10273 problems found
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.
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 \[x^5 - 5x^4 - 10x^3 + 10x^2 + 5x - 1 = 0.\]
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 \(k\).
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.
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 \(ABC\).
Explain what you understand by a convergent series. Investigate for what ranges of values of \(x\) the following series are convergent:
The coordinates \((x,y)\) of a point on a curve are given in terms of a parameter \(t\) by the equations \[ x=a(1-t^2), \quad y=at(1-t^2). \] Sketch the curve. Find the radius of curvature of each of the two branches of the curve at the double point. Find the area of the loop.
Three light rods are freely jointed at their extremities to form an equilateral framework \(ABC\). Particles of mass \(m\) are attached at \(A\) and \(B\), and the whole framework can rotate without friction in a vertical plane about \(C\), which is fixed. The framework is set in motion; prove that, when the angular displacement of the rod \(AB\) from its lower horizontal position is \(\theta\), the thrust in this rod is \(\dfrac{mg \cos\theta}{\sqrt{3}}\).
Show that the feet of the four normals which can be drawn from the point \((\xi, \eta)\) to the conic \(Ax^2+By^2=1\) lie on the rectangular hyperbola \[(A-B)xy - A\eta x + B\xi y = 0.\] Show that the normals to the conic \(aa'x^2+bb'y^2=1\) at the extremities of the chords \(ax+by-1=0\) and \(a'x+b'y+1=0\) are concurrent. Find the coordinates of the point where they meet.
If \(u_0 = 1\) and \(u_n = \dfrac{2u_{n-1}+3}{u_{n-1}+2}\), prove that, as the positive integer \(n\) tends to infinity, \(u_n\) tends to the limit \(\sqrt{3}\). (You may find it useful to prove that \({u_n}^2 < 3\) and that \(u_{n+1} > u_n\).)