It is known that the circumcircle of a triangle of tangents to a parabola passes through the focus of the parabola; reciprocate this result with respect to a circle whose centre is on the directrix of the parabola, and hence prove that, if \(A, B, C, D\) are any four points on a rectangular hyperbola, the circle through the feet of the perpendiculars from \(D\) to the sides of the triangle \(ABC\) passes through the centre of the rectangular hyperbola. Deduce that
The homogeneous coordinates \((x, y, z)\) of a point are so chosen that the equation of the line at infinity is \(px+qy+rz=0\) and the equation of the circle with respect to which the triangle of reference \(\Delta\) is self-polar is \(x^2+y^2+z^2=0\); prove that
Define "convergent sequence of real numbers." Prove that, if \(a_n \to a\) and \(b_n \to b\) as \(n\to\infty\), then \(a_n b_n \to ab\). If \(c_n\) is a sequence of positive numbers and if \[\frac{c_{n+1}^2}{c_n^2} \to c^2,\] shew that \[c_n^{1/n} \to |c|.\] If \(a_n < b_n\) and if \(a_n \to a\), \(b_n \to b\), shew that \[a \le b.\]
Shew how the H.C.F. of two polynomials \(f(x)\) and \(g(x)\) may be found without solving the equations \(f(x)=0\) and \(g(x)=0\). If \(d(f)\) denotes "the degree of \(f(x)\)," shew that necessary and sufficient conditions that \(f(x)\) and \(g(x)\) have a common factor are that polynomials \(F(x)\) and \(G(x)\) exist such that \(d(F) \le d(g)-1\), \(d(G) \le d(f)-1\), and \(Ff=Gg\). Hence find the values of \(a\) for which the equation \[x^5+x^2+x+a=0\] has a repeated root.
If \(u, v\) are positive and \(p>1\), shew that
\[\frac{u^p}{v^{p-1}} \ge pu - (p-1)v.\]
By writing first \(u = \frac{x}{x+y}, v = \frac{\xi}{\xi+\eta}\) and then \(u = \frac{y}{x+y}, v=\frac{\eta}{\xi+\eta}\) with positive \(x,y,\xi,\eta\) deduce that
\[ \frac{x^p}{\xi^{p-1}} + \frac{y^p}{\eta^{p-1}} \ge \frac{(x+y)^p}{(\xi+\eta)^{p-1}}. \]
Now substitute \(x=a_k, y=y_k, \xi = \sum_{k=1}^n a_k, \eta = \sum_{k=1}^n y_k\) and deduce
\[ \left(\sum_{k=1}^n a_k^p\right)^{\frac{1}{p}} + \left(\sum_{k=1}^n y_k^p\right)^{\frac{1}{p}} \ge \left\{\left(\sum_{k=1}^n(a_k+y_k)\right)^p\right\}^{\frac{1}{p}}.\]
Under what conditions does equality hold?
What is the corresponding result when \(0
Shew how to find \(\lim_{x\to 0} \frac{f(x)}{g(x)}\), when \(f(0)=0\) and \(g(0)=0\). Find the limit as \(x \to 0\) of
An elastic string of modulus \(\lambda\) and density \(\rho\) per unit length when unstretched lies in the form of a semicircle of radius \(a\) on the upper half of a smooth circular cylinder whose axis is horizontal. If \(T\) is the tension of the string at a point whose radius makes an angle \(\phi\) with the upward vertical, the angle to the same point of the string if lying unstretched symmetrically across the cylinder being \(\theta\), shew that \[ \frac{dT}{d\theta} + g\rho a \sin\phi = 0,\] where \[ T = \lambda \left(\frac{d\phi}{d\theta}-1\right). \] If \(\lambda = 2g\rho a\), deduce that \[ T = \lambda\left(\sqrt{2}\cos\frac{\phi}{2}-1\right), \] and that the unstretched length of the string is \(2a\sqrt{2}\log(\sqrt{2}+1)\).
Two particles, each of weight \(W\), are joined by a light elastic string of natural length \(l\) and modulus of elasticity \(W\). They are held on a rough plane inclined at \(60^\circ\) to the horizontal at a distance \(d(>l)\) apart, so that the string lies along a line of greatest slope. If the coefficient of friction between each particle and the plane is \(\sqrt{3}/2\), and the particles are released simultaneously, examine whether, and if so how, equilibrium is broken in the cases (i) \(d=\frac{3}{2}l\); (ii) \(d=\frac{5}{2}l\).
A particle of mass \(m\) is slightly disturbed from rest at the highest point of a smooth uniform hemisphere of radius \(a\) and mass \(M\) whose base is free to move on a smooth horizontal plane. Shew that, while the particle is in contact with the hemisphere, it describes an elliptic path in space, and find an equation in \(\cos\alpha\) to determine the angle \(\alpha\) which the radius to the particle makes with the vertical when the particle leaves the sphere. By sketching the graph \(y=x^3-3kx+2k\), where \(k>1\), or otherwise, shew that this equation in \(\cos\alpha\) has a unique root between \(0\) and \(1\), and shew that, whatever the ratio of \(M/m\), the particle leaves the hemisphere before the radius to it makes an angle \(\cos^{-1}(\frac{2}{3})\) with the vertical.
Find expressions for the tangential and normal components of the acceleration of a particle moving in a plane curve. A thin tube is in the form of the curve whose coordinates are given parametrically by the equations \(x=a(\theta+\sin\theta)\), \(y=a(1-\cos\theta)\), where the line \(y=0\) is horizontal and the line \(x=0\) is vertically upwards. A particle of mass \(m\) is joined by a light elastic string, of natural length \(a\sqrt{2}\) and modulus of elasticity \(mg/2\sqrt{2}\), lying inside the tube, to the point given by \(\theta=\frac{\pi}{2}\). It is held at rest at the point given by \(\theta=0\), and then released. Prove that the particle comes to rest again just as the string becomes slack. Prove also that the pressure on the particle due to the curve acts upwards throughout the motion, and find its value at the point whose parameter is \(\theta\).