A thin-walled cylindrical tube of radius \(a\) and weight \(W_1\) stands with its axis vertical on a smooth horizontal plane. Within the tube is placed a solid rod of weight \(W_2\) and of total length \(2(l+b)\) which exceeds \(2a\), formed of a cylinder of length \(2l\) and radius \(b\) terminating in hemispherical ends. If the coefficient of friction between the rod and the tube be \(\mu\) and the rod does not project above the top of the tube, shew that the system will overturn if \[ W_2\cos\theta(l\sin\theta - \mu a) > W_1a(\sin\theta + \mu\cos\theta), \text{ where } \theta = \cos^{-1}\frac{a-b}{l}. \]
Shew that the area of a segment of a circle of radius \(r\) cut off by a chord of length \(2c\), where \(c/r\) is small, is approximately \[ \frac{2}{3}\frac{c^3}{r} + \frac{1}{5}\frac{c^5}{r^3}. \]
Prove that, if \(\alpha + \beta + \gamma = 360^\circ\), \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma - 2\cos\alpha\cos\beta\cos\gamma = 1. \] If the cosines of the angles of a plane quadrilateral are \(c_1, c_2, c_3, c_4\), prove that \[ \Sigma c_1^4 - 2\Sigma c_1^2 c_2^2 + 4 c_1^2 c_2^2 c_3^2 - 4c_1c_2c_3c_4 (\Sigma c_1^2 - 2) = 0. \]
Prove that a variable conic through four fixed points meets a fixed line in pairs of points in involution. Five points \(A, B, H, K, P\) are given on a line \(l\). In a plane through \(l\), arbitrary lines \(a, b, p\) are drawn through \(A, B, P\) respectively; \(p\) meets \(a,b\) in \(X, Y\). Through \(X, Y, H, K\) is drawn an arbitrary conic meeting \(a, b\) again in \(U, V\). Shew that \(UV\) meets \(l\) in a fixed point determined completely by the series of points \(A, B, H, K, P\).
\(A\) and \(B\) are two pegs on the same horizontal and at distance \(d\) apart. A square picture frame of side \(d\) is supported by elastic strings \(AA'\), \(BB'\) attached to the upper corners of the frame, and it hangs in equilibrium with both strings vertical. It is pulled down vertically and then released, and it is found that it oscillates vertically without rotation. Examine whether either of the following statements is necessarily true:
A chord of the curve \(y=f(x)\), parallel and near to the tangent at the point \(P(\xi, \eta)\), meets the curve at \(Q\) and \(R\) near to \(P\). Prove that the gradient of the line joining \(P\) to the middle point of \(QR\) is approximately equal to \[ f'(\xi) - \frac{1}{6} \frac{f'''(\xi)}{f''(\xi)}. \] Note: The formula provided in the exam paper is extremely hard to read. The transcription is based on a standard result for this problem which may not match the original paper exactly.
Prove that \[ \cos\alpha + \cos(\alpha+\beta) + \cos(\alpha+2\beta) + \dots + \cos\{\alpha+(n-1)\beta\} = \cos\left(\alpha + \frac{n-1}{2}\beta\right) \sin\frac{n\beta}{2} \text{cosec}\frac{\beta}{2}. \] Shew that, if \(\alpha = \frac{\pi}{17}\), then \[ 16 \cos3\alpha \cos5\alpha \cos7\alpha \cos11\alpha = -1. \]
Prove that two conics which intersect in four distinct points have one and only one common self-polar triangle. Two parabolas touch the sides of a triangle \(ABC\) and intersect one another in \(P, Q, R, S\). Prove that the line joining any two of the points \(P, Q, R, S\) passes through one of the vertices of the triangle formed by the lines through the vertices of \(ABC\) parallel to the opposite sides.
A solid cylinder of weight \(w\) and of radius \(R\) rests with its axis vertical on a rough horizontal plane. If the coefficient of friction between the surfaces in contact be \(\mu\), shew that the couple required to rotate the cylinder about its axis is \(\frac{2}{3}\mu w R\), assuming that the normal pressure on the base of the cylinder is uniformly distributed over the area of the base. Find also what couple will be required if it be assumed that the normal pressure per unit area of the base varies inversely as the distance from the axis of the cylinder.
Find \[ \lim_{x\to 0} \frac{(1+x)^{1/x}-e}{x}. \]