Problems

If \(y = \frac{x^4+x^2-12}{x^4-4}\), determine the range of values possible for \(y\) when \(x\) is real. Sketch the graph of the function.

A solid in the form of a ring is generated by rotating a plane area possessing an axis of symmetry about an axis in its plane parallel to the axis of symmetry and not cutting the boundary of the area. Show that the radius of gyration \(\kappa\) of the ring about the axis of rotation is given by \(\kappa^2 = c^2 + \lambda^2\), where \(c\) is the distance of the centroid of the area from the axis of rotation, and \(\lambda\) is the radius of gyration of the area about its axis of symmetry. (Note: The question states \(\kappa^2=c^2+3\lambda^2\). The '3' is faint. Standard theorem is \(I = I_{cm} + Md^2\), so \(MK^2=M\lambda^2+Mc^2\), giving \(K^2=c^2+\lambda^2\). Let's stick with the scan.) Scan shows \(\kappa^2=c^2+3\lambda^2\). If the area is not symmetrical, show that for a uniform body having the same shape as the solid, the Moment of Inertia differs from \(MK^2\) by a term which is independent of \(c\).

1. Evaluate \[ \int_0^{2\pi} \frac{\cos(n-1)x - \cos nx}{1-\cos x} \, dx, \] where \(n\) is a positive integer.
2. Find the limit of \(\frac{1}{x^2}-\cot^2 x\), as \(x \to 0\).

A plane quadrilateral is formed by the four straight lines \(l_i\) (\(i=1,2,3,4\)), and the point of intersection of \(l_i, l_j\) is denoted by \(A_{ij}\). Prove that the circumcircles of the four triangles formed by sets of three of the lines have a common point. Prove also that the orthocentre \(O_1\) of the triangle with vertices at \(A_{23}, A_{34}, A_{42}\) is on the radical axis of the circles whose diameters are \(A_{14}A_{23}\) and \(A_{12}A_{34}\). Deduce that

1. the orthocentres of the four triangles formed by threes of the lines are collinear;
2. the middle points of the three diagonals of the quadrilateral are collinear.

The equation of a conic, in general homogeneous coordinates, is \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \quad (1) \] A variable line through a fixed point \(O\), whose coordinates are \((x',y',z')\), meets the conic in \(P, P'\). Prove that the locus of the harmonic conjugate of \(O\) with respect to \(P, P'\) is a line \(\Lambda\) whose equation is \[ (ax'+hy'+gz')x + (hx'+by'+fz')y + (gx'+fy'+cz')z = 0. \] Shew that, if \(O\) lies on the conic, then \(\Lambda\) passes through \(O\), and conversely. Deduce that the condition that the line \(lx+my+nz=0\) should touch the conic is \[ \begin{vmatrix} a & h & g & l \\ h & b & f & m \\ g & f & c & n \\ l & m & n & 0 \end{vmatrix} = 0. \] Interpret this equation geometrically when the left-hand side of equation (1) is (i) the product of two linear factors, (ii) a perfect square.

If \(\alpha_1, \alpha_2, \dots, \alpha_n\) are the roots of the equation \[ f(x) = x^n + a_1 x^{n-1} + \dots + a_n = 0, \] prove that \[ \frac{f(x)}{x-\alpha_1} + \frac{f(x)}{x-\alpha_2} + \dots + \frac{f(x)}{x-\alpha_n} = nx^{n-1} + (n-1)a_1x^{n-2} + \dots + a_{n-1}. \] If \(s_k\) denotes the sum of the \(k\)th powers of the roots, prove the relation \[ s_k + a_1 s_{k-1} + a_2 s_{k-2} + \dots + a_{k-1}s_1 + ka_k = 0 \quad (k=1, \dots, n). \] Obtain a similar relation for \(k>n\) and shew that, if \(k>n\), \[ \begin{vmatrix} s_k & s_{k-1} & \dots & s_{k-n} \\ s_{k+1} & s_k & \dots & s_{k-n+1} \\ \vdots & \vdots & \ddots & \vdots \\ s_{k+n} & s_{k+n-1} & \dots & s_k \end{vmatrix} = 0. \] Find the value of \(s_k\) for the equation \[ x^n + x^{n-1} + \dots + x + 1 = 0. \]

If \(x > 1\) and \(m\) is a positive integer greater than 1, prove that \[ \frac{x^m-1}{m} - \frac{x^{m-1}-1}{m-1} > 0, \] and hence that \(x^m - 1 > m(x-1)\). Generalise this result to the case in which \(m\) is a rational number greater than 1. Prove that, if \(\alpha, \beta\) are positive rational numbers whose sum is 1, and \(a,b\) are positive, then \[ a^\alpha b^\beta < \alpha a + \beta b, \] unless \(a=b\).

Explain what is meant by the statement \[ \phi(n) \to a \text{ as } n \to \infty, \] where \(n\) is a positive integer. Prove that, if \(\phi(n) \to a\) and \(\psi(n) \to b\) as \(n \to \infty\), then \[ \phi(n)\psi(n) \to ab. \] Determine the behaviour, as \(n \to \infty\), of two of the following:

1. \(n^r x^n\), where \(r\) is an integer, positive, negative or zero;
2. \(\sqrt[n]{x}\), where \(x\) is positive;
3. \(\sqrt[n]{n}\).

If \(Q_n(x) = (1+x^2)^{\frac{n}{2}+1} \frac{d^n y}{dx^n}\), where \(y=\frac{1}{\sqrt{(1+x^2)}}\), prove that \(Q_n(x)\) is a polynomial of degree \(n\) satisfying the following relations:

1. \(Q_{n+1} = (1+x^2)Q_n' - (2n+1)xQ_n\);
2. \(Q_{n+1} + (2n+1)xQ_n + n^2(1+x^2)Q_{n-1} = 0\);
3. \(Q_n' + n^2 Q_{n-1} = 0\);
4. \((1+x^2)Q_n'' - (2n-1)xQ_n' + n^2 Q_n = 0\).

Shew from first principles that necessary and sufficient conditions of equilibrium of a system of co-planar forces are that the sums of the resolved parts of the forces in any two distinct directions should separately vanish, and that the sum of the moments of the forces about any point in the plane should vanish. If the moments of a coplanar system about two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are respectively \(G_1\) and \(G_2\), and if the resolved part along \(AB\) of the resultant of the system is \(F\), shew that the ordinate of the point in which the resultant cuts the line \(x=0\) is \[ \begin{vmatrix} y_1 & x_1 & G_1 \\ y_2 & x_2 & G_2 \\ (x_2-x_1) & y_2-y_1 & -Fl \end{vmatrix} \div \begin{vmatrix} x_2-x_1 & G_2-G_1 \\ y_2-y_1 & -Fl \end{vmatrix} \] where \(l=AB\).