Shew that, if \[ (b-c)^2(x-a)^2 + (c-a)^2(x-b)^2 + (a-b)^2(x-c)^2 = 0, \] and no two of \(a, b, c\) are equal, then \[ x = \tfrac{1}{3} \{a+b+c \pm (a^2+b^2+c^2-bc-ca-ab)^{\frac{1}{2}}\}. \] Shew that one root of the equation \(x^3 = 100(x-1)\) is approximately 1.0103, and determine the other roots, correct to two places of decimals.
Prove that if \(ABCD\) is a quadrilateral then in general the sum of the rectangles \(AB.CD\) and \(BC.AD\) is greater than the rectangle \(AC.BD\). State and prove under what condition the two quantities become equal. \(P\) and \(Q\) are two points within a quadrilateral \(ABCD\) such that the triangles \(BPC\), \(CQD\), \(BAD\) are similar, corresponding vertices being in the order named. Prove that \(APCQ\) is a parallelogram.
Three forces \(P, Q, R\) act along the sides \(BC, CA, AB\) of a triangle \(ABC\), and are in equilibrium with three forces \(P', Q', R'\) acting along \(HA, HB, HC\), where \(H\) is the orthocentre of the triangle. Prove that \begin{gather*} P\sec A + Q\sec B + R\sec C = 0, \\ PP'\operatorname{cosec} 2A + QQ'\operatorname{cosec} 2B + RR'\operatorname{cosec} 2C = 0, \\ \text{and } PP' + QQ' + RR' - (QR' + Q'R)\cos A - (RP' + R'P)\cos B - (PQ' + P'Q)\cos C = 0. \end{gather*}
Prove that the line joining a point \(P\) on the circumcircle of a triangle to the orthocentre of the triangle is bisected by the pedal line (or Simson's line) of \(P\) with respect to the triangle. Hence or otherwise shew that if \(ABCD\) is a quadrilateral, and \(AB, CD\) meet at \(P\), and \(BC, AD\) at \(Q\), then the orthocentres of the triangles \(QAB, QCD, PBC, PAD\) are collinear.
Four articles are distributed to four persons, with no restriction as to how many any person may receive. Shew that the probability that two will be given to one person and one to each of two others is \(\frac{9}{32}\). Find the probability of the same distribution if there are four articles but five persons.
Find the equation of the line joining the two points \(P\) and \(Q\) in which the circles \begin{align*} (x-a)^2 + y^2 &= a^2 \\ x^2 + (y-b)^2 &= b^2 \end{align*} intersect. Shew that the circle described on \(PQ\) as diameter is \[ (a^2+b^2)(x^2+y^2) = 2ab(bx+ay). \] Find the length of the tangent from the point \((\lambda b, \lambda a)\) to any one of these circles.
Nine equal light rods are smoothly jointed together at their ends so that three form a triangle \(BCD\), three others join the points \(B, C, D\) to a common point \(A\), and the remaining three join the points \(B, C, D\) to another common point \(E\). The framework is suspended from \(A\) and a load \(W\) hangs from \(E\). Find the stresses in all the rods.
\(ABCDEFG\) is a regular heptagon inscribed in a circle of radius 1. Shew that the distance between the centroids of the triangles \(ABD, CEF\) is \(\frac{1}{3}\sqrt{7}\).
Shew that for all real values of \(x\) and \(\theta\) the expression \(\dfrac{x^2+x\sin\theta+1}{x^2+x\cos\theta+1}\) lies between \(\dfrac{4-\sqrt{7}}{3}\) and \(\dfrac{4+\sqrt{7}}{3}\).
Shew that the equation of the normal at a point \((\alpha, \beta)\) of the curve \(f(x,y)=0\) is \[ (x-\alpha) \frac{\partial}{\partial \beta} f(\alpha, \beta) = (y-\beta) \frac{\partial}{\partial \alpha} f(\alpha, \beta). \] Hence shew that from any point \((\xi, \eta)\) three normals, of which either one or all three must be real, can be drawn to the parabola \(y^2 = 4ax\). Prove that the area of the triangle formed by joining the feet of these three normals is \[ \{4a(\xi-2a)^3 - 27a^2\eta^2\}^{\frac{1}{2}} \] and deduce the equation to the locus of centres of curvature of the parabola.