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 \(u_{n+1} > u_n\).)
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.
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 \(\dfrac{mg \cos\theta}{\sqrt{3}}\).
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.
If \[y = \frac{\log \{x + \sqrt{(1+x^2)}\}}{\sqrt{(1+x^2)}},\] verify that \[(1+x^2)\frac{dy}{dx} + xy = 1.\] Assuming that \(y\) can be expanded in a series of ascending powers of \(x\), prove that the series is \[x - \frac{2}{3}x^3 + \frac{2.4}{3.5}x^5 - \dots + (-1)^n \frac{2.4\dots2n}{3.5\dots(2n+1)}x^{2n+1} + \dots.\]
Show that the four points \((ct_i, c/t_i)\) \((i=1, 2, 3, 4)\) of the rectangular hyperbola \(xy=c^2\) are concyclic if and only if \(t_1 t_2 t_3 t_4 = 1\). \(A, B, C, D\) are the four points of intersection of a circle and a rectangular hyperbola. Prove that the six perpendiculars from the mid-points of the sides of the quadrangle \(ABCD\) to the opposite sides are concurrent.
The barrel of a gun of mass \(M\) is horizontal and of length \(l\); whilst a shell of mass \(m\) is being discharged from the gun the propelling gases exert a constant force \(P\) on the shell, and this force ceases as soon as the shell leaves the gun. From the instant of firing until the gun is brought to rest, recoil is resisted by a constant damping force \(R(
Prove Varignon's theorem, that the sum of the moments of two coplanar forces about any point in their plane is equal to the moment of their resultant about that point. Investigate whether the converse holds, namely, that if Varignon's theorem is true, then the resultant of two coplanar forces must be given by the parallelogram construction. A force in a given plane has moments \(G_1, G_2, G_3\), respectively, about points whose coordinates referred to axes in the plane are \((a_1, b_1), (a_2, b_2)\) and \((a_3, b_3)\). Determine the components \((X,Y)\) of the force referred to these axes and show that the force meets the \(x\)-axis in the point whose abscissa is \[ \begin{vmatrix} b_1 & a_1 & G_1 \\ b_2 & a_2 & G_2 \\ b_3 & a_3 & G_3 \end{vmatrix} \div \begin{vmatrix} b_1 & 1 & G_1 \\ b_2 & 1 & G_2 \\ b_3 & 1 & G_3 \end{vmatrix}. \]
Prove that, if \(x=r\cos\theta, y=r\sin\theta\) and \(\phi\) is any function of \(r\) and \(\theta\), \[\frac{\partial\phi}{\partial x} = \cos\theta \frac{\partial\phi}{\partial r} - \frac{\sin\theta}{r}\frac{\partial\phi}{\partial\theta},\] and obtain a corresponding expression for \(\partial\phi/\partial y\). Prove that, if \(\phi = r^{-n}\sin n\theta\), then \[\frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} = 0.\]
\(P, Q, R\) are points on the sides \(BC, CA, AB\) of a triangle \(ABC\), and are not collinear. \(QR\) meets \(BC\) in \(L\), \(RP\) meets \(CA\) in \(M\), \(PQ\) meets \(AB\) in \(N\). Show that \(L, M, N\) are collinear if and only if \(AP, BQ, CR\) are concurrent. If \(AP, BQ, CR\) meet in \(O\), the line \(LMN\) may be called the polar of \(O\) with respect to the triangle \(ABC\). Show that in this case \(LMN\) is also the polar of \(O\) with respect to the triangle \(PQR\). Finally, if \(l, m, n\) are the polars of \(A\) with respect to \(OBC\), of \(B\) with respect to \(OCA\) and of \(C\) with respect to \(OAB\), show that \(LMN\) is the polar of \(O\) with respect to the triangle whose sides are \(l, m, n\).