Problems

(i) If the remainders when a polynomial \(f(x)\) is divided by \((x-a)(x-b)\) and by \((x-a)(x-c)\) are the same, shew that \[ (b-c)f(a) + (c-a)f(b) + (a-b)f(c) = 0. \] (ii) When \(x\) and \(y\) are eliminated from the equations \[ x^2-y^2 = ax-by; \quad 4xy=bx+ay; \quad x^2+y^2=1, \] prove that \[ (a+b)^{\frac{2}{3}} + (a-b)^{\frac{2}{3}} = 2. \]

In the series \(u_0+u_1x+u_2x^2+\dots\) any three successive coefficients are connected by the relation \[ u_{r+1} + pu_r + qu_{r-1}=0; \] shew how to find the sum to \(n\) terms. Assuming that the series \[ 2+\frac{7}{5}x + \frac{91}{125}x^2 + \dots \] is of this type, find the \(n\)th term and the sum to infinity.

With the usual notation for the radii of the inscribed and escribed circles of the triangle \(ABC\), prove that

1. [(i)] \(a = \frac{\Delta}{r} - \frac{\Delta}{r_1}\),
2. [(ii)] \(\frac{r_1}{a}\cos^2\frac{A}{2} = \frac{r_2}{b}\cos^2\frac{B}{2} = \frac{r_3}{c}\cos^2\frac{C}{2} = \cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\).

If the radius of the escribed circle which touches \(BC\) is equal to the radius of the circumcircle, prove that \[ \cos A = \cos B + \cos C. \]

Prove that \[ \sin(\alpha+\beta)+\sin(\alpha+2\beta)+\dots+\sin(\alpha+n\beta) = \frac{\sin(\alpha+\frac{n+1}{2}\beta)\sin\frac{n\beta}{2}}{\sin\frac{\beta}{2}} \] and deduce the sum of \[ \sin\theta - \sin2\theta + \sin3\theta - \dots - \sin 2r\theta. \] Shew also that \[ \cos^2x + \cos^22x + \dots + \cos^2nx = \frac{2n-1}{4} + \frac{\sin(2n+1)x}{4\sin x}. \]

The side \(a\) and angle \(A\) of the triangle \(ABC\), whose area is \(\Delta\), are constant. Shew that, when the other sides and angles undergo slight variations \(\delta b\) etc.,

1. [(i)] \(\delta b \sec B + \delta c \sec C = 0\),
2. [(ii)] \(\delta\Delta = \frac{1}{2}a^2\frac{\sin(C-B)}{\sin A}\delta B\).

Prove that \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{1}{v^2}\left(v\frac{du}{dx} - u\frac{dv}{dx}\right). \] Differentiate \[ \frac{x\sin^{-1}x}{(1-x^2)^{\frac{1}{2}}} + \frac{1}{2}\log(1-x^2), \] and find the value of \[ \frac{d^8}{dx^8}\left(\frac{\sin^3x \cos x}{256}\right) \] when \(x=\frac{\pi}{12}\).

Shew that the altitude of the right circular cone of maximum volume which can be inscribed in a sphere of radius \(a\) is \(4a/3\). What is the value of the altitude (in terms of \(a\)) when the area of the curved surface is a maximum?

A curve is given by the parametric equations \[ x = 3\cos\theta - \cos3\theta, \quad y = 3\sin\theta - \sin3\theta. \] Shew that the angle which the tangent at any point makes with the \(x\) axis is \(2\theta\). If \(s\) is the length of the arc of the curve measured from the point for which \(\theta=0\), prove that \[ s = 12 \sin^2\frac{\theta}{2}. \]

Sketch the curve whose equation is \[ y^2=c^2\frac{(x-a)}{(b-x)} \quad (b>a) \] and shew that the area enclosed by the curve and its asymptote is \(\pi c(b-a)\).

(i) Evaluate \[ \int \frac{(x-3)dx}{4x^2+5x+1}. \] (ii) Given \(\log_{10}e = 0.4343\), prove that \[ \int_1^3 x\log_x\left(1+\frac{1}{x}\right)dx = 1.601. \] (iii) Find a reduction formula for \(\int_0^{\pi/2}\sin^n x\,dx\) where \(n\) is a positive integer, and evaluate \[ \int_0^{\pi/2} \sin^6 x\,dx. \]