Given is the graph of the relation \(y = ax^{2} + bx + c\) where a, b and c are constants. Use the graph to :

(a) find the roots of the equation \(ax^{2} + bx + c = 0\);

(b) determine the values of constants a, b and c in the relation using the values of the coordinates P and Q and hence write down the relation illustrated in the graph

(c) find the maximum value of y and the corresponding value of x at this point.

(d) find the values of x when y = 2.

Given that \(\log_{10} 2 = 0.3010\) and \(\log_{10} 3 = 0.4771\), calculate without using mathematical tables or calculator, the value of :

(a) \(\log_{10} 54\) ;

(b) \(\log_{10} 0.24\).

(a) Simplify : \(\frac{1}{3^{5n}} \times 9^{n - 1} \times 27^{n + 1}\)

(b) The sum of the ages of a woman and her daughter is 46 years. In 4 years' time, the ratio of their ages will be 7 : 2. Find their present ages.

The sides of a rectangular floor are xm and (x + 7)m. The diagonal is (x + 8)m. Calculate, in metres :

(a) the value of x ;

(b) the area of the floor.

The diagram above shows the bar charts representing the number of vehicles manufactured by a company in January, February and March, 1992.

(a) How many vehicles were produced in February?

(b) What fraction of the vehicles manufactured in February were cars?

(c) How many buses were produced altogether from January to March, 1992?

(d) What is the ratio in the lowest term of the number of lorries produced in February to that in March?