Shew that straight lines which are parallel to the same straight line are parallel to one another.
If two triangles have the three sides of the one equal to the three sides of the other each to each, prove that the triangles are congruent.
Prove the geometrical proposition corresponding to the algebraic identity \[ a^2-b^2=(a+b)(a-b). \]
Define a tangent to a circle. Prove that the tangent at any point of a circle and the radius through the point are perpendicular to one another.
Resolve 6981975 into prime factors and find what square number is nearest to it.
Determine the value of \(\dfrac{2068 \times \cdot02682}{\cdot4109}\) to four places of decimals.
Find, to the nearest penny, the difference between the simple and the compound interest on £350 for 5 years at 3\% per annum.
A man sells a farm of 74 acres 3 roods 10 poles at £21. 6s. 8d. per acre and invests the proceeds in 2\(\frac{1}{2}\)\% Consols at 76. Find what income he secures by this investment.
Factorise:
Simplify \(\dfrac{2x+5}{6x+7} - \dfrac{2x-1}{6x+5} - \dfrac{32x+33}{36(x+1)^2-1}\). and prove that, if \(a+b+c=0\), then \(a^3+b^3+c^3=3abc\).