PAST QUESTIONS 2010

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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. Factorize \((m + n)(2x - y) - x(m + n)\)

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   Question 1.a.

   \((m + n)(2x - y) - x(m + n)\)

   \(\Rightarrow (m + n)(2x - y - x)\)

   \(\Rightarrow (m + n)(2x - x - y)\)

   \(\Rightarrow (m + n)(x - y)\)

   
   
   

2. \(A\) and \(B\) are subsets of a universal set

   \(\hspace{0.5cm}\mathbb{U} = \) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}
   \(\hspace{0.5cm}\)such that
   \(\hspace{0.5cm}\)\(A =\) {even numbers} and
   \(\hspace{0.5cm} B =\) {multiples of 3}.

   \(\hspace{0.5cm} i)\) List the elements of the sets \(A, B, (A \cap B), (A \cup B)\) and \((A \cup B)^\prime\)

   \(\hspace{0.5cm} ii)\) Illustrate the information in \((i)\) on a Venn diagram.

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   Question 1.b.i.

   \(\mathbb{U} = \) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}

   \(A =\) {even numbers}

   \(\Rightarrow\)\(A =\) {2, 4, 6, 8, 10, 12, 14, 16, 18}

   \(B =\) {multiples of 3}

   \(\Rightarrow B =\) {3, 6, 9, 12, 15, 18}

   \(A \cap B\) \(=\) {6, 12, 18}

   \(A \cup B\) \(=\) {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}

   \((A \cup B)^\prime\) \(=\) {1, 5, 7, 11, 13, 17}

   
   

   Question 1.b.ii.

   
   
   
   

3. Find the values of \(x\) and \(y\) in the vector equation:

   \(\hspace{0.1cm} \begin{pmatrix}5 \\ 3\end{pmatrix} + 2\begin{pmatrix}x \\ y\end{pmatrix} - \begin{pmatrix}1 \\ -7\end{pmatrix} = 0\)

   Show Solution

   Question 1.c.

   \(\begin{pmatrix}5 \\ 3\end{pmatrix} + 2\begin{pmatrix}x \\ y\end{pmatrix} - \begin{pmatrix}1 \\ -7\end{pmatrix} = 0\)

   Solving for \(x\):

   \(5 + 2x - 1 = 0\)

   \(\Rightarrow\) \(2x + 4 = 0\)

   \(\Rightarrow\) \(2x = -4\)

   \(\Rightarrow\) \(\dfrac{2x}{2} = \dfrac{-4}{2}\)

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

   Solving for \(y\):

   \(3 + 2y - (-7) = 0\)

   \(\Rightarrow\) \(3 + 2y + 7 = 0\)

   \(\Rightarrow\) \(2y + 10 = 0\)

   \(\Rightarrow\) \(2y = -10\)

   \(\Rightarrow\) \(\dfrac{2y}{2} = \dfrac{-10}{2}\)

   \(\Rightarrow\) \(y = -5\)

   \(\therefore\) \(\underline{x \ \text{is} -2 \ \text{and} \ y \ \text{is} -5}\)

   
   
   

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Question 2

1. Find the sum of 2,483.65, 701.532 and 102.7, leaving your answer to one decimal place.

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   Question 2.a.

   Finding the sum:

   
   
   
   

2. In the quadrilateral \(ABCD\), \(|AB| = 3\) cm, \(|BC| = 4\) cm, \(|CD| = 12\) cm and angle \(ABC = 90^\circ\) and angle \(ACD = 90^\circ\).

   

   Calculate:

   \(\hspace{0.5cm} i)\) the perimeter of \(ABCD\)

   \(\hspace{0.5cm} ii)\) the area of \(ABCD\).

   Show Solution

   Question 2.b.i.

   Perimeter = distance around an object.

   Perimeter of \(ABCD\)

   \(\Rightarrow\) \(|AB| + |BC| + |CD| + |DA|\)

   \(\Rightarrow\) \(3 + 4 + 12 + 13\)

   \(\Rightarrow\) \(32 \ cm\)

   \(\therefore\) the perimeter of \(ABCD\) is 32 cm

   
   

   Question 2.b.ii.

   Area of a triangle

   \(\Rightarrow\) \(\frac{1}{2} \times length \times height\)

   Area of \(\triangle ABC\)

   \(\Rightarrow\) \(\frac{1}{2} \times 3 \times 4\)

   \(\Rightarrow\) \(\frac{1}{2} \times 12\)

   \(\Rightarrow\) \(6\) cm\(^2\)

   Area of \(\triangle ACD\)

   \(\Rightarrow\) \(\frac{1}{2} \times 5 \times 12\)

   \(\Rightarrow\) \(\frac{1}{2} \times 60\)

   \(\Rightarrow\) \(30\) cm\(^2\)

   Total area of \(ABCD\)

   \(\Rightarrow\) \(6 + 30\)

   \(\Rightarrow\) \(36\) cm\(^2\)

   \(\therefore\) the area of ABCD is \(36\) cm\(^2\)

   
   
   

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Question 3

1. Evaluate \(\dfrac{2^7 \times 3^4 \times 5^3}{2^3 \times 3^2 \times 5^2}\), leaving your answer in standard form.

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   Question 3.a.

