PAST QUESTIONS 2023

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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. Given the sets \(A =\) {multiples of 3 less than 12}, \(B =\) {integers between 4 and 8} and \(C =\) {4, 5, 7}, find:

   \((i)\) \(A \cap B\);

   \((ii)\) \((A \cup B) \cap C\);

   \((iii)\) \((A \cap B) \cup C\).

   Show Solution

   Question 1.a.i

   \(A =\) {multiples of 3 less than 12}

   \(\Rightarrow A = \) \(\{ 3, 6, 9\}\)

   \(B =\) {integers between 4 and 8}

   \(\Rightarrow B = \) \(\{ 5, 6, 7\}\)

   \(C =\) {4, 5, 7}

   \( A \cap B\)

   \(\Rightarrow \{6\}\)

   
   

   Question 1.a.ii

   \(A \cup B\)

   \(\Rightarrow \{3, 5, 6, 7, 9\}\)

   \((A \cup B) \cap C\)

   \(\Rightarrow \{5, 7\}\)

   
   

   Question 1.a.iii

   \(A \cap B\)

   \(\Rightarrow \{6\}\)

   \((A \cap B) \cup C\)

   \(\Rightarrow \{4, 5, 6, 7\}\)




2. Simplify: \(1\frac{3}{4} - 2\frac{5}{6} - 1\frac{9}{10} + 4\frac{7}{8}\)

   Show Solution

   Question 1.b.

   \(1\frac{3}{4} - 2\frac{5}{6} - 1\frac{9}{10} + 4\frac{7}{8}\)

   \(\Rightarrow \frac{7}{4} - \frac{17}{6} - \frac{19}{10} + \frac{39}{8}\)

   \(\Rightarrow \dfrac{30(7) - 20(17) - 12(19) + 15(39)}{120}\)

   \(\Rightarrow \dfrac{210 - 340 - 228 + 585}{120}\)

   \(\Rightarrow \dfrac{227}{120}\)

   \(\Rightarrow 1\frac{107}{120}\)

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Question 2

1. Simplify: \(15(4 - 6) \times 49 \div 7\).

   Show Solution

   Question 2.a.

   \(15(4 - 6) \times 49 \div 7\)

   \(\Rightarrow 15(-2) \times 49 \div 7\)

   \(\Rightarrow -30 \times 49 \div 7\)

   \(\Rightarrow -30 \times 7\)

   \(\Rightarrow -210\)




2. Expand and simplify: \(b(12a - 3) - (a - b)(3 + b)\).

   Show Solution

   Question 2.b.

   \(b(12a - 3) - (a - b)(3 + b)\)

   \(\Rightarrow b(12a - 3) - \left(a(3 + b) - b(3 + b)\right)\)

   \(\Rightarrow 12ab - 3b - \left(3a + ab - 3b - b^2\right)\)

   \(\Rightarrow 12ab - 3b - 3a - ab + 3b + b^2\)

   \(\Rightarrow 12ab - ab - 3a - 3b + 3b + b^2\)

   \(\Rightarrow 11ab - 3a + b^2\)

   \(\Rightarrow b^2 + 11ab - 3a\)




3. Akosua walked for 3 hours at a rate of \(1\frac{1}{2}\) km per hour from her village to Paamu to take a bus to Quamu. If the bus travelling at \(15\frac{1}{2}\) km per hour takes \(2\) hours to travel from Paamu to Quamu.

   \(\hspace{0.5cm} i)\) what is the distance from Akosua's village to Quamu?

   Show Solution

   Question 2.c.i

   Akosua's village to Paamu:

   time spent \(= 3\) hours

   speed \(= 1\frac{1}{2}\) km/h

   distance \(= \text{speed} \times \text{time}\)

   \(\Rightarrow \text{distance} = 1\frac{1}{2} \times 3\) km

   \(\Rightarrow \text{distance} = \frac{3}{2} \times 3\) km

   \(\Rightarrow \text{distance} = \frac{9}{2}\) km

   \(\Rightarrow \text{distance} = 4\frac{1}{2}\) km

   
   

   From Paamu to Quamu:

   speed \(= 15\frac{1}{2}\) km/h

   time spent \(= 2\) hours

   distance \(= \text{speed} \times \text{time}\)

   \(\Rightarrow \text{distance} = 15\frac{1}{2} \times 2\) km

   \(\Rightarrow \text{distance} = \frac{31}{2} \times 2\) km

   \(\Rightarrow \text{distance} = 31\) km

   
   

   From Akosua's village to Quamu:

   distance \(= 4\frac{1}{2} + 31\) km

   \(\Rightarrow \text{distance} = 4\frac{1}{2} + 31\) km

   \(\Rightarrow \text{distance} = 35\frac{1}{2}\) km

   \(\Rightarrow \text{distance} = 35.5\) km

   \(\therefore\) the distance from Akosua's village to Quamu is 35.5 km

   \((ii)\) how long would it take a man, riding a bicycle at 5 km per hour, to travel from Akosua's village to Quamu?

