BECE 1993 Mathematics Past Questions and Answers

Paper 1: Objectives

1. Expand (2a + b)(a + 2b).
   1. A. 2a + 26°
   2. B. 2a² + 6b
   3. C. 5a² + 2a²
   4. D. 2b² + 2a + 4ab
   5. E. 2a² + 5ab + 2b²
   Answer: E

   (2a + b)(a + 2b) = 2a(a + 2b) + b(a + 2b) = 2a² + 5ab + 2b².
2. Find the missing number in the following binary operation:
   
   1100110 − ******* = 111011
   1. A. 111011
   2. B. 101001
   3. C. 100011
   4. D. 101110
   5. E. 101011
   Answer: E

   Let x be the missing number. 1100110 − x = 111011, so x = 1100110 − 111011 = 101011 (in base ten: 102 − 59 = 43).
3. If x = {1, 3, 5, 7, 9, 11, 13, 15}, find the truth set of x - 3 ≥ 10.
   1. A. {15}
   2. B. {13, 15}
   3. C. {11, 13, 15}
   4. D. {9, 11, 13, 15}
   5. E. {7, 9, 11, 13, 15}
   Answer: B

   The truth set is {13, 15} since only values greater than or equal to 13 satisfy x - 3 ≥ 10.
4. Which of these has the least number of lines of symmetry?
   1. A. An equilateral triangle
   2. B. A rectangle
   3. C. A square
   4. D. A circle
   5. E. An isosceles triangle
   Answer: E

   An isosceles triangle has only 1 line of symmetry, the least among the options.
5. Find $2\frac{1}{2}\%$ of ₵2,000.00.
   1. A. ₵40.00
   2. B. ₵50.00
   3. C. ₵100.00
   4. D. ₵800.00
   5. E. ₵5,000.00
   Answer: B

   $2\frac{1}{2}\%=\frac{5}{200}$. $\frac{5}{200}\times ₵2,000=₵50.00$.
6. Find the highest common factor (HCF) of 18, 36, and 120.
   1. A. 2⁵
   2. B. 2 × 3 × 5
   3. C. 2 × 3
   4. D. 2² × 3
   5. E. 3²
   Answer: C

   Prime factors in lowest powers: HCF = 2 × 3 = 6.
7. Arrange the fractions $\frac{3}{4},\frac{2}{3},\frac{3}{5}$ in ascending order.
   1. A. $\frac{2}{3},\frac{3}{4},\frac{3}{5}$
   2. B. $\frac{3}{4},\frac{2}{3},\frac{3}{5}$
   3. C. $\frac{3}{5},\frac{2}{3},\frac{3}{4}$
   4. D. $\frac{3}{5},\frac{3}{4},\frac{2}{3}$
   5. E. $\frac{3}{4},\frac{3}{5},\frac{2}{3}$
   Answer: C

   LCM of 4, 3, 5 is 60: $\frac{3}{4}=\frac{45}{60},\frac{2}{3}=\frac{40}{60},\frac{3}{5}=\frac{36}{60}$. Ascending: $\frac{3}{5},\frac{2}{3},\frac{3}{4}$.
8. Make b the subject of the equation 1/a + 1/b = c.
   1. A. b = -a - c
   2. B. b = a + ac
   3. C. b = 1 - ac
   4. D. b = a / (1 - ac)
   5. E. b = 1 / (a - c)
   Answer: D

   Rearrange and solve for b: b = a / (1 - ac).
9. A man has three children aged 9, 12, and 24 years. Find the ratio of their ages.
   1. A. 1 : 2 : 3
   2. B. 1 : 2 : 4
   3. C. 2 : 3 : 6
   4. D. 3 : 4 : 8
   5. E. 3 : 6 : 8
   Answer: D

   Ratio = 9 : 12 : 24 simplifies to 3 : 4 : 8.
10. Ten students take 9 days to complete a task. How many days would 15 students take?
    1. A. 5 days
    2. B. 6 days
    3. C. 13 days
    4. D. 14 days
    5. E. 16 days
    Answer: B

    Inverse relationship: 9 days × (10 students / 15 students) = 6 days.

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11. Which of the following inequalities is/are true?
    
