Problems

Prove that, if \(f(x)\) is a function whose differential coefficient \(f'(x)\) is positive throughout a given interval, then \(f(x_2)>f(x_1)\), if \(x_2>x_1\), where \(x_1, x_2\) are any two values of \(x\) in the interval. \par Prove that \[ x - \frac{x^3}{6} + \frac{x^5}{120} > \sin x \] for all positive values of \(x\), and that \[ \left(1-\frac{x^2}{2}+\frac{x^4}{24}\right)\sin x > \left(x-\frac{x^3}{6}+\frac{x^5}{120}\right)\cos x \] when \(0

(i) If \(u=xyz\), where \(x, y, z\) are connected by the relations \[ yz+zx+xy=a, \quad x+y+z=b \quad (a, b \text{ being constants}), \] prove that \[ \frac{du}{dx} = (x-y)(x-z). \] (ii) If \(\xi, \eta\) are functions of \(x, y\) such that \(\xi=e^x \cos y, \eta=e^x \sin y\), and \(x,y\) are functions of \(r, \theta\) such that \(x=e^r \cos\theta, y=e^r \sin\theta\), where \(r\) is a function of \(\theta\), prove that \[ \frac{d\xi}{d\eta} = \frac{\frac{dr}{d\theta}-\tan(y+\theta)}{1+\frac{dr}{d\theta}\tan(y+\theta)}. \]

If \(y=x(1-x)/(1+x^2)\),

1. [(i)] find the maximum and minimum values of \(y\);
2. [(ii)] find the points of inflexion of the curve;
3. [(iii)] sketch a graph showing clearly the points determined in (i), (ii), and also the position of the curve relative to the line \(y=x\).

(i) Prove that, if \(n\) is a positive integer, \[ \int_0^{\pi/2} e^{\lambda x} \cos nx dx = \frac{1}{\lambda^2+n^2}\{\lambda e^{\lambda\pi/2}-1\}, \] where \(\lambda\) has one of the values \(\pm 1, \pm n\), and classify the cases. \par (ii) Find the area bounded by the parabola \(y^2=ax\) and the circle \(x^2+y^2=2a^2\).

Prove that the circles which circumscribe the four triangles formed by four straight lines have a common point, and that the orthocentres of the triangles are collinear.

The points A, B, C lie on a straight line, and P is a point not on the line. The centres of the circles PBC, PCA and PAB are X, Y and Z. Prove that the four points P, X, Y, Z lie on a circle.

Prove that the equation of the normal at a point on the parabola \(x=am^2, y=2am\) is \(y+mx=2am+am^3\). \par Three normals are drawn to the parabola from the point (X, Y). Prove that if two of them are at right angles the length of the third is \(3\sqrt{(Y^2+a^2)}\).

Prove that if QOQ', ROR' are chords of a conic in fixed directions the ratio QO.OQ' : RO.OR' is constant for all positions of O. \par QOQ', ROR' are normals to an ellipse at Q, R intersecting at O at right angles and meeting the curve again at Q', R'. Prove that QQ', RR' are proportional to the distances of the centre from the tangents at Q and R.

A straight line through a fixed point P cuts a conic in A, B. Prove that the locus of the harmonic conjugate of P with regard to A, B is a straight line, the polar of P. \par C, D are conjugate points on the polar of P. Any line through P cuts the conic in A, B; AD cuts BC in E and AC cuts BD in F. Prove that E, F lie on the conic and that EF goes through P.

``Two conics are inscribed in the same triangle ABC touching BC at the same point. If from any point on BC lines are drawn touching the conics at P, Q, then PQ passes through A.'' \par State the dual theorem and prove either the theorem or its dual.