   Show Solution

   Question 2.c.ii

   \(\text{distance} = 35\frac{1}{2}\)

   \(\text{speed} = 5\) km/h

   \(\text{time} = \dfrac{\text{distance}}{\text{speed}}\)

   \(\Rightarrow \text{time} = \dfrac{35\frac{1}{2}}{5}\)

   \(\Rightarrow \text{time} = \dfrac{71}{2} \div 5\)

   \(\Rightarrow \text{time} = \dfrac{71}{2} \times \dfrac{1}{5}\)

   \(\Rightarrow \text{time} = \dfrac{71}{10}\)

   \(\Rightarrow \text{time} = 7\frac{1}{10}\)

   \(\Rightarrow \text{time} = 7.1\) hours

   Converting 0.1 hours to minutes:

   \(\Rightarrow 0.1 \times 60\) minutes

   \(\Rightarrow 6\) minutes

   \(\therefore\) it will take the man 7 hours 6 minutes.

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Question 3

1. \(i)\) Express \(8 \times 32 \times 4 \times 2\) in the form \(2^m\).

   \(\hspace{0.5cm} ii)\) Using your answer in \((a)(i)\), state the value of \(m\).

   Show Solution

   Solution




2. \(i)\) Factorize the expression \(\pi n^2k - \frac{1}{4}\pi n^2Q\).

   \(\hspace{0.5cm} ii)\) Use your answer in \((b)(i)\) to find the value of the expression when \(\pi = \frac{22}{7}\), \(n = 2\), \(k = 19\) and \(Q = 20\).

   Show Solution

   Solution




3. Gifty and Justina shared an amount of Gh₵ 418.00. If Gifty had 20\(\%\) more than Justina, how much did Justina receive?

   Show Solution

   Question 3c

   Gh₵ 418.00 \(\Rightarrow 100\%\)

   let Justina's \(\%\) share \(= x\)

   then Gifty's \(\%\) share \(= x + 20\)

   \(\Rightarrow x + x + 20 = 100\)

   \(\Rightarrow 2x + 20 = 100\)

   \(\Rightarrow 2x = 100 - 20\)

   \(\Rightarrow 2x = 80\)

   \(\Rightarrow \dfrac{2x}{2} = \dfrac{80}{2}\)

   \(\Rightarrow x = 40\)

   \(\therefore\) Justina's percentage share was 40\(\%\)

   
   

   If \(100\% =\) Gh₵ 418.00

   \(\Rightarrow 40\% = \dfrac{40\%}{100\%} \times\) Gh₵ 418.00

   \(\hspace{1.2cm} = \dfrac{2}{5} \times\) Gh₵ 418.00

   \(\hspace{1.2cm} = \dfrac{Gh₵ 836}{5}\)

   \(\hspace{1.2cm} = Gh₵ 167.20\)

   \(\therefore\) Justina's share was Gh₵ 167.20

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Question 4

1. If \(4m - 2(3 + 2m) + m(2m + 4) = 0\), find the values of \(m\).

   Show Solution

   Solution




2. At a political rally, there were 240 women, 200 men, 160 boys and 120 girls.

   \(\hspace{0.5cm} i)\) Draw a pie chart to illustrate the information.

   \(\hspace{0.5cm} ii)\) What percentage of the people at the rally were females?

   Show Solution

   Solution

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Question 5

Madam Esi used \(\frac{1}{4}\) and \(\frac{2}{3}\) of her \(x\) acres of land to cultivate mangoes and oranges respectively.

1. Express, in terms of \(x\), the number of acres of the land she used to cultivate:

   \(\hspace{0.5cm} i)\) mangoes;

   \(\hspace{0.5cm} ii)\) oranges.

2. If madam Esi used 20 more acres of land to cultivate oranges than mangoes, find the value of \(x\).

3. How many acres of land was used to cultivate mangoes?

4. Calculate, correct to the nearest whole number, the percentage of the land that was not used.

Show Solution

Solution

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Question 6

1. Using a scale of 2 cm to 2 units on both axes, draw on a graph sheet two perpendicular axes, \(Ox\) and \(Oy\), for the interval \(-10 \leq x \leq 10\) and \(-10 \leq y \leq 10\).

2. On the same graph sheet, draw:

   \(\hspace{0.5cm} i)\) a quadrilateral \(ABCD\) with vertices \(A(2, 4)\), \(B(2, 8)\), \(C(8, 8)\) and \(D(8, 4)\);

   \(\hspace{0.5cm} ii)\) the image \(A_1B_1C_1D_1\) of \(ABCD\) under a translation by vector \(\begin{pmatrix}-5 \ -2\end{pmatrix}\), where \(A \rightarrow A_1\), \(B \rightarrow B_1\), \(C \rightarrow C_1\) and \(D \rightarrow D_1\);

   \(\hspace{0.5cm} iii)\) the image \(A_2B_2C_2D_2\) of \(ABCD\) under a reflection in the \(y\)-axis, where \(A \rightarrow A_2\), \(B \rightarrow B_2\), \(C \rightarrow C_2\) and \(D \rightarrow D_2\).

3. \(i)\) What type of quadrilateral is \(ABCD\)?

   \(\hspace{0.5cm} ii)\) Find the gradient of \(\overline{A_2B_1}\).

   Show Solution

   Solution

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