    I. 5/6 > 3/4
    
    II. 3/4 > 2/5
    
    III. 1/2 > 3/8
    1. A. I only
    2. B. II only
    3. C. III only
    4. D. I and II only
    5. E. I and III only
12. Find the image of P in the mapping below:
    
    1 → 3
    
    2 → 5
    
    3 → 7
    
    P → ?
    1. A. P² + 2
    2. B. P² + 1
    3. C. 2P + 1
    4. D. P + 2
    5. E. P² - 2
13. A car travels 50 km in an hour. How far does it travel in 30 minutes?
    1. A. 1,500 km
    2. B. 100 km
    3. C. 80 km
    4. D. 25 km
    5. E. 20 km
14. In an enlargement, the area of an object was multiplied by 144. Find the scale factor.
    1. A. 12
    2. B. 36
    3. C. 48
    4. D. 72
    5. E. 144
15. Mr. Yevu saved €2,500 at 25% interest per annum for 4 years. Find the interest.
    1. A. €625.00
    2. B. €2,500.00
    3. C. €3,125.00
    4. D. €5,000.00
    5. E. €10,000.00
16. The table below gives the number of goals scored by a football team in a league season:
    

    Number of goals scored in a match	0	1	2	3	4	5
    Frequency	1	7	6	4	1	1

    
    Find the total number of goals scored by the team.
    1. A. 41
    2. B. 40
    3. C. 20
    4. D. 19
    5. E. 15
17. The table below gives the number of goals scored by a football team in a league season:
    

    Number of goals scored in a match	0	1	2	3	4	5
    Frequency	1	7	6	4	1	1

    
    What is the mean number of goals scored by the team?
    1. A. 7
    2. B. 6
    3. C. 4
    4. D. 2
    5. E. 1
18. The probability of getting a head is 1/3. Find the probability of getting a tail.
    1. A. 1/3
    2. B. 2/3
    3. C. 1/2
    4. D. 1/4
    5. E. 0
19. What is the probability of obtaining a tail when a coin is tossed?
    1. A. 1/2
    2. B. 2/3
    3. C. 1/3
    4. D. 3/4
    5. E. 0
20. Given that $\mathbf{a}=\left(\begin{array}{c}-2\\ 3\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}2\\ -5\end{array}\right)$, find $\mathbf{a}+2\mathbf{b}$.
    1. A. $\left(\begin{array}{c}-2\\ 3\end{array}\right)$
    2. B. $\left(\begin{array}{c}2\\ 13\end{array}\right)$
    3. C. $\left(\begin{array}{c}0\\ -2\end{array}\right)$
    4. D. $\left(\begin{array}{c}6\\ 13\end{array}\right)$
    5. E. $\left(\begin{array}{c}2\\ -7\end{array}\right)$
21. Find in base ten the value of 4 in 143_five.
    1. A. 48
    2. B. 40
    3. C. 25
    4. D. 20
    5. E. 9
22. If the interior angle of a regular polygon is 120°, how many sides does it have?
    1. A. 5
    2. B. 6
    3. C. 7
    4. D. 8
    5. E. 9
23. Simplify $(3\frac{1}{2}+7)÷(4\frac{1}{3}-3)$
    1. A. $6\frac{7}{8}$
    2. B. $7$
    3. C. $7\frac{7}{8}$
    4. D. $10\frac{1}{2}$
    5. E. $14$
24. Find the solution set of $n-\frac{2}{3}>\frac{1}{3}-n$.
    1. A. $\{n:n>-1\}$
    2. B. $\{n:n=0\}$
    3. C. $\{n:n>\frac{1}{3}\}$
    4. D. $\{n:n>\frac{1}{2}\}$
    5. E. $\{n:n>1\}$
25. 
    
    Calculate the size of the angle marked x in the diagram with [AE] = [ED] = [DC] = [CE].
    1. A. 150°
    2. B. 120°
    3. C. 90°
    4. D. 60°
    5. E. 30°
26. Five times a number is four more than the number. Find the number.
    1. A. -3/2
    2. B. 1
    3. C. -2
    4. D. -1/3
    5. E. -1
27. A basket contains 450 oranges. If each orange costs ₵15.00, find the total cost of the oranges.
    1. A. ₵30.00
    2. B. ₵465.00
    3. C. ₵435.00
    4. D. ₵675.00
    5. E. ₵6,750.00
28. A bottle of soft drink costs ₵200.00. The commission on one bottle is 2%. Find the commission on 24 bottles.
    1. A. ₵96.00
    2. B. ₵296.00
    3. C. ₵400.00
    4. D. ₵4,704.00
    5. E. ₵4,800.00
29. If 9/21 is equivalent to 1 + x/7, find x.
    1. A. 1
    2. B. 2
    3. C. 4
    4. D. 5
    5. E. 6
30. Write 39.975 km correct to three significant figures.
    1. A. 39 km
    2. B. 39.975
    3. C. 49 km
    4. D. 40.0 km
    5. E. 40.9 km
31. Find x, if $\frac{1}{x}+\frac{1}{3}=1$.
    1. A. $-\frac{3}{2}$
    2. B. $-\frac{2}{3}$
    3. C. $\frac{2}{3}$
    4. D. $\frac{3}{4}$
    5. E. $\frac{3}{2}$
32. The bearing of Aboku from Bebeka is 055°. What is the bearing of Bebeka from Aboku?
    1. A. 035°
    2. B. 055°
    3. C. 125°
    4. D. 235°
    5. E. 305°
33. 
    
