Prove that the value of the determinant \[ \begin{vmatrix} t_1+x & a+x & a+x & a+x \\ b+x & t_2+x & a+x & a+x \\ b+x & b+x & t_3+x & a+x \\ b+x & b+x & b+x & t_4+x \end{vmatrix} \] is \(A + Bx\), where \(A\) and \(B\) are independent of \(x\). Show further that if \[ f(t) = (t_1-t)(t_2-t)(t_3-t)(t_4-t) \] then \(A = \dfrac{af(b)-bf(a)}{a-b}\) and \(B = \dfrac{f(b)-f(a)}{a-b}\).
Show that, for all real values of \(x\) and \(\theta\), the value of the expression \[ \frac{x^2+x \sin \theta+1}{x^2+x \cos \theta+1} \] lies between \(\dfrac{4-\sqrt{7}}{3}\) and \(\dfrac{4+\sqrt{7}}{3}\).
Solution: \begin{align*} && y &= \frac{x^2+x \sin \theta+1}{x^2+x \cos \theta+1} \\ && y' &= \frac{(2x+\sin\theta)(x^2+x\cos \theta+1)-(2x+\cos \theta)(x^2+x\sin\theta+1)}{(x^2+x\cos \theta+1)^2} \\ &&&= \frac{2x^2(\cos\theta-\sin \theta)+(\sin \theta-\cos\theta)(x^2+1)}{(x^2+x\cos \theta+1)^2} \\ &&&= \frac{(\cos \theta-\sin\theta)(x^2-1)}{(x^2+x\cos \theta+1)^2} \end{align*} If \(\sin \theta = \cos \theta\) then the expression is exactly equal to \(1\), so there is nothing to check. (Other than \(1\) is in our range). Therefore if \(\cos \theta \neq \sin \theta\) the turning points are \(x = \pm 1\), which are \(\frac{2\pm\sin \theta}{2\pm\cos \theta}\) Differentiating again, this time wrt to \(\theta\), we get \begin{align*} && y' &= \frac{(\pm \cos \theta)(2\pm \cos \theta)-\mp \sin \theta(2 \pm \sin \theta)}{(2 \pm \cos \theta)^2} \\ &&&= \frac{\pm 2(\cos \theta+\sin \theta)+(\cos^2 \theta+\sin^2 \theta)}{(2 \pm \cos \theta)^2} \\ &&&= \frac{\pm 2(\cos \theta + \sin \theta)+1}{(2 \pm \cos \theta)^2} \end{align*} Therefore \(\cos(\theta+\phi) = \frac{1}{2\sqrt{2}}\)
Prove that the locus of the poles of a fixed line \(l\) with respect to conics of a confocal family is a straight line which is the normal at the point of contact to that member of the family which touches \(l\). \(Q\) is a point on the tangent at \(P\) to a conic and \(T, T'\) are the points of contact of the tangents from \(Q\) to a confocal conic. Prove that \(TP, PT'\) are equally inclined to \(QP\).
Explain the process of reciprocation with respect to a conic, with notes on the special case when the conic is a circle. A conic has a given focus \(S\), passes through a given point \(P\) and touches a given line \(l\). Prove that its directrix (corresponding to \(S\)) envelops a conic which passes through \(S\).
Interpret the equation \(S+\lambda t^2=0\), where \(S=0\) and \(t=0\) are the equations of a conic and one of its tangents. Two chords \(AB\) and \(CD\) of a conic \(S\) meet in the point \(O\), and one of the tangents from \(O\) to \(S\) touches \(S\) at \(P\). Another conic \(S'\) is drawn through \(A, B, C, D\) to touch at \(P'\) the harmonic conjugate \(OP'\) of \(OP\) with respect to the lines \(AB, CD\). Prove that there exists a conic which has four-point contact with \(S\) at \(P\) and four-point contact with \(S'\) at \(P'\).
A rectangle \(R\) has centre \(M\) and sides \(2a, 2b\). A point \(O\) is taken on the line through \(M\) perpendicular to the plane of \(R\) such that \(MO=h\). Lines are drawn from \(O\) through all points of the perimeter of \(R\). Prove that these lines cut off on the sphere with centre \(O\) and unit radius an area \[ 4 \sin^{-1} \frac{ab}{\sqrt{\{(a^2+h^2)(b^2+h^2)\}}}. \]
On the tangent at \(P\) to a plane curve \(\Gamma\) a point \(P_1\) is taken so that \(PP_1=a\), where \(a\) is a fixed positive number and \(PP_1\) is drawn in the direction of increasing \(s\) (the arc of \(\Gamma\) measured from a fixed point). As \(P\) describes \(\Gamma\), \(P_1\) describes a curve \(\Gamma_1\). If \(s_1\) is the arc of \(\Gamma_1\) measured from a fixed point to \(P_1\), prove that \[ \frac{ds_1}{ds} = \frac{R}{\rho}, \] where \(\rho=CP\) and \(R=CP_1\), \(C\) being the centre of curvature of \(\Gamma\) at \(P\). Prove that the centre of curvature of \(\Gamma_1\) at \(P_1\) is the point \(C_1\) on \(CP_1\) such that \[ \frac{CC_1}{C_1P_1} = \frac{a\rho}{R^2} \frac{d\rho}{ds}. \]
Prove that \[ \int_0^1 \frac{t^4(1-t)^4}{1+t^2} dt = \frac{22}{7} - \pi. \] Evaluate \(\displaystyle\int_0^1 t^4(1-t)^4 dt\) and deduce that \[ \frac{22}{7} - \frac{1}{1260} > \pi > \frac{22}{7} - \frac{1}{630}. \]
A uniform beam, of weight \(W\) and length \(l\), is clamped horizontally at one end, and a vertical force \(W\) is applied upwards at a point \(X\) at a distance \(x\) from the clamp. Find the bending moment at a point \(Y\) at distance \(y\) from the clamp, and calculate, in terms of \(x\), the greatest bending moment at any point of the beam. Find where \(X\) must be in order to make this greatest bending moment as small as possible.
Two particles each of mass \(m\), moving in a plane, attract each other with a force of magnitude \(\lambda r^{-2}\), where \(\lambda\) is constant and \(r\) is the distance between the particles. Prove that (i) the (linear) momentum of the system is constant in magnitude and direction, (ii) the angular momentum about the centre of mass is constant, (iii) the sum of the kinetic energy of the particles and the potential energy is constant. If initially \(r=r_0\), one particle is at rest, and the other is moving with velocity \(v_0\) at right angles to the join of the particles, show that the stationary value of \(r\) other than the value \(r_0\) is given by \[ mv_0^2 r_0^2 \left(\frac{1}{r} + \frac{1}{r_0}\right) = \lambda. \]