Shew that, if \[ e^x \sin x = a_0 + \frac{a_1}{1!}x + \frac{a_2}{2!}x^2 + \dots + \frac{a_n}{n!}x^n + \dots, \] then \(a_{4n}=0\), \(a_{4n+1} = (-1)^n 4^n\); and determine \(a_{4n+2}\) and \(a_{4n+3}\).
Prove that the circumcircle of the triangle formed by the feet of the three normals from \((h, k)\) to the parabola \(y^2=4ax\) passes through the vertex of the parabola, and that the coordinates of its centre are \(\left( a+\frac{h}{2}, \frac{k}{4} \right)\).
Show that the path of a particle moving freely under gravity is a parabola, and that the velocity at any point is the same as would be acquired by the particle falling vertically from rest at the level of the directrix. A ball is thrown from a point on the ground so as to pass over a wall of height 25 feet situated at a distance of 60 feet from the thrower. The velocity is the least that will allow the ball to clear the wall. Show that the ball again reaches the ground in about 2.8 seconds.
Shew that, if \(n\) straight lines are drawn in a plane in such a way that no two are parallel and no three meet in a point, then the plane is divided into \(\frac{1}{2}(n^2+n+2)\) regions. How many of these regions stretch to infinity?
Find the maxima and minima values of \[ \frac{x+y-1}{x^2+2y^2+2}. \]
From the point \(P(\alpha\cos\theta, \beta\sin\theta)\) of the ellipse \(S' \equiv \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} - 1 = 0\) tangents are drawn to the ellipse \(S \equiv \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1 = 0\) to meet \(S'\) again in points \(Q, R\). \par Shew that the equation of the line \(QR\) is \[ \frac{x}{\alpha}(a^2\beta^2+a^2b^2-\alpha^2b^2)\cos\theta + \frac{y}{\beta}(\alpha^2\beta^2-a^2b^2+\alpha^2b^2)\sin\theta + (a^2b^2-\alpha^2b^2-a^2\beta^2)=0. \]
A smooth sphere of mass \(m\) is resting on a smooth horizontal inelastic table. A second sphere of mass \(M\) is dropped vertically so as to strike the first with velocity \(V\), the line of centres at the moment of impact making an angle \(\theta\) with the vertical. If the coefficient of restitution between the spheres is \(e\), show that the velocity communicated to the first sphere is \[ (1+e)MV\frac{\sin\theta\cos\theta}{m+M\sin^2\theta}. \]
Two coplanar forces \(X, Y\) are parallel to the (rectangular) axes of \(x\) and \(y\) respectively, their points of application being \((a,0)\) and \((0,b)\). If the forces are rotated through the same angle, shew that their resultant always passes through the fixed point whose coordinates are \[ x = \frac{aX^2+bXY}{X^2+Y^2}, \quad y=\frac{bY^2+aXY}{X^2+Y^2}. \]
Trace the curve \[ y^4 - 4axy^2 + 3a^2x^2 - x^4 = 0, \] and shew that it has tangents parallel to the axis of \(x\) at two points for which \(x^4 = \frac{3}{4}a^4\).
A rectangular hyperbola \(H\) with centre \(O\) cuts a line \(l\) in two points \(P, Q\). \(X\) is the pole with respect to \(H\) of the line joining the remaining intersections of \(H\) with the circle \(OPQ\). \(O\) and \(l\) are fixed and \(H\) is allowed to vary. Shew that \(X\) remains fixed.