\) and another line, shew that the equation of this other line is \[ (a^2+b^2)p' - 2(aa'+bb')p = 0. \]
Through a point \(K\) inside a triangle \(ABC\) a line \(XX'\) is drawn parallel to \(BC\) to meet the other sides of the triangle in \(X, X'\) respectively; and corresponding lines \(YY', ZZ'\) are drawn through \(K\) parallel to \(CA, AB\) respectively. Prove that the area of the convex hexagon which has \(XX', YY', ZZ'\) for diagonals is \[ (p^2+q^2+r^2+qr+rp+pq)\Delta, \] where \((p,q,r)\) are the areal coordinates of \(K\) referred to \(ABC\) and \(\Delta\) is the area of the latter triangle. Hence, or otherwise, prove that the area of the hexagon cannot be less than \(2\Delta/3\).
A system of coaxal circles has real limiting points: prove that its reciprocal with respect to a circle whose centre is at either limiting point is a system of confocal conics. Determine the properties of the system of confocal conics corresponding to the following properties of the system of coaxal circles:
A uniform cube of weight \(W\) and edge \(2a\) is placed upon a rough plane and a uniform sphere of weight \(W'\) and diameter \(2a\) rests upon the plane, touching the cube at the centre of one of its faces \((F)\). The plane is gradually tilted from a horizontal position round a line lying in the plane and parallel to the face \(F\) of the cube. Shew that if \(\mu\) be the coefficient of friction for every contact and if \(\mu<1\), then equilibrium will be broken by the cube slipping and the sphere rolling down the plane when the angle of inclination of the plane to the horizontal is \(\alpha\), where \[ \tan\alpha = \frac{\mu W}{W + (1-\mu)W'}. \]
A plane convex quadrilateral \(ABCD\) formed by four rigid rods \(AB, BC, CD, DA\) smoothly jointed at the angular points is kept in equilibrium by stretched elastic strings of tensions \(T_1, T_2, T_3, T_4\), the first having its ends attached to points on \(AD, AB\) respectively, the second to \(BA, BC\) and so on. Shew that \[ M_1 . \text{area } BCD - M_2 . \text{area } CDA + M_3 . \text{area } DAB - M_4 . \text{area } ABC = 0, \] where \(M_1\) is the numerical value of the moment of \(T_1\) about \(A\), \(M_2\) that of \(T_2\) about \(B\), and so on.
A triangle \(ABC\) is formed of three weightless rods and masses \(m_1, m_2\) and \(m_3\) are attached to the vertices \(A, B, C\) respectively; initially the system is moving uniformly on a smooth horizontal plane with velocity \(u\), parallel to \(BC\). Impulses act upon the masses along \(AB, BC\) and \(CA\) respectively, proportional to these lengths: investigate the subsequent motion of the system finding the time of a complete revolution and also the position of the vertex \(A\) at any time. What is the condition that the velocity of \(A\) shall never vanish?
A switchback railway consists of straight stretches smoothly joined by circular arcs, the whole lying in a vertical plane. Shew that a car started on a level stretch and running freely will leave the track if the downward gradient exceed \(\cos^{-1}\frac{2}{3}\) at any point; but that if braking is available up to half the weight of the car, gradients of about \(77^\circ\) are admissible. A level and a straight descending stretch are smoothly joined by an arc of radius \(a\). Two equal cars without brakes are joined by a cable of length \(2a\). Shew that the greatest admissible gradient such that the second car does not leave the track is \(\cos^{-1}\frac{1}{17}\). [Neglect the size of the cars, and resistance to motion other than braking.]