Problems

\(AB\) is a fixed chord of a circle, and \(KL\) is a variable chord of fixed length; \(AK\) and \(BL\) intersect at \(P\). Prove that \(P\) lies on one or other of two fixed circles.

Assuming that the coefficients in the Cartesian equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] are all real, \(a,h,b\) not being all zero, and that imaginary points are ignored, prove that the equation represents either (1) an ellipse, or (2) a hyperbola, or (3) a parabola, or (4) two intersecting straight lines, or (5) one point, or (6) two parallel straight lines, or (7) one straight line. Prove also that there are two further cases in which the equation represents no real point. Give an example (the simpler the better) of each of these nine cases; and state clearly, in tabular form or otherwise, in terms of the coefficients \(a,h,b\dots\), necessary and sufficient conditions for the occurrence of each of the nine cases.

Four light rods, similar in all respects, are hinged together to form a rhombus \(ABCD\), and \(AC, BD\) are joined by elastic strings of natural length \(AB\). The rhombus is suspended at \(A\), and a weight is attached to \(C\) equal to a third of the weight which would stretch either string to twice its natural length. Shew that in equilibrium \(3AC=4BD\).

Shew that if a triangle be self-polar with regard to a rectangular hyperbola its in- and ex-centres lie on the hyperbola.

Prove that \[ e + \frac{2}{e} = \sum_{n=0}^{\infty} \frac{5n+1}{2n+1}. \]

Give an account of the principal properties of determinants, and indicate their application to the solution of simultaneous linear equations.

Two uniform heavy rods \(AB, AC\), each of length \(2a\), are rigidly connected at \(A\) at right angles to each other, and rest on a fixed rough cylinder of radius \(c\), the plane of the rods being perpendicular to the axis of the cylinder. Shew that, if \(c < \frac{a}{\sqrt{2}}\), \(\tan\epsilon\) is the coefficient of friction between the rods and the cylinder, and \(\alpha\) is the angle which the bisector of the angle \(BAC\) makes with the vertical in limiting equilibrium, then \[ c\sin(\alpha+2\epsilon)=(a-c)\sin\alpha. \]

\(l_ix + m_iy + n_i = 0\), (\(i=1,2,3\)), are the equations of three lines. \(N_i\) is the cofactor of \(n_i\) in \[ \begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}. \] Prove that the necessary and sufficient condition for the origin to lie in an acute angle between the lines \(i=1,2\), is that \[ (l_1l_2 + m_1m_2)n_1n_2 < 0; \] and that for the origin to lie inside the triangle formed by the three lines, the necessary and sufficient conditions are that \(n_1N_1, n_2N_2, n_3N_3\) have the same sign.

If \(\alpha, \beta, \gamma\) are the roots of \[ x^3 - 6x^2 + 18x - 36 = 0, \] prove that \begin{align*} \alpha^2 + \beta^2 + \gamma^2 &= 0, \\ \alpha^3 + \beta^3 + \gamma^3 &= 0. \end{align*}

The functions \(f\) and \(\phi\) are supposed to have as many derivatives as may be required over the ranges considered. Shew that, if \(\phi\) is subject to the conditions \[ \phi'(0)=0, \quad \phi(h) = \phi'(h) = \phi''(h) = 0, \quad \dots\dots\dots\dots\dots(1) \] then \[ \int_0^h F(t)\phi''''(t)dt = F(h)\phi'''(h) - F(0)\phi'''(0) + \int_0^h F''(t)\phi''(t)dt, \] where \[ F(t)=f(c+t)+f(c-t), \] \(c\) and \(h\) being real constants and \(h\) positive. Shew that there is a polynomial in \(t\) of the fourth degree satisfying (1), and deduce the formula \[ \int_{c-h}^{c+h} f(x)dx = \frac{h}{3}[f(c-h)+4f(c)+f(c+h)] - \frac{1}{72}\int_0^h (h-t)^3(h+3t)\{f''''(c+t)+f''''(c-t)\}dt. \] Deduce that, if \(f\) is a polynomial of the third or lower degree, the value (\(\alpha\)) of the integral on the left is given exactly by the first term (\(\alpha'\)) on the right, and that if \(f\) is a general function whose fourth derivative satisfies the inequalities \[ l < f''''(x) < L, \quad (c-h \le x \le c+h)\dots\dots(2) \] then limits to the error involved in taking \(\alpha'\) as an approximation to \(\alpha\) are given by \[ \frac{1}{90}h^5l \le \alpha'-\alpha \le \frac{1}{90}h^5L. \] Deduce that, if we approximate to the definite integral \[ A = \int_a^b f(x)dx, \quad (b>a) \] by Simpson's rule, i.e. by dividing \((a,b)\) into \(2n\) parts each of length \(h\) (\(2nh=b-a\)), by points \[ a=x_0, x_1, \dots, x_{2n}=b, \] and taking as an approximation the sum \[ A' = \frac{h}{3}[y_0+y_{2n}+2(y_2+y_4+\dots+y_{2n-2})+4(y_1+y_3+\dots+y_{2n-1})], \] where \(y_r = f(x_r)\), then limits to the error will be given by \[ \frac{1}{180}(b-a)h^4l \le A'-A \le \frac{1}{180}(b-a)h^4L, \] provided that (2) is satisfied throughout the interval \(a \le x \le b\).