Problems

Find for what ranges of values of \(\theta\) between \(0\) and \(\pi\) each of the following inequalities is satisfied:

1. [(i)] \(\tan 3\theta > 3\tan\theta\);
2. [(ii)] \(\cos 5\theta > 16\cos^5\theta\).

If \(y = a + x \log y\), where \(x\) is small, prove that \(y\) is approximately equal to \[ a + x\log a + \frac{x^2}{a}\log a \] and obtain the term in \(x^3\) in the expansion.

Find the limits, as \(x\) tends to \(\frac{1}{2}\), of the following expressions:

1. [(i)] \(\displaystyle \frac{(1-x)^m - x^m}{(1-x)^n - x^n}\), where \(m, n\) are positive integers;
2. [(ii)] \(\displaystyle \frac{(x-4x^4)^{\frac{1}{3}} - (\frac{1}{2})^\frac{1}{3}}{1-(8x^3)^{\frac{1}{2}}}\).

Prove that, if \(x\) is positive, \[ \frac{2x}{2+x} < \log(1+x) < x. \] Prove also that, if \(a\) and \(h\) are positive, \[ \log(a+\theta h) - \log a - \theta\{\log(a+h) - \log a\}, \] considered as a function of \(\theta\), has a maximum for a value of \(\theta\) between \(0\) and \(\frac{1}{2}\).

Prove that, if \(r_1, r_2\) denote the distances from two fixed points \(O_1, O_2\), of a variable point \(P\) in a fixed plane through \(O_1O_2\), and \(\theta_1, \theta_2\) the angles \(PO_1O_2, \pi - PO_2O_1\), then curves of the two families \begin{align*} r_1^m r_2^n &= \lambda, \\ m\theta_1 + n\theta_2 &= \mu, \end{align*} where \(m\) and \(n\) are constants and \(\lambda, \mu\) are variable parameters, cut orthogonally.

A function \(f(x)\) is defined, for \(x \ge 0\), by \[ f(x) = \int_{-1}^1 \frac{dt}{\sqrt{\{1-2xt+x^2\}}}. \] Prove that, if \(0 \le x \le 1\), \(f(x)=2\). What is the value of \(f(x)\) if \(x > 1\)? Has the function \(f(x)\) a differential coefficient for \(x=1\)?

Find the asymptotes of the curve \[ (x+y-1)^3 = x^3+y^3, \] and prove that they meet the curve only at infinity. Prove also that there is no point on the curve for values of \(x\) between \(a\) and \(1\), where \(a\) is the real root of the equation \[ 3a^3+3a^2-3a+1=0. \] Give a sketch showing the general form of the curve.

State the theorems of Ceva and Menelaus and prove one of them together with its converse. Corresponding pairs of sides of the triangles \(ABC, PQR\) intersect in points \(A', B', C'\) (i.e. \(BC, QR\) intersect in \(A'\)). If \(AA', BB', CC'\) are concurrent, shew that a sufficient condition for the concurrence of \(PA', QB', RC'\) is that the intersections of \(BQ, CR\), and of \(BR, CQ\), shall lie in \(B'C'\).

Prove that a circle \(C\) will invert into a circle \(C'\) or a straight line, and that two points inverse with respect to \(C\) will invert into points inverse with respect to \(C'\). What happens when the inverse curve is a straight line? Shew that in general eight circles can be drawn to touch three given coplanar circles having real points of intersection.

Find the condition that the two pairs of straight lines represented by the equations \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] shall be harmonic conjugates with respect to each other. Hence deduce the equation of the bisectors of the angles between the first pair. Find the equations of the principal axes of the conic \[ 13x^2+37y^2-32xy-14x-34y-35=0. \]