Problems

The determinant \(D_n\), with \(n\) rows and columns, has elements as follows: $$d_{r,r} = a, \quad d_{r,r+1} = +1, \quad d_{r+1,r} = -1 \quad \text{(all } r\text{),}$$ other elements zero. Find a recurrence relation connecting \(D_n\), \(D_{n-1}\) and \(D_{n-2}\). Hence show that, for even \(n\), \(D_n\) has the value \(\cosh(n+1)\theta/\cosh\theta\), where \(\theta = \sinh^{-1}\frac{1}{2}a\). Determine the value of \(D_n\) for odd \(n\).

1. [(i)] The variables \(x\) and \(y\) are connected by the equation \(f(x, y) = 0\). Calculate \(dy/dx\) and \(d^2y/dx^2\) in terms of the partial derivatives of \(f\) with respect to \(x\) and \(y\).
2. [(ii)] The equation \(\phi(x, y, z) = 0\) can be regarded as defining \(x\) as a function of \(y\) and \(z\), or \(y\) as a function of \(x\) and \(z\), or \(z\) as a function of \(x\) and \(y\). Prove that $$\left(\frac{\partial x}{\partial y}\right)_z \cdot \left(\frac{\partial y}{\partial z}\right)_x \cdot \left(\frac{\partial z}{\partial x}\right)_y = -1,$$ where \((\partial x/\partial y)_z\) for instance means the partial derivative of \(x\) with respect to \(y\), keeping \(z\) constant.

Let \(f(x, y, a, b, c) = 0\) be the equation of a circle having its centre at \((a, b)\) and radius \(c\). Regarding this equation as defining \(y\) as a function of \(x\), find the values for arbitrary values of \(a\), \(b\), and \(c\) obtained by the second-order condition satisfied by such functions. Give the equation satisfied by \(x\) regarded as a function of \(y\), \(a\), \(b\) and \(c\) under the same conditions.

If \(A\) is a fixed point of a conic \(S\) and \(B\) any other fixed point in the plane, show that the locus of a point \(P\) such that \(PA\) and \(PB\) are conjugate lines with respect to \(S\) is also a conic \(S'\). Name any special points through which \(S'\) passes and any special lines it touches.

A quadrilateral has sides \(ABC\), \(AB'C'\), \(A'BC'\) and diagonal lines \(A'B'C'\), \(A'B'C\) and \(XYM\). By considering the triangles \(A'B'C'\) and \(XYM\) prove that \(LA'\), \(MB\) and \(NC'\) are concurrent. Deduce that there is a quadrangle having as sides the six lines \(LA\), \(LA'\), \(MB\), \(MB'\), \(NC\), \(NC'\). Identify its diagonal point-triangle.

If the probability that an event occurs in a single trial is \(p\), show that the probability that it occurs exactly \(r\) times in \(n\) trials is equal to the term containing \(p^r q^{n-r}\) in the binomial expansion of \((p + q)^n\), where \(q = 1 - p\). Calculate \(m\), the mean value of \(r\), and also the mean value of \((r - m)^2\).

\(ABC\) is a triangle and \(O\) any point, not necessarily in its plane. The points \(L\), \(M\), \(N\) divide the sides \(BC\), \(CA\), \(AB\) in the ratio \(\lambda : 1\), \(\mu : 1\), \(\nu : 1\) respectively. The vector \(\overrightarrow{OA}\) (the position vector of \(A\)) is denoted by \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are similarly defined. Express in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) the position vectors of \(L\), \(M\) and \(N\). Show that, if \(\lambda\mu\nu = 1\), the lines \(AL\), \(BM\) and \(CN\) are concurrent, and state the position vector of their point of intersection.

In order to steer a car, the short axles carrying the front wheels are turned about vertical pins at \(A\) and \(B\) through angles \(\alpha\) and \(\beta\). If the curvature of the path of the mid-point of \(CD\) is \(\kappa\), find approximate expressions for the values that \(\alpha\) and \(\beta\) should have to avoid side-slip, neglecting \(\kappa^3\) and higher powers. (\(AC = a\), \(CD = AB = b\).) Two arms \(AE\) and \(BF\) of length \(c\), making a fixed angle \(\theta\) with the front axles, are connected by a bar \(EF\) (not drawn). Find the best angle to choose for \(\theta\), given that \(c^2\) is small enough to be neglected.

A light strut of length \(a\) is freely pivoted at one end \(A\), and the other end \(B\) carries a light small pulley. A light rope passes over the pulley; one end is fixed at a height \(h\) vertically above \(A\), and the other end carries a weight. Discuss the positions of equilibrium and their stability.

A uniform cylinder of mass \(m\) and radius \(a\) is hung from a fixed point by a very long light string fastened to a point on it. The cylinder is released from rest with the string wound half a turn round it, as in the left-hand diagram, and descends with its axis remaining horizontal and parallel to its original position. What points in the cylinder have zero velocity when it has reached the position shown in the second diagram? Find the angular velocity and the tension in the string when the axis reaches its lowest point.