   \(\dfrac{2^7 \times 3^4 \times 5^3}{2^3 \times 3^2 \times 5^2}\)

   \(\Rightarrow\) \(\dfrac{2^7}{2^3} \times \dfrac{3^4}{3^2} \times \dfrac{5^3}{5^2}\)

   \(\Rightarrow\) \(2^{(7-3)} \times 3^{(4-2)} \times 5^{(3-2)}\)

   \(\Rightarrow\) \(2^4 \times 3^2 \times 5^1\)

   \(\Rightarrow\) \(16 \times 9 \times 5\)

   \(\Rightarrow\) \(16 \times 45\)

   \(\Rightarrow\) \(720\)

   \(\Rightarrow\) \(7.20 \times 10^2\) in standard form.

   
   
   

2. Kwame rode a bicycle for a distance of \(x\) km and walked for another \(\frac{1}{2}\) hour at a rate of 6 km per hour. If Kwame covered a total distance of 10 km, find the distance \(x\) he covered by bicycle.

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   Question 3

   Distance covered by bicycle \(= x\) km

   Time spent walking \(= \frac{1}{2}\) hour

   Walking speed \(= 6\) km/h

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

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

   \(\Rightarrow \text{distance walked} = 6 \times \frac{1}{2}\)

   \(\Rightarrow \text{distance walked} = 3\) km

   
   

   Total distance covered

   \(\Rightarrow x + 3 = 10\)

   \(\Rightarrow x = 10 - 3\)

   \(\Rightarrow x = 7\) km

   \(\therefore\) Kwame covered 7 km by bicycle.

   
   
   

3. A rectangular tank of length 22 cm, width 9 cm and height 16 cm is filled with water. The water is poured into a cylindrical container of radius 6 cm. Calculate the

   \(\hspace{0.5cm} (i)\) volume of the rectangular tank.

   \(\hspace{0.5cm} (ii)\) depth of water in the cylindrical container. [Take \(\pi = \frac{22}{7}\)]

   Show Solution

   Question 3.c.i

   \(\text{length, L} = 22\) cm

   \(\text{width, W} = 9\) cm

   \(\text{height, H} = 16\) cm

   \(\text{Volume} = L \times W \times H\)

   Volume of tank:

   \(\Rightarrow 22 \times 9 \times 16\)

   \(\Rightarrow 3,168\) cm\(^3\)

   \(\therefore\) the volume of the rectangular tank is \(3,168\) cm\(^3\)

   
   

   Question 3.c.ii

   Volume of water \(= 3,168\) cm\(^3\)

   Base radius \(= 6\) cm

   \(\Rightarrow 3,168 = \pi r^2 h\)

   \(\Rightarrow 3,168 = \frac{22}{7} \times 6^2 \times h\)

   \(\Rightarrow 3,168 = \frac{22 \times 6^2 \times h}{7}\)

   \(\Rightarrow 3,168 \times 7 = 22 \times 36 \times h\)

   \(\Rightarrow h = \frac{3,168 \times 7}{22 \times 36}\)

   \(\Rightarrow h = 4 \times 7\)

   \(\Rightarrow h = 28\) cm

   \(\therefore\) the depth of the water is 28 cm

   
   
   

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Question 4

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

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2. The area of a trapezium is \(31.5\) cm\(^2\). If the parallel sides are of length \(7.3\) cm and \(5.3\) cm, calculate the perpendicular distance between them.

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3. The marks scored by four students in a Mathematics test are as follows:

   \(\hspace{0.5cm}\) Esi \(\hspace{0.95cm}\) \( - \) \(\hspace{0.5cm}\) 92
   \(\hspace{0.5cm}\) Seth \(\hspace{0.6cm}\) \( - \) \(\hspace{0.5cm}\) 85
   \(\hspace{0.5cm}\) Mary \(\hspace{0.5cm}\) \( - \) \(\hspace{0.5cm}\) 65
   \(\hspace{0.5cm}\) Efe \(\hspace{0.9cm}\) \( - \) \(\hspace{0.5cm}\) \(x\)

   \(\hspace{0.5cm} i)\) Write down an expression for the mean (average) of the marks.

   \(\hspace{0.5cm} ii)\) If the mean is less than \(80\), write a linear inequality for the information.

   \(\hspace{0.5cm} iii)\) Find the possible marks Efe scored in the test. Represent your answer on the number line.

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Question 5

1. Solve \(\dfrac{4x - 3}{2} = \frac{8x - 10}{8} + 2\frac{3}{4}\)

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2. Using a scale of 2 cm to 1 unit on both axes, draw two perpendicular lines \(OX\) and \(OY\) on a graph sheet. Mark the \(x\)-axis from \(-5\) to \(5\) and the \(y\)-axis from \(-6\) to \(6\).

   \(\hspace{0.5cm} i)\) Plot the points \(A(2, 3)\) and \(B(-3, 4)\) and join them with a long straight line.

   \(\hspace{0.5cm} ii)\) Plot on the same graph sheet , the points \(C(4, 2)\) and \(D(-2, -3)\) and join them with a long straight line to meet the line, and through, \(AB\).

   \(\hspace{0.5cm} iii)\) Measure the angle formed between the lines \(AB\) and \(CD\)

   \(\hspace{0.5cm} iv)\) Find the coordinates of the point at which the lines through \(AB\) and \(CD\) meet.

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Question 6

The following table shows the frequency distribution of the number of letters in the surnames of some students in a school.

1. From the distribution, determine

   \(\hspace{0.5cm} i)\) the mode;

   \(\hspace{0.5cm} ii)\) the mean.

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2. If a student is selected at random, find the probability that his\her name will contain more than 7 letters.

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3. Draw a bar chart for the distribution.

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