PAST QUESTIONS 1990

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Highest score: \(\mathbf{\frac{35}{40}}\)

Malvin Agyemang

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Try the questions first, using not more than 15 minutes for each question, and watch the accompanying videos to see how the questions are solved.

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Question 1

1. List the members of each of the sets

\( B=\){Whole numbers from 20 to 30} and
\( D=\){factors of 63}
List the members of:
\((i)\) \(B \cap D\)
\((ii)\) \(B \cup D\)

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Solution

\(B=\) {Whole numbers from 20 to 30}

\(\Rightarrow B =\) {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}

\(D=\) {factors of 63}

\(\Rightarrow D =\) {1, 3, 7, 9, 21, 63}




\((i)\) \(B \cap D =\) {21}

\((ii)\) \(B \cup D =\) {1, 3, 7, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}





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Watch the video below for a walk-through of the solution.








2. In a class of 60 students, 46 passed Mathematics and 42 passed English Language. Every student passed at least one of the two subjects.

\((i)\) Illustrate the information on a Venn Diagram.

\((ii)\) How many students passed in both subjects?

Let \(n\) represent the number of students who passed in both subjects.


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Solution

Let \(\mathbb{U} \Rightarrow\) the total students in the class.

Let \(M \Rightarrow\) students who passed Mathematics.

Let \(E \Rightarrow\) students who passed English.

Let \(n \Rightarrow\) number of students who passed in both subjects.

\(n(\mathbb{U}) = 60\)

\(n(M) = 46\)

\(n(E) = 42\)




\((i)\)





\((ii)\) \(46 -n + n + 42 - n = 60\)

\(46 + 42 - n = 60\)

\(88 - n = 60\)

\(88 - 60 = n\)

\(\Rightarrow n = 28\)

\(\therefore\) 28 students passed in both subjects.





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Question 2

1. Factorise completely:

\(3a^2 + 2ab - 12ac - 8bc\)

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Solution

\(3a^2 + 2ab - 12ac - 8bc\)

\(a(3a + 2b) - 4c(3a + 2b)\)

\(\Rightarrow (3a + 2b)(a - 4c)\)





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Watch the video below for a walk-through of the solution.








2. Solve \(\frac{x}{4} + \frac{3}{5} = \frac{3x}{2} - 2\)

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Solution

\(\frac{x}{4} + \frac{3}{5} = \frac{3x}{2} - 2\)

Multiplying through by LCM of 20:

\(\Rightarrow \frac{x}{4}(20) +\) \(\frac{3}{5}(20) =\) \(\frac{3x}{2}(20) - 2(20)\)

\(\Rightarrow 5x + 3(4) = 3x(10) - 40\)

\(\Rightarrow 5x + 12 = 30x - 40\)

Grouping like terms:

\(\Rightarrow 12 + 40 = 30x - 5x\)

\(\Rightarrow 52 = 25x\)

\(\Rightarrow \frac{52}{25} = \frac{25x}{25}\)

\(\Rightarrow x = \frac{52}{25}\)

\(\Rightarrow x = 2\frac{2}{25}\)

\(\therefore x\) is \(2\frac{2}{25}\)





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Watch the video below for a walk-through of the solution.








3. Find the solution set of \(x + 3 > 19 - 3x\), where \(x\) is a real number.
   Illustrate your answer on the number line.

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Solution

\(x + 3 > 19 - 3x\)

Grouping like-terms:

\(\Rightarrow x + 3x > 19 - 3\)

\(\Rightarrow 4x > 16\)

\(\Rightarrow \frac{4x}{4} > \frac{16}{4}\)

\(\Rightarrow x > 4\)

\(\therefore\) the solution set is {\(x:x > 4, x \in\) real numbers}






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Question 3

1. Using a ruler and a pair of compasses only:

\((i)\) Construct a triangle \(PQR\) such that \(|PQ| = 6cm\), \(|QR| = 4cm\) and \(\angle PQR=90°\).

\((ii)\) Construct perpendicular bisectors (mediatos) of \(PQ\) and \(QR\). Name the intersection of the mediators \(O\)

\((iii)\) Draw a circle with \(O\) as center and \(OQ\) as radius.



2. Measure

\((i)\) \(\lvert PR \rvert \)

\((ii)\) \(\angle QPR\)

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Question 4

The following is the result of a survey conducted in a class of a Junior Secondary School to find the favourite soft drink of each pupil in the class.

1. Draw a bar chart showing this information, using a scale of 2cm to 1 unit on the vertical axis.
2. How many pupils are in the class?
3. What percentage of pupils in the class prefer Club Cola?

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