A survey conducted revealed that four out of every twenty taxi drivers do not have a valid driving license. If 6 drivers are selected at random, calculate, correct to three decimal places, the probability that

(a) exactly 2 ;

(b) more than 3 ;

(c) at least 5; have valid driving license.

The table shows the frequency distribution of marks scored by some candidates in an examination.

(a) Draw the cumulative frequency curve of the distribution.

(b) Use your graph to estimate the :

(i) semi-interquartile range of the distribution; (ii) percentage of candidates who passed with distinction if the least mark for distinction was 72.

A bag contains 4 red, 6 blue and 8 green identical marbles.

(a) If three marbles are drawn at random, without replacement, calculate the probability that :

(i) all will be green ; (ii) all will have the same colour.

(b) If each marble is replaced before another is drawn, calculate the probability that all will have the same colour.

(a) An object P of mass 6.5kg is suspended by two light inextensible strings, AP and BP. The strings make angles 50° and 60° respectively with the downward vertical.

(i) Express the forces acting on P in component form; (ii) If P is at rest, write down the vector equation connecting all the forces; (iii) Calculate, correct to one decimal place, the tensions in the strings.

(b) A particle of mass 5 kg moves with initial velocity \(\frac{1}{2} m/s\) and final velocity \(\frac{3}{4} m/s \). Find the magnitude of its change in momentum.

(a) \(m \begin{pmatrix} 2 \\ 1 \end{pmatrix} + n \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \begin{pmatrix} 5 \\ -4 \end{pmatrix}\) where m and n are scalars. Find the value of (m + n).

(b) A(-1, 3), B(2, -1) and C(5, 3) are the vertices of \(\Delta\) ABC.

(i) Express in column notation, the unit vectors parallel to AB and AC.

(ii) Use a dot product to calculate \(\stackrel \frown{BAC}\), correct to the nearest degree.