A particle starts from rest at any point \(P\) in the arc of a smooth cycloid whose axis vertical and vertex \(A\) downwards; prove that the time of descent to the vertex is \(\pi\sqrt{\frac{a}{g}}\), where \(a\) is the radius of the generating circle. \par Shew also that if the particle is projected from \(P\) downwards along the curve with velocity equal to that with which it reaches \(A\) when starting from rest at \(P\), it will now reach \(A\) in half the time taken in the preceding case.
Shew that, if \[ \frac{x^2}{a} + \frac{y^2}{b} = x+y \quad \text{and} \quad \frac{a^2}{x} + \frac{b^2}{y} = a+b, \] either \[ \frac{x}{a} + \frac{y}{b} + 1 = 0 \quad \text{or} \quad x+y=a+b. \]
Find the coefficients \(A, B, C\) in order that the equation \[ ax^2+2bx+c = A(x-p)^2+2B(x-p)(x-q)+C(x-q)^2 \] may be an identity. \par Prove that \[ AC-B^2 = (ac-b^2)/(p-q)^2. \]
Find the number of combinations of \(n\) things taken \(r\) together. \par From the first \(6n\) integers 3 are selected whose sum is odd. Shew that the number of ways in which this can be done is equal to twice the sum of the squares of the first \(3n-1\) integers.
Show that the series \[ m + \frac{m(m-1)}{1!} + \frac{m(m-1)(m-2)}{2!} + \dots \] is convergent when \(m\) is a positive fraction, and sum it to infinity. Sum the series \[ 2-5m+8\frac{m(m-1)}{2!}-11\frac{m(m-1)(m-2)}{3!}+\dots \] when it is convergent.
Solution: The first expression is \begin{align*} \sum_{r=0}^{\infty} m\binom{m-1}{r}x^r &= m(1+x)^{m-1} \end{align*} If \(m > 0\) then we have an alternating series, which converges when \(x \to 1\), therefore the sum is \(m2^{m-1}\). Our sum is \begin{align*} && S_x &= \sum_{r=0}^{\infty} (-1)^r(3r+2)\binom{m}{r}x^r \\ &&&= \sum_{r=0}^{\infty}(3r+2)\binom{m}{r}(-x)^r \\ &&&= 3\sum_{r=0}^{\infty}r\binom{m}{r}(-x)^r +2\sum_{r=0}^{\infty}\binom{m}{r}(-x)^r \\ &&&= -3x\sum_{r=0}^{\infty}r\binom{m}{r}(-x)^{r-1}+2(1-x)^m \\ &&&=3xm(1-x)^{m-1}+2(1-x)^m\\ &&&= (1-x)^{m-1}(3xm+2(1-x)) \\ &&&= ((3m-2)x+2)(1-x)^{m-1} \end{align*} but as \(x \to 1\) this is zero unless \(m=1\), in which case we have \(3\)
State the chief properties of the complete quadrilateral. \par An ellipse being drawn, give a construction for drawing the two tangents which have a given direction.
Prove that the radius of curvature of a central conic is a third proportional to the perpendicular from the centre on the tangent, and the conjugate semi-diameter. \par At the extremities \(P, Q\) of two conjugate diameters of an ellipse the circles of curvature are drawn, meeting the ellipse in \(X, Y\). Prove that \(PY, QX\) are parallel.
Prove that the equations \[ \frac{x}{5-6t-3t^2} = \frac{y}{5+8t-t^2} = \frac{1}{1+t^2} \] represent a circle of radius 5.
Shew that from any point four normals can be drawn to an ellipse, and that their feet lie on a rectangular hyperbola through the point. \par Prove that the perpendicular from the point on a diameter of the ellipse meets the conjugate diameter on this hyperbola.
Find the length of the latus rectum, and the coordinates of the focus, of the parabola \[ (x\cos\alpha+y\sin\alpha)^2 = 4a(x\cos\beta+y\sin\beta+c). \] Prove that the equation of the tangents from the origin is \[ c(x\cos\alpha+y\sin\alpha)^2 + a(x\cos\beta+y\sin\beta)^2 = 0. \]