A uniform solid circular cylinder of weight \(W\) is placed on top of a fixed horizontal circular cylinder (of different radius) and has contact with it along its highest generator. The upper cylinder is slightly displaced and begins to roll down in contact with the lower cylinder. If the coefficient of friction between the surfaces is \(3/8\), prove that slipping begins when the inclination to the vertical of the plane through the axes is \(\cos^{-1}4/5\). Show that at this instant the reaction between the cylinders is of magnitude \(W\sqrt{73}/10\) in a direction inclined to the vertical at an angle \(\tan^{-1}12/41\).
Find the value of the \(n\)-rowed determinant of the form \[ \begin{vmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{vmatrix} \] whose element in the \(i\)th row and \(j\)th column is 1 if \(i\) and \(j\) differ by 0 or 1, and 0 otherwise.
The cubic equation \[ x^3 + px^2 + qx + r = 0, \quad \text{(I)} \] has roots \(\alpha\), \(\beta\) and \(\gamma\). Form the cubic equation with roots \(2\alpha - \beta - \gamma\), \(-\alpha + 2\beta - \gamma\), \(-\alpha - \beta + 2\gamma\), expressing the coefficients in terms of \(p\), \(q\) and \(r\). Find the condition that the roots of (I) should be in arithmetical progression. Find also the condition that the roots of (I) should be in geometrical progression.
The joins of a point \(P\) to the vertices \(X, Y, Z\) of a triangle meet the opposite sides in \(L, M, N\). \(MN\) meets \(YZ\) in \(A\), \(NL\) meets \(ZX\) in \(B\), and \(LM\) meets \(XY\) in \(C\). Prove that \(A, B\) and \(C\) lie on a straight line \(p\). Show that, if \(P\) moves along a straight line, the envelope of \(p\) is in general a conic inscribed in \(XYZ\); discuss the special case in which the locus of \(P\) passes through \(X\).
Define the polar of a point with respect to a conic, and prove that, if the polar of a point \(P\) passes through a point \(Q\), then the polar of \(Q\) passes through \(P\). Such points are called conjugate. Similarly two lines, each passing through the pole of the other, are called conjugate. \(P_1\) and \(P_2\) are given points and \(S\) is a given conic. Show that the locus of the point of intersection of two lines, one through \(P_1\) and the other through \(P_2\), conjugate with respect to \(S\), is a conic \(S'\) passing through \(P_1\) and \(P_2\). Discuss the form of \(S'\) if (i) \(P_1\) and \(P_2\) are conjugate with respect to \(S\), or (ii) \(P_1P_2\) touches \(S\).
\(LM\) and \(L'M'\) are lines not in the same plane; \(N\) and \(N'\) are points on \(LM\) and \(L'M'\) respectively such that \(LM : MN = L'M' : M'N'\). Prove that \(LL'\), \(MM'\) and \(NN'\) are parallel to a plane. \(L'', M''\) and \(N''\) are points on \(LL'\), \(MM'\) and \(NN'\) respectively such that \[ LL' : L'L'' = MM' : M'M'' = NN' : N'N''. \] Prove that \(L'', M''\) and \(N''\) are collinear.
Prove Simpson's formula \(\frac{1}{3}h (y_0 + 4y_1 + y_2)\) for the area bounded by a curve of the type \(y = A + Bx + Cx^2\), two ordinates of heights \(y_0, y_2\) and the axis \(y=0\), where \(y_1\) is the height of the mid-ordinate and \(h\) is the interval between successive ordinates. To approximate to the area under a curve for which \(y_0=0\) and the tangent at this point of intersection with \(y=0\) is perpendicular to \(y=0\), it is sometimes convenient to fit a curve of the type \(y^2 = x^2(a+bx)\) to the points \((0,0)\), \((h, y_1)\), \((2h, y_2)\). Show that the corresponding formula for the area is \(\frac{2}{15}h (4\sqrt{2} y_1 + y_2)\). Illustrate these rules by finding approximately the area of a quadrant of a circle of radius \(a\). The area is to be divided into strips of breadth \(\frac{1}{4}a\) by lines parallel to a bounding radius; for the two longer strips use Simpson's rule.
Prove that, in the parabola \(y^2 = 4ax\), the length of arc between the vertex and the point where the tangent makes an angle \(\psi\) (\(0 < \psi < \frac{1}{2}\pi\)) with the axis is \[ a \log (\cot \psi + \operatorname{cosec} \psi) + a \cot \psi \operatorname{cosec} \psi. \] The parabola rolls without slipping on a fixed straight line. Prove that its focus describes a curve given by the equations \[ x = a \log (\cot \psi + \operatorname{cosec} \psi), \quad y = a \operatorname{cosec} \psi. \] Prove further that the equation of this curve is \(y = a \cosh (x/a)\).
If \(y = \log_e \{x + \sqrt{(1 + x^2)}\}\), prove that \[ (1 + x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = 0. \] Prove, by induction or otherwise, that \[ (1+x^2) \frac{d^ny}{dx^n} + (2n - 3) x \frac{d^{n-1}y}{dx^{n-1}} + (n-2)^2 \frac{d^{n-2}y}{dx^{n-2}} = 0. \] Hence obtain the series expansion of \(y\) in ascending powers of \(x\).
A particle moves in a medium that resists the motion with a force proportional to the speed. Prove that, if the particle falls vertically under gravity, the speed approaches a limiting value, \(w\) say. Show that, if the particle is projected vertically upwards with a speed \(V\) that is small compared with \(w\), it reaches its starting point again after a time \[ \frac{2V}{g} \left(1 - \frac{1}{3} \frac{V}{w}\right), \] approximately.