Problems

A heavy sphere rests on a rough plane inclined at an angle \(\theta\) to the horizontal. The sphere is to be kept from rolling down the plane by a rough cubical block, of negligible weight and edge less than the radius of the sphere, resting on the plane, the edge of the block in contact with the sphere being horizontal. The angles of friction between the block and the plane and between the block and the sphere are \(\lambda_1\) and \(\lambda_2\) respectively, both angles being less than \(45^\circ\). If the points of contact of the sphere with the block and with the plane are \(P_1\) and \(P_2\), shew that the size of the block must be limited by the relation \(\lambda_1 + \lambda_2 \gtreqless \phi \gtreqless \theta\), where \(\phi\) is the angle subtended by \(P_1P_2\) at the centre of the sphere. Explain the significance of the two limits for \(\phi\).

Prove that if a hexagon is inscribed in a conic the points of intersection of pairs of opposite sides are collinear. A conic passes through a given point \(A\) and touches two given lines \(b, c\) at given points \(B, C\). Shew how to construct the tangent at \(A\), and the point where any given line through \(A\) meets the conic again.

Shew that, if \(c_r\) is the coefficient of \(x^r\) in the expansion of \((1+x)^n\), where \(n\) is a positive integer, \[ c_0^2 + c_1^2 + c_2^2 + \dots + c_n^2 = \frac{(2n)!}{(n!)^2}. \] Find the value of \(c_0^2 - c_1^2 + c_2^2 - \dots + c_n^2\), where \(n\) is even.

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 \[ \frac{2}{3}\frac{c^3}{r} + \frac{1}{5}\frac{c^5}{r^3}. \]

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 \[ W_2\cos\theta(l\sin\theta - \mu a) > W_1a(\sin\theta + \mu\cos\theta), \text{ where } \theta = \cos^{-1}\frac{a-b}{l}. \]

Prove that the reciprocal of a circle with respect to a point \(S\) is a conic with one focus at \(S\). Determine what relations between \(S\) and the circle decide whether the conic is an ellipse, parabola, or hyperbola. Given four points \(S, A, B, C\), prove that in general four conics may be drawn through \(A, B, C\) having \(S\) as focus; and that three of them are hyperbolas while the remaining one may be a hyperbola, parabola, or ellipse.

Prove that, if \(n\) is a positive integer, the number of solutions of the equation \(x + 2y + 3z = 6n\), for which \(x, y, z\) are positive integers or zero, is \(3n^2 + 3n + 1\). Find the corresponding number of solutions of the equation \(x + 2y + 3z = 6n + 1\).

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.

\(A\) 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:

1. [(a)] The centre of gravity of the frame lies on the vertical line bisecting \(AB\).
2. [(b)] The strings have the same natural length.

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 \(A, B, H, K, P\).