    The diagram above shows the construction of:
    1. A. the perpendicular bisector of the line
    2. B. an angle of 45° at the point A
    3. C. an angle of 45° at the point B
    4. D. an angle of 90° at the point O
    5. E. an angle of 90° at the point B
34. 
    
    The dimensions of the rectangle are given in base two. Find its perimeter.
    1. A. 100 cm
    2. B. 101 cm
    3. C. 110 cm
    4. D. 1001 cm
    5. E. 1010 cm
35. 
    
    The area of circle with center O is 120 cm². Angle AOB is 60°. Find the area of sector AOB.
    1. A. 2 cm²
    2. B. 3 cm²
    3. C. 6 cm²
    4. D. 20 cm²
    5. E. 60 cm²
36. 
    
    In the diagram, ABCD is a parallelogram. If angle ABC=110° and DEF=40°, find the angle marked θ.
    1. A. 30°
    2. B. 40°
    3. C. 50°
    4. D. 70°
    5. E. 110°
37. 
    
    In the diagram below AC || DF. Angle CBT is 40° and angle DET is 140°. Find y.
    1. A. 320°
    2. B. 280°
    3. C. 220°
    4. D. 100°
    5. E. 80°
38. 
    
    In the diagram below, the angle of elevation of K from M is?
    1. A. 17°
    2. B. 73°
    3. C. 90°
    4. D. 107°
    5. E. 163°
39. 
    
    In the diagram below, ACD is an isosceles triangle with [AD] = [AC] and DC || BE. Find the angle marked x.
    1. A. 55°
    2. B. 62.5°
    3. C. 110°
    4. D. 117.5°
    5. E. 125°
40. Three girls Ama, Adjoa, and Abena measured the sides of three right-angled triangles. Ama’s measurements: 80 mm, 40 mm, 50 mm. Adjoa’s: 50 mm, 120 mm, 130 mm. Abena’s: 20 mm, 30 mm, 40 mm. Whose measurements were correct?
    1. A. Ama's only
    2. B. Abena's and Ama's
    3. C. Adjoa's and Abena's
    4. D. Ama's and Adjoa's only
    5. E. Adjoa's only

Paper 2: Theory

1. a) Simplify $\frac{2a+4b}{3}-\frac{3(a-b)}{2}$.
   
   b) Solve $5(a-5)-\frac{1}{2}(2a+6)=4$.
   
   c) If $\mathbf{r}=\left(\begin{array}{c}3\\ 1\end{array}\right)$ and $\mathbf{q}=\left(\begin{array}{c}-2\\ 1\end{array}\right)$, calculate $6(\mathbf{r}+2\mathbf{q})$.
2. Using a ruler and a pair of compasses only,
   
   a) construct a triangle ABC such that |BA| = 10 cm, ∠ABC = 90° and ∠BAC = 30°. Measure the length BC.
   
   b) i) Bisect the angle ACB to meet BA at D.
   
   ii) What type of triangle is CDA?
   
   c) Calculate the area of triangle ABC.
3. Olu bought a radio for ₵65,000. After one year, the radio was valued at 75% of the cost price.
   
   a) What was the price of the radio after one year?
   
   b) If he sold the radio for ₵55,700, calculate his profit or loss over the cost price.
4. a) Using a scale of 2 cm to 1 unit on both axes, draw two perpendicular lines OX and OY on a graph sheet.
   
   b) On this graph sheet, mark the x-axis from −5 to 5 and the y-axis from −6 to 6.
   
   c) Plot on the same graph sheet the points A(−2, 4) and B(4, −5). Join the points A and B with the help of a ruler.
   
   d) Using the graph, find:
   
   i) the gradient (slope) of the line AB.
   
   ii) the value of x when y = 0.
   
   iii) the value of y when x = 2.
   
   e) Plot on the same graph sheet the points C(−3, −1) and D(3, 3). Join the points C and D. Measure the angle between the lines AB and CD. What is the gradient of the line CD?
5. The ages of 20 school children were recorded as follows:
   
   13, 9, 15, 17, 13, 9, 11, 9, 11, 15, 17, 15, 11, 9, 9, 11, 15, 11, 11, 11.
   
   a) Make a frequency table for the data using the ages 9, 11, 13, 15, and 17.
   
   b) Use your table to calculate the mean age (correct to the nearest whole number).

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