If \(y=\frac{5x}{(4-x)(x-9)}\), shew that no real values of \(x\) can be found which will give \(y\) values between \(\frac{4}{5}\) and 5. Shew that the equation \(x^3-3a^2x+2b^3=0\) has three real roots if \(a\) is greater than \(b\).
If \(q_r\) denote the number of combinations of \(n\) things \(r\) at a time, prove from first principles that the number of combinations of \((n+1)\) things \(r\) at a time is \(q_r+q_{r-1}\). Prove also that \(q_1-\frac{1}{2}q_2+\frac{1}{3}q_3-\dots+(-1)^{n-1}\frac{1}{n}q_n=1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}\).
Prove the law of formation of the successive convergents to the continued fraction \[ \frac{a_1}{b_1+}\frac{a_2}{b_2+}\dots. \] Prove that, if \(u_n\) is the \(n\)th convergent of the continued fraction \[ 1-\frac{1}{2-}\frac{1}{3-}\frac{1}{2-}\frac{1}{3-}\frac{1}{2-}\dots, \] then \[ u_{n+1} = \frac{2u_n+1}{2u_n+4}. \]
Five weightless rods \(AB, BC, CD, DA\) and \(AC\) smoothly jointed at their ends form a framework, in which \(AB=AC=AD=2BC=2CD\). The framework is suspended from \(A\) with \(AD\) vertical, a weight 10lb. hanging from \(B\) and a horizontal force acting at \(D\). Find the stress in each of the rods.
Prove that the centre of gravity of three uniform rods in the form of a triangle coincides with the centre of the circle inscribed in the triangle formed by joining their middle points. Find the position of the centre of gravity of the portion of a solid circular cylinder between its circular base and an oblique section made by a plane touching its base.
State the Principle of Virtual Work. A circular ring of weight \(w\) and radius \(a\) hangs vertically over a smooth horizontal plank of width \(2b\), and is held so that its plane makes an angle \(\theta\) with the vertical planes through the edges of the plank, by a couple in the plane of the plank. Prove that the moment of the couple is \[ \frac{wb^2\cos\theta}{(a^2\sin^2\theta-b^2)^{\frac{3}{2}}}. \]
Explain how to find the relative velocity of two particles moving with given velocities in the same plane. If particles \(P\) and \(Q\) are travelling in straight lines with uniform velocities \(u\) and \(v\) towards a point \(O\), shew that the least distance between them during the motion is \(\frac{u-v}{v}p\), where \(p\) is the perpendicular distance of \(O\) from the direction of the relative velocity of \(P\) to \(Q\) when they are equidistant from \(O\).
If \(ABC, DEF\) are two triangles self-conjugate with respect to a given conic, prove that the points \(A,B,C,D,E,F\) lie on a conic. By a suitable choice of \(DEF\) deduce a proposition about rectangular hyperbolas.
Let \(A,B,C,A',B',C'\) be any six points in space and \(O\) any point of the line of intersection of the planes \(ABC, A'B'C'\). Let \(OX\) be the line through \(O\) intersecting \(BC'\) and \(B'C\); \(OY\) the line intersecting \(CA', C'A\); and \(OZ\) the line intersecting \(AB', A'B\). Prove that \(OX, OY, OZ\) lie in a plane.
Prove that an infinity of triangles can be inscribed in the conic \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) whose sides touch the conic \(\frac{x^2}{a'^2}+\frac{y^2}{b'^2}=1\) if \(\pm \frac{a'}{a} \pm \frac{b'}{b} \pm 1 = 0\).