Problems

Prove that if \(y^3+3ax^2+x^3=0\), then \[ \frac{d^2y}{dx^2} + \frac{2a^2x^2}{y^5} = 0. \] Shew that the curve given by the above equation is everywhere concave to the axis of \(x\), and that there is a point of inflexion where \(x=-3a\).

Shew that the locus of the intersections of pairs of tangents to the curve \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta) \] which are at right angles to one another is the curve \[ x=a\{\theta+\frac{1}{2}\pi(1+\cos\theta)\}, \quad y=\frac{1}{2}\pi a \sin\theta. \] Draw both curves.

Evaluate \[ \int_0^{\frac{1}{2}\pi} \cos^3 x dx, \quad \int_0^{\frac{1}{4}\pi} \frac{dx}{3+2\cos x}, \quad \int_0^1 \frac{x+2}{(4+x^2)(1-x)}dx. \]

Find the area between the curve \[ y^2(3a-x)=x^3 \] and its asymptote.

A roof whose slope is inclined at \(30^\circ\) to the horizontal runs into another roof whose slope is inclined at \(45^\circ\), their horizontal ridge lines being inclined to one another at an angle of \(60^\circ\). Find the inclination to the horizontal of the line of intersection of the two slopes.

Points \(P\) and \(Q\) are taken upon two opposite sides \(AB\), \(CD\) of a square \(ABCD\). Shew that, if the diagonals of the trapezium \(APQD\) meet in \(X\) and those of \(BPQC\) meet in \(Y\), \(XY\) will pass through the centre of the square.

A smooth sphere is suspended from a fixed point by a string of length equal to its radius. To the same point a second string is attached which after passing over the sphere supports a weight equal to that of the sphere. Shew that the string which supports the sphere makes an angle \(\sin^{-1}(\frac{1}{4})\) with the vertical.

From a point \(P\) outside a circle two lines \(PAB, PDC\) are drawn, cutting the circle at \(A, B, C, D\). Prove that \(PA.PB\) is equal to \(PC.PD\). \par If \(AC, DB\) meet in \(Q\), show that the circles \(PBD, PAC, QAB, QCD\) all pass through a point which lies on \(PQ\).

Prove that the curve in which a right circular cone is cut by a plane possesses the following properties of a conic:

1. [(i)] the focus-directrix property, \(SP=e.PM\),
2. [(ii)] the focal property, \(SP+S'P = \text{constant}\),
3. [(iii)] the property that \(PN^2\) is proportional to \(AN.NA'\), where \(PN\) is the perpendicular from \(P\) upon the major axis \(AA'\).

Prove that the locus of the vertices of right circular cones, which have a given ellipse as a plane section, is a hyperbola whose foci lie at the ends of the major axis of the ellipse. \par A number of tetrahedra have the same base \(ABC\), and the point where the inscribed sphere touches the face \(ABC\) is the same for all. Prove that the point where the sphere escribed to \(ABC\) touches the face \(ABC\) is also the same for all the tetrahedra.

A sphere is divided by two parallel planes into three portions of equal volume; find to three places of decimals the ratio of the thickness of the middle portion to the diameter of the sphere.