Problems

Establish the result for the radius of curvature at any point of a plane curve whose tangential-polar \((p, \psi)\) equation is given. If in such a curve the intercept on any tangent between the point of contact and the foot of the perpendicular from the origin on to the tangent is \(p+a\), where \(a\) is a constant, find the angle between the tangents at the points for which \(p=a\) and \(p=2a\) respectively. Find the radius of curvature at the first of these points.

1. Shew that the coefficient of \(x^{n-1}\) in the expansion in a series of ascending powers of \(x\) of \(\dfrac{\cot\theta-x}{1+(\cot\theta-x)^2}\) is \(\cos n\theta\sin^n\theta\).
2. If \(y=(\tan^{-1}x)^2\), eliminate \(\tan^{-1}x\) between the expressions for \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\). Deduce a linear relation between the values of \(\dfrac{d^ny}{dx^n}, \dfrac{d^{n-2}y}{dx^{n-2}}, \dfrac{d^{n-4}y}{dx^{n-4}}\) when \(x=0\).

If \(x>1\), prove that \begin{align*} x^3+3x+2+6x\log x &> 6x^2, \\ x^4+8x+12x^2\log x &> 8x^3+1. \end{align*}

Find the asymptotes of the curve \(xy^2 = 4(x-a)(x-b)\), where \(b>a>0\). Sketch the curve, and find the area bounded by the curve and the lines \(x=0, x+b=0\).

If \(y = \sin^p x \cos^q x \sqrt{1-k^2\sin^2 x}\) and \(p, q, k\) are constants, find \(\sqrt{1-k^2\sin^2 x}\dfrac{dy}{dx}\). Hence, by taking suitable values for \(p\) and \(q\), express \(I_m = \int_0^{\frac{\pi}{2}} \dfrac{\sin^m x\,dx}{\sqrt{1-k^2\sin^2 x}}\) in terms of \(I_{m-2}\) and \(I_{m-4}\).

If \(r\) denotes distance from a focus of an ellipse, find the mean value of \(r\) with respect to angular distance from the major axis for points on the perimeter of the ellipse. Determine also for the ellipse the mean value of \(r\) with respect to area, stating the result in terms of the eccentricity \(e\) and the semi-latus rectum \(\lambda\).

It is known that the circumcircle of a triangle of tangents to a parabola passes through the focus of the parabola; reciprocate this result with respect to a circle whose centre is on the directrix of the parabola, and hence prove that, if \(A, B, C, D\) are any four points on a rectangular hyperbola, the circle through the feet of the perpendiculars from \(D\) to the sides of the triangle \(ABC\) passes through the centre of the rectangular hyperbola. Deduce that

1. the locus of centres of rectangular hyperbolas through three fixed points is a circle;
2. given four points on a circle the Simson Lines of each point with respect to the triangle formed by the other three points are concurrent.

The homogeneous coordinates \((x, y, z)\) of a point are so chosen that the equation of the line at infinity is \(px+qy+rz=0\) and the equation of the circle with respect to which the triangle of reference \(\Delta\) is self-polar is \(x^2+y^2+z^2=0\); prove that

1. the centroid of \(\Delta\) has coordinates \((\frac{1}{p}, \frac{1}{q}, \frac{1}{r})\),
2. the orthocentre of \(\Delta\) has coordinates \((p, q, r)\),
3. the circumcircle of \(\Delta\) has for its equation \(p(q^2+r^2)yz + q(r^2+p^2)zx + r(p^2+q^2)xy=0\),
4. the circumcentre of \(\Delta\) has coordinates \((\frac{q^2+r^2}{p}, \frac{r^2+p^2}{q}, \frac{p^2+q^2}{r})\),

Define "convergent sequence of real numbers." Prove that, if \(a_n \to a\) and \(b_n \to b\) as \(n\to\infty\), then \(a_n b_n \to ab\). If \(c_n\) is a sequence of positive numbers and if \[\frac{c_{n+1}^2}{c_n^2} \to c^2,\] shew that \[c_n^{1/n} \to |c|.\] If \(a_n < b_n\) and if \(a_n \to a\), \(b_n \to b\), shew that \[a \le b.\]

Shew how the H.C.F. of two polynomials \(f(x)\) and \(g(x)\) may be found without solving the equations \(f(x)=0\) and \(g(x)=0\). If \(d(f)\) denotes "the degree of \(f(x)\)," shew that necessary and sufficient conditions that \(f(x)\) and \(g(x)\) have a common factor are that polynomials \(F(x)\) and \(G(x)\) exist such that \(d(F) \le d(g)-1\), \(d(G) \le d(f)-1\), and \(Ff=Gg\). Hence find the values of \(a\) for which the equation \[x^5+x^2+x+a=0\] has a repeated root.