Kuulchat - BECE MATHEMATICS 2017

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OBJECTIVE TEST

If Q = {1,3,5,7,9,10,11,13,15} and T = {1,2,3,5,6,7,10,11,12}, find Q ∪ T.

{1,2,3,5,7,10,11}

{1,3,5,7,9,11,13,15}

{1,2,3,4,5,6,7,8,9,10,11,12,13}

{1,2,3,5,6,7,9,10,11,12,13,15}

If 21:2x = 7:12, find the value of x.

Given that

, find the value of p.

Simplify 3q x 12pq.

A. 15pq²

15p²q

36pq²

36p²q

If A = {2,6,8} and B = {4,6,8,10}, which of the following statements is true?

A ⊂ B

A ∩ B = {2,6,8}

A ∪ B = {2,4,6,8,10}

A ⊃ B

Find the product of 4xy⁴ and x²yz.

4x³y⁴z

4x³y⁵z

4x²y⁴z

4x²y⁴

The sum of the interior angles of a regular polygon with 10 sides is

144^o.

900^o.

1,440^o.

1,800^o.

Solve

The ages of the members of a social club are 20 years, 55 years, 60 years and 25 years. Find the mean age of the members of the club.

20 years

30 years

40 years

50 years

10.

Evelyn saved GH₵ 35.48 every month for 8 months. How much did she save?

GH₵ 183.60

GH₵ 280.63

GH₵ 283.20

GH₵ 283.84

11.

Evaluate

0.012

0.12

1.2

12.0

12.

A woman deposited an amount of GH₵ 50,000.00 at a bank for 2 years at a rate of 20% per annum. Find the simple interest.

GH₵ 1,000.00

GH₵ 2,000.00

GH₵ 10,000.00

GH₵ 20,000.00

13.

What is the total cost of x shirts at GH₵ 5.00 each and y shirts at GH₵ 1.50 each?

A. 5x + 1.5y

5y + 1.5x

5(x + 1.5y)

1.5(5x + y)

14.

At a meeting attended by 23 people, the females were 7 more than the males. How many males were there?

15.

Find the value of x in the diagram.

28^o

30^o

34^o

60^o

16.

How many lines of symmetry does a rhombus have?

17.

In 1995, 215 boys and 185 girls were admitted into a Senior Secondary School. Find, correct to the nearest whole number, the percentage of girls admitted.

46%

47%

53%

54%

18.

Simplify

19.

Solve 25x + 450 ≤ 3000.

x ≥ 102

x ≤ 102

x ≥ 138

x ≤ 138

20.

Give that

and

find a + b.

21.

Mr. Agyekum has 11 of the GH₵ 20.00 notes, 15 of the GH₵ 10.00 notes and 6 of the GH₵ 5.00 notes. How much does Mr. Agyekum have altogether?

GH₵ 280.00

GH₵ 320.00

GH₵ 360.00

GH₵ 400.00

22.

A man travelled a distance of 1.5 km in 30 minutes. What distance can he cover in 50 minutes, travelling at the same speed?

2.2 km

2.5 km

2.8 km

3.2 km

23.

In the diagram, UVW is an isosceles triangle. |UV| = |UW| and angle VUW = 70^o. Find angle UVW.

70^o

60^o

55^o

35^o

24.

Arrange the following in descending order:

25.

The point D (4,3) is reflected in the y-axis. Find the coordinates of its image.

(-4,-3)

(-3,4)

(-4,3)

(3,-4)

26.

Simplify:

27.

Divide 64.5 by 0.015, leaving the answer in standard form.

4.3 x 10⁴

4.3 x 10³

4.3 x 10²

4.3 x 10

28.

The point Q(-2,3) is rotated anticlockwise about the origin through an angle of 90^o. Find the coordinates of its image.

(-3,-2)

(-3,2)

(3,-2)

(3,2)

29.

Elias bought five books. Their mean price was GH₵ 3.25. The total cost for four of the books was GH₵ 11.75. What was the cost of the fifth book?

GH₵ 3.50

GH₵ 4.00

GH₵ 4.20

GH₵ 4.50

30.

Tins of milk each of volume 77 cm³ and weight 170 g were packed into an empty carton of volume 1540 cm³ and weight 500 g.

How many tins of milk can be packed to fill the carton?

31.

Tins of milk each of volume 77 cm³ and weight 170 g were packed into an empty carton of volume 1540 cm³ and weight 500 g.

What is the weight of the carton when packed with the tins of milk?

2.06 kg

2.94 kg

3.90 kg

8.50 kg

32.

A piece of cloth is 8.4 m long. If 30 cm is needed to sew a napkin, how many napkins can be sewn from this piece of cloth?

33.

Express

as a decimal fraction.

0.3200

0.3125

0.3676

0.3222

34.

A match box contains 40 sticks. If 15 of them are spolit, find the probability that a stick chosen at random is not spoilt?

35.

The number of pupils who attended hospital from eight classes on a particular day are:

1,5,3,1,7,5,1,1.

Find the median number.

36.

The number of pupils who attended hospital from eight classes on a particular day are:

1,5,3,1,7,5,1,1.

What is the modal number?

37.

The number of pupils who attended hospital from eight classes on a particular day are:

1,5,3,1,7,5,1,1.

