Problems

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.

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.

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.

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 \[ \lambda_1\mu_1 r_1^2 + \dots + \lambda_4\mu_4 r_4^2 = 0. \]

Prove that \[ 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{1}{6}n(n+1)(2n+1), \] and evaluate \[ 1^3 + 3^3 - 5^3 + 7^3 - 9^3 + \dots - 37^3 + 39^3. \]

\(O\) is the middle point of a straight line \(AB\) of length \(2a\). \(P\) moves so that \(AP.BP = c^2\). Shew that the radius of curvature at \(P\) of the locus is \[ 2c^2r^3/(3r^4 + a^4 - c^4), \] where \(r=OP\).

Six equal heavy rods each of weight \(W\) are freely hinged at their ends and form a regular hexagon \(ABCDEF\) which when hung up by the point \(A\) is kept from altering its shape by two light rods \(BF\) and \(CE\). Prove that the thrusts of these rods are \[ \frac{5\sqrt{3}W}{2} \quad \text{and} \quad \frac{\sqrt{3}W}{2}. \] What would be the thrusts in the rods if a weight \(W\) were attached at \(D\)?

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 \(QP\).

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 \(\frac{1}{2}b^2\).

Three points \(A, B, C\) being chosen at random on a circle of radius \(a\), shew that the mean value of the area of the triangle \(ABC\) is \[ \frac{3a^2}{2\pi}. \]