(a) The point P(3, -5) is rotated through an angle 60° anticlockwise about the origin. (i) Obtain the matrix for the rotation ; (ii) Find the image P' of the point P under the rotation.

(b) A linear transformation is given by \(N : (x, y) \to (2x + 3y, 3x - y)\).

(i) Write down the matrix N of the transformation ; (ii) If \(N^{2} + aN + bI = 0\), where a, b \(\in\) R, \(I\) is the \(2 \times 2\) matrix and \(0\) is the \(2 \times 2\) null matrix, find the values of a and b.

A survey indicated that 65% of the families in an area have cars. Find, correct to three decimal places, the probability that among 7 families selected at random in the area

(a) exactly 5 ;

(b) 3 or 4 ;

(c) at most 2 of them have cars.

The following table shows the distribution of marks obtained by some students in an examination.

(a) Construct a cumulative frequency table for the distribution

(b) Draw an ogive for the distribution

(c) Use your graph in (b) to determine : (i) semi- interquartile range ; (ii) number of students who failed, if the pass mark for the examination is 37 ; (iii) probability that a student selected at random scored between 20% and 60%.

(a) Five female and seven male teachers applied for 4 vacancies in a Junior High School. The teachers are equally qualified. Find the number of ways of employing the 4 teachers, if : (i) there is no restriction ; (ii) at least 2 of them are females.

(b) The table shows the positions awarded to 7 contestants by Judges X and Y in a competition.

(i) Calculate, correct to one decimal place, the Spearman's rank correlation coefficient.

(ii) Interpret your answer in b(i) above.

The position vectors of points P, Q and R with respect to the origin are \((4i - 5j), (i + 3j)\) and \((-5i + 2j)\) respectively. If PQRM is a parallelogram, find:

(a) the position vector of M ;

(b) \(|\overrightarrow{PM}|\) and \(|\overrightarrow{PQ}|\) ;

(c) the acute angle between \(\overrightarrow{PM}\) and \(\overrightarrow{PQ}\), correct to 1 decimal place ;

(d) the area of PQRM.