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731

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

Marks 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99
Frequency 50 50 40 60 100 100 50 25 15 10

(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%.

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732

(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.

Contestant P Q R S T U V
Judge X 2 7 1 3 6 5 4
Judge Y 4 6 2 3 1 1 5

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

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

 

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733

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.

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734

(a) Forces \(F_{1} = \begin{pmatrix} -5 & 4 \end{pmatrix} N; F_{2} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} N; F_{3} = \begin{pmatrix} 2 & -1 \end{pmatrix} N\) and \(F_{4} = \begin{pmatrix} 3 & -5 \end{pmatrix} N\) act on a body. Find the : 

(i) resultant of these forces ; (ii) fifth force that will keep the body in equilibrium.

(b) A body moving at 20 ms\(^{-1}\) accelerates uniformly at 2.5 ms\(^{-2}\) for 4 seconds. It continues the journey at the speed for 8 seconds, before coming to rest in t seconds with a uniform retardation. If the ratio of the acceleration to the retardation is 3 : 4, 

(i) sketch the velocity- time graph of the journey ; (ii) find t ; (iii) find the total distance of the journey.

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735

(a) An object is thrown up a smooth plane inclined at an angle of 30° to the horizontal. If the plane is 15m long and the object comes to rest at the top, find the :

(i) initial speed of the object ; (ii) time taken to reach the top.

(b)  

Force of magnitudes \(5 N, 5\sqrt{3} N, 10 N, 5\sqrt{3} N\) and \(5 N\) act on a body P, of mass 5 kg as shown in the diagram. Find the :

(i) magnitude of the resultant force ; (ii) acceleration of the body.

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