The following data gives the lengths, in cm, of 30 pieces of iron rods :

45 55 65 60 61 68 59 54 64 76 50 68 72 68 80 67 70 62 79 67 64 63 71 59 64 53 57 74 55 57

(a) Using class intervals of 45 - 49, 50 - 54, 55 - 59, ... construct a frequency table of the data.

(b) Draw the histogram for the distribution

(c) Calculate the mean of the distribution

(d) What is the probability of selecting  an iron rod whose length is in the modal class?

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.