Problems

Prove that in an equation with real coefficients imaginary roots occur in pairs of the type \(\lambda \pm i\mu\). \par If \(\alpha, \beta, \gamma\) are the roots of \[ x^3-px^2+qx-r=0, \] express \((\alpha^2-\beta\gamma)(\beta^2-\gamma\alpha)(\gamma^2-\alpha\beta)\) in terms of the coefficients.

Differentiate \[ x^x, \quad \sin^{-1}\frac{x}{\sqrt{a^2-x^2}}, \quad \log\frac{x^2+x\sqrt{2}+1}{x^2-x\sqrt{2}+1}. \] Prove that, if \(y=\frac{\sin x}{\sqrt{x}}\), \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\tfrac{1}{4})y=0. \]

Shew how to evaluate the indeterminate forms \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). \par Find the limit when \(\theta=\frac{\pi}{2}\) of \((\log(\sec\theta+\tan\theta)).(\log(\operatorname{cosec}\theta+\cot\theta))\).

Find the equations of the tangent and normal at any point of the curve \[ x=3\sin t-2\sin^3t, \quad y=3\cos t-2\cos^3t. \] Prove that the evolute is \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=2^{\frac{2}{3}}\).

Evaluate the integrals

1. [(1)] \(\int \frac{(x+1)dx}{(x-1)\sqrt{1+x-x^2}}\),
2. [(2)] \(\int \sinh ax \cos bx dx\),
3. [(3)] \(\int \frac{\cos x dx}{(17+8\cos 2x)^2}\),
4. [(4)] \(\int_0^{\frac{\pi}{2}} \sin^{2m}\theta \cos^{2n}\theta d\theta\).

Find formulae giving the length of an arc of a plane curve whose equation is given in terms of (1) \(p\) and \(\psi\), (2) \(p\) and \(r\), where \(r\) is the radius vector from a fixed point, \(p\) the perpendicular from that point on the tangent and \(\psi\) the inclination of \(p\) to a fixed line. \par Prove that the length of that part of the curve \(p=r-a\) which lies between the circles \(r=a\) and \(r=2a\) is \(a(\sqrt{3}-\frac{\pi}{3})\).

Prove that in an obtuse-angled triangle the square on the side opposite the obtuse angle is greater than the sum of the squares on the other two sides by twice the rectangle contained by one of these sides and the projection of the other on it. \item[*3.] Prove that the angle which an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference.

Shew how to construct a mean proportional to two given straight lines and prove the validity of your construction.

A merchant buys teas at 2s. 1d. and 1s. 8d. per lb, and mixes them in the proportion of 7 lbs. of the former to 3 lbs. of the latter; find roughly what profit per cent. he makes by selling the mixture at 2s. 4d. per lb.

Find correct to the nearest shilling the income obtained by investing \pounds2172 in 4\(\frac{1}{2}\) per cent. bonds at 97\(\frac{3}{4}\).