Seven slips of paper, three red and four blue, are placed in a bag. Shew that if three slips are drawn at random from the bag, the chances are six to one against all three having the same colour. If the three slips which have been drawn are then replaced, and three slips are again drawn at random from the bag, shew that the chances are slightly more than seventy-one to one against the slips drawn on both occasions being all six of the same colour.
\(ABCD\) is a rectangle, and a circle touches \(AC\) at \(A\). Prove that the polar of \(B\) with respect to the circle passes through \(D\).
Forces \(P\cos A, P\cos B, P\cos C\) act along the sides \(CB, AC, AB\) of a triangle \(ABC\), in the directions named. Show that the resultant passes through \(D\) and \(E\), the feet of the perpendiculars from \(A\) and \(B\) on \(BC\) and \(CA\) respectively. Find the magnitude of the resultant.
\(ABC\) is any triangle, \(X\) any point. Shew that there exists a point \(X'\) such that \[ B\hat{A}X' = X\hat{A}C, \quad C\hat{B}X' = X\hat{B}A, \quad A\hat{C}X' = X\hat{C}B. \] Prove that if \(BC, CA, AB\) subtend equal angles at \(X\), then the feet of the perpendiculars from \(X'\) on \(BC, CA, AB\) are the vertices of an equilateral triangle.
Prove that \[ 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{1}{6}n(n+1)(2n+1), \] and evaluate \[ 1^3 + 3^3 - 5^3 + 7^3 - 9^3 + \dots - 37^3 + 39^3. \]
If \(s_1=0, s_2=0, s_3=0, s_4=0\) are the equations (each in the standard form \(x^2+y^2+2gx+2fy+c=0\)) of four circles (of radii \(r_1, r_2, r_3, r_4\) respectively) every two of which cut orthogonally, shew that the two circles \begin{align*} \lambda_1 s_1 + \dots + \lambda_4 s_4 &= 0, \\ \mu_1 s_1 + \dots + \mu_4 s_4 &= 0 \end{align*} will cut orthogonally if \[ \lambda_1\mu_1 r_1^2 + \dots + \lambda_4\mu_4 r_4^2 = 0. \]
An isosceles triangle rests with its plane vertical and its vertex downwards between two smooth pegs in the same horizontal line and at a distance \(a\) apart. If the base of the triangle is \(3a\) and its vertical angle is \(2\theta\), show that equilibrium is possible when the base makes an angle \(\cos^{-1}(\cos^2\theta)\) with the horizontal.
Solve completely the equation \[ \sin 5x = \cos 4x. \] Deduce that one root of the equation \[ 16s^4 + 8s^3 - 12s^2 - 4s + 1 = 0 \] is given by \(s=\sin\frac{\pi}{18}\). Express the other three roots as sines of certain angles.
If \begin{align*} \alpha + \beta + \gamma &= a, \\ \alpha^2 + \beta^2 + \gamma^2 &= b, \\ \alpha^3 + \beta^3 + \gamma^3 &= c, \end{align*} find \(\alpha\beta\gamma\) and \(\alpha^4+\beta^4+\gamma^4\) in terms of \(a\), \(b\) and \(c\). Verify that when \(a=0\), they are respectively \(\frac{1}{6}c\) and \(\frac{1}{2}b^2\).
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, \(QT, QT'\) are the tangents from \(Q\) to a confocal conic. Shew that \(PT, PT'\) are equally inclined to \(QP\).