Prove that, if \[ a_r = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{r!}, \] then \[ \frac{a_0}{1!} + \frac{a_1}{2!} + \frac{a_2}{3!} + \dots \text{ to infinity} = 2\sqrt{e}. \] Sum the infinite series \[ \frac{x}{2} + \frac{2x^2}{3} + \frac{3x^3}{4} + \dots. \]
Prove that, having given \(c^2=a^2d\), the product of a pair of the roots of the equation \[ x^4+ax^3+bx^2+cx+d=0 \] is equal to the product of the other pair. Solve the equation \[ x^4-x^3-16x^2-2x+4=0. \]
Having given \[ \begin{vmatrix} \sin\theta & \cos\theta & \sin x \cos a \\ \cos\theta & -\sin\theta & \cos x \\ 1 & 1 & 1 \end{vmatrix} = 0, \] shew that if \(\theta\) and \(x\) are small \[ \theta = x \cos a - \frac{1}{2}x^2\sin^2a + \dots, \] determining the next term of the series.
\(P\) is any point on an ellipse, and \(PQ, PR\) are chords cutting the major axis at points equidistant from the centre. The tangents at \(Q\) and \(R\) intersect in \(T\). Prove that \(PT\) is bisected by the minor axis.
An equilateral triangle has its angular points on the rectangular hyperbola \(xy=a^2\). Shew that the abscissae of the angular points are connected by the relations \begin{align*} (x_1+x_2+x_3)x_1x_2x_3 + 3a^4 &= 0, \\ a^4(x_2x_3+x_3x_1+x_1x_2) + 3x_1^2x_2^2x_3^2 &= 0. \end{align*} Shew also that the locus of the middle points of the sides of such triangles is \[ 3(x^2+y^2)^2 = 16xy(xy-a^2). \]
Prove that the radius of curvature at any point of a curve \(y=f(x)\) is \[ \frac{\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}}}{\frac{d^2y}{dx^2}}. \] Shew that, if in a curve \(x^2=a^2(\sec\phi+\tan\phi)\), where \(\phi\) is the angle which the tangent makes with the axis of \(x\), then the radius of curvature is \(\frac{1}{2}a\sec^2\phi\).
Trace the curve \(r=a(\sin\theta-\cos 2\theta)\), and find the area of the loop which passes through the point \((2a, \frac{\pi}{2})\).
A triangular prism, of mass \(M\), rests with one face on a smooth horizontal plane, the other faces each making an angle \(\alpha\) with the plane. Two smooth particles, whose masses are \(m\) and \(m'\), slide down the inclined faces. Find the acceleration of the prism on the plane, and shew that the ratio of the accelerations of the particles relative to the prism is \[ \frac{M+m-m'\cos 2\alpha}{M+m'-m\cos 2\alpha}. \]
A smooth parabolic tube is fixed in a vertical plane with its vertex downwards. A particle starts from rest at the extremity of the latus rectum (\(4a\) in length), and slides down the tube; express as a definite integral the time taken to reach the vertex, and shew that this time is approximately \(2.7 \sqrt{\frac{a}{g}}\) seconds.
Prove that the feet of the perpendiculars drawn on the sides of a triangle from any point of the circumcircle are collinear. If \(O\) is a point on a circle the circles on the chords \(OP, OQ, OR\) as diameters intersect in three points other than \(O\) which are collinear.