Problems

Draw a diagram to illustrate the truth of the algebraical identity \[ (a-b)(a+b) = a^2-b^2. \] \item[*3.] Prove that the angle at the centre of a circle is double any angle at the circumference standing on the same arc.

Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the bisected angle.

A milkman buys eggs at 10 for 3s. and sells them at \(4\frac{1}{2}\)d. each; what is his profit per cent.?

A man sells out £800 of Swedish Bonds at \(118\frac{1}{4}\) and reinvests the proceeds in five per cent. National War Bonds at par; what income does he now receive?

Simplify:

1. [*(1)] \(\frac{ab(a+b)+a^3+b^3}{ab(a-b)-a^3+b^3}\).
2. [(2)] \((2a^2 - 2ab\sqrt{3}+3b^2)(2a^2+2ab\sqrt{3}+3b^2)\).

\item[*10.] Solve the equations:

1. [(1)] \(\frac{b-c}{x+a}+\frac{c+a}{x} = \frac{a+b}{x-c}\).
2. [(2)] \(4x^2 - 12x = 71\).

At an election the majority was 1184, which was one-fifth of the total number of votes; how many votes did each side poll?

Find the sum of 24 terms of the series \(4\frac{1}{2}+3\frac{3}{4}+3+\dots\). The sum of eight terms of an arithmetical progression is 24 and its fifth term is 1; find the first term and the common difference.