By factorizing the determinant, or otherwise, show that \[ \begin{vmatrix} x & y & z & u \\ u & x & y & z \\ z & u & x & y \\ y & z & u & x \end{vmatrix} = (x^2+z^2-2yu)^2 - (u^2+y^2-2zx)^2. \] Express \[ \{(x^2+z^2 - 2yu)^2 - (u^2 + y^2 - 2zx)^2\} \{(X^2 + Z^2 - 2YU)^2 - (U^2 + Y^2 - 2ZX)^2\} \] in the form \((A^2+C^2-2BD)^2 - (D^2+B^2-2CA)^2\), giving explicit expressions for \(A, B, C, D\) in terms of \(x, y, z, u\) and \(X, Y, Z, U\).
Let \(N(n)\) denote, for any given integer \(n\) (positive, zero, or negative) the number of solutions of the equation \[ x+2y+3z=n \] in non-negative integers \(x, y, z\) (so that \(N(n)=0\) for \(n<0\), \(N(0)=1\), \(N(1)=1\), \(N(2)=2\), etc.). By considering the coefficient of \(t^n\) in the expansion of \[ \frac{1-t^6}{(1-t)(1-t^2)(1-t^3)} \] in ascending powers of \(t\), or otherwise, prove that \[ N(n) - N(n-6) = n \quad (n>0), \] and write down the corresponding formula for \(n=0\). Defining the integers \(q, r\) by \[ n = 6q+r \quad (0 \le r < 6), \] obtain an expression (or expressions) for \(N(n)\) (\(n \ge 0\)) in terms of \(n\) and \(r\). Show that, for every \(n \ge 0\), \(N(n)\) is the integer nearest to \(\frac{1}{12}(n+3)^2\).
Prove the following inequalities:
If \(z = \frac{y}{x} f(x+y)\) and subscripts denote partial differentiations, show that \begin{align*} xz_x + yz_y &= \frac{y}{x}(x+y)f'(x+y); \\ x^2z_{xx} + 2xyz_{xy} + y^2z_{yy} &= \frac{y}{x}(x+y)^2f''(x+y), \end{align*} in which \(f'(t)\) stands for \(df(t)/dt\), and so on. Find also the value of \[ x^3z_{xxx} + 3x^2yz_{xxy} + 3xy^2z_{xyy} + y^3z_{yyy}. \]
A conic \(K\) touches four straight lines \(a, b, c, d\) at \(A, B, C, D,\) respectively. Prove that there is a conic \(S\) through the six points \(A, B, C, D, ab, cd\) (where \(ab\) is the intersection of \(a\) and \(b\)), and a conic \(\Sigma\) touching the six lines \(a, b, c, d, AB, CD\). Show that the tangents to \(K\) at its four points of intersection with \(\Sigma\) touch \(S\).
Show that the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] should represent two straight lines is \[ \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0. \] Show further that necessary conditions that these lines should be real are \(h^2 \ge ab\); \(f^2 \ge bc\); \(g^2 \ge ca\). If \(\Delta=0\), prove that the point of intersection of the lines is \[ (hf-bg)/(ab-h^2); \quad (gh-af)/(ab-h^2). \]
If \(Q=ax^2+2bx+c\), and \[ I_n = \int \frac{dx}{Q^{n+1}}, \] show by differentiating \((Ax+B)/Q^n\) (where \(A, B\) are adjustable constants), or otherwise, that \[ 2n(ac-b^2)I_n = \frac{ax+b}{Q^n} + (2n-1)aI_{n-1}. \] Obtain a similar formula of reduction for \[ J_n = \int \frac{x\,dx}{Q^{n+1}}. \] Evaluate \[ \int_0^1 \frac{dx}{(x^2-x+1)^3}. \]
From the parallelogram of forces show that, if two couples acting in a plane are in equilibrium, their moments are equal and opposite. Show, conversely, that two co-planar couples of equal and opposite moment are in equilibrium. A force acting in a plane has moments \(M_1, M_2, M_3\) about points whose coordinates referred to axes \(Ox, Oy\) in the plane are \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) respectively. Show that the equation of the line of action of the force is \[ x \begin{vmatrix} 1 & y_1 & M_1 \\ 1 & y_2 & M_2 \\ 1 & y_3 & M_3 \end{vmatrix} + y \begin{vmatrix} x_1 & 1 & M_1 \\ x_2 & 1 & M_2 \\ x_3 & 1 & M_3 \end{vmatrix} = \begin{vmatrix} x_1 & y_1 & M_1 \\ x_2 & y_2 & M_2 \\ x_3 & y_3 & M_3 \end{vmatrix}. \]
A light inelastic string, of length \(2l\), is fixed at its upper end; it carries a particle of mass \(m\) at its mid-point and a particle of mass \(M\) at its lower end. The particles move in a vertical plane so that the upper and lower portions of the string make angles \(\theta\) and \(\phi\) respectively with the vertical, and on the same side of it. If the angular displacements \(\theta, \phi\) are small, write down the equations of motion of \(m\) and \(M\), neglecting quantities of order higher than the first. Show that solutions of these equations can be obtained by assuming that \(\phi=k\theta\), where \(k\) is a constant. In particular, describe the corresponding motions when \(m=3M\).
Justify the rule for writing down the equations of motion of a rigid lamina in a plane (sometimes referred to as ``the principle of the independence of the motions of translation and rotation''). A uniform straight rod, of mass \(M\) and length \(2l\), moves in a vertical plane, with the lower end of the rod sliding on a horizontal table. At any instant the rod makes an acute angle \(\theta\) with the vertical. Find the vertical acceleration of the mass-centre in terms of \(\theta\) and its time-derivatives. If the table is smooth and the rod is released from rest when \(\theta=\beta\), find the initial angular acceleration of the rod and the initial value of the reaction between the rod and the table.