Problems

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\(\frac{3}{4}\).

Resolve \[ 12x^2+x-35 \quad \text{and} \quad bc-ca-ab+a^2 \] each into two factors and \((6x^2-x)^2 - 7(6x^2-x)+10\) into four factors. \item[*10.] Solve the equations:

1. [(1)] \((a-b)x-ay=b(2a-3b)\); \((x-y)=6b\).
2. [(2)] \(3x+5y=x^2-2y^2=1\).

\item[*11.] When eggs are lowered 1\(\frac{1}{2}\)d. per dozen, one dozen more can be bought for 30s. Find the price per dozen.

Find the sum of 18 terms of the series \(10+8\frac{3}{5}+7\frac{1}{5}+\dots\). \par Find also what term of the series is equal to \(-\frac{1}{5}\).