Calculate the mean

38.

The distance from the centre of a circle to any point on it is called

circumference.

diameter.

radius.

sector.

39.

Express 1352 as a product of prime factors.

2³ x 13³

2³ x 13²

2² x 13³

2² x 13²

40.

Which of the following statements about sets is true?

Every set is a subset of the null set.

The universal set is the subset of the null set.

The intersection of two sets is always null set.

The universal set is the union of all its subsets.

THEORY QUESTIONS

In a class of 30 girls, 17 play football, 12 play hockey and 4 play both games.

Draw a Venn diagram to illustrate the given information.

How many girls play:

one or two of the games;

none of the two games?

In the diagram, ABCD is a circle of radius 14 cm and centre O. Line BO is perpendicular to line AC. Calculate, the total area of the shaded portions.

[Take π = $\frac{22}{7}$ ]

Show Solution

Let U = {girls in the class}

n(U) = 30

Let F = {girls who play football}

n(F) = 17

Let H = {girls who play hockey}

n(H) = 12

Number of girls who played only football = Number of girls who played football - Number of girls who play both football and hockey

Number of girls who played only football = 17 - 4 = 13

Number of girls who played only hockey = Number of girls who played hockey - Number of girls who play both football and hockey

Number of girls who played only hockey = 12 - 4 = 8

Number of girls who do not play any of the games is the number outside the two circles.

Number of girls who played none of the games = Number of total girls - sum of the numbers in the circles

Number of girls who played none of the games = 30 - (13+4+8)

Number of girls who played none of the games = 30 - 25

Number of girls who played none of the games = 5

Note: When you sum everything in the rectangle, you should get the total number in the universal set (U)

At least one game is the area of the two circles

One or two games = 13 + 4 + 8 = 25

Neither or none of the two games is the area outside the two circles

none of the two games = 5

Whenever you are asked to calculate the area or volume of a shaded portion, follow the steps below:

Step 1: Calculate the area/volume of the bigger portion
Step 2: Calculate the area/volume of the smaller portion
Step 3: Subtract the area/volume of the smaller portion from the bigger portion to get the area/volume of the shaded portion

Area of the shaded portion = Area of half of the circle - Area of the triangle

Area of a circle = πr²

Area of half of the circle = $\frac{1}{2}$ x πr²

Area of a triangle = $\frac{1}{2}$ x bh

Area of the shaded portion = $\frac{1}{2}$ x πr² - $\frac{1}{2}$ x bh

The radius of the circle = 14 cm

The height (h) of the triangle is also the radius = 14 cm

The base (b) of the triangle = 14cm + 14 cm = 28 cm

NOTE: Radius is the distance from the centre of the circle to any point on the circumference of the circle.

Area of the shaded portion = $\frac{1}{2}$ x $\frac{22}{7}$ x 14 x 14 - $\frac{1}{2}$ x 28 x 14

Area of the shaded portion = 22 x 14 - 28 x 7

Note for 22 x 14: 2 cancels itself and 14, 7 times, 7 cancels itself and 7, 1 time

Note for 28 x 7: 2 cancels itself and 14, 7 times

Area of the shaded portion = 308 - 196

Area of the shaded portion = 112 cm²

Note: Don't forget the unit. You will lose marks for not adding the unit to your final answer.

Two consecutive odd numbers are such that seven times the smaller, sutracted from nine times the bigger, gives 144. Find the two numbers.

A paint manufacturing company has a machine which fills 24 tins with paint in 5 minutes.

How many tins will the machine fill in

1 minute, correct to the nearest whole number?

1 hour?

How many hours will it take to fill 1440 tins?

Given that s = $\frac{n}{2}$ [2a + (n - 1)d], a = 3,d = 4 and n = 10, find the value of s.

Show Solution

Odd numbers are numbers which are not divisible by 2. E.g. 1, 3, 5, 7, 9, 11,...

You could realize that the difference/interval between each consecutive numbers is 2. Hence if you know one odd number, it's next consecutive number is just the addition of 2

Let x = the first odd number

The next consecutive odd number = x + 2

seven times the smaller number (x) = 7 x x = 7x

nine times the bigger (x + 2) = 9 x (x + 2) = 9 x x + 9 x 2 = 9x + 18

seven times the smaller, sutracted from nine times the bigger = 9x + 18 - 7x = 2x + 18

Gives 144 → 2x + 18 = 144

Solve for the value of x to calculate for the first odd number

2x + 18 = 144

2x = 144 - 18

2x = 126

Divide both sides of the equation by 2

$\frac{2x}{2}$ = $\frac{126}{2}$

x = 63

The first odd number = 63

The second odd number = 63+2 = 65

A paint manufacturing company has a machine which fills 24 tins with paint in 5 minutes.

How many tins will the machine fill in

If 5 minutes = 24 tins

1 minute = $\frac{24 tins x 1}{5}$ = 4.8 = 5 tins to the nearest whole number

	4.8
5	24 -20 40 -40 00

Since the statement for the time for the machine is given in minutes, change the hour to minutes.

1 hour = 60 minutes

5 minutes = 24 tins
60 minutes = $\frac{24 tins x 60}{5}$ = 24 x 12 = 288 tins

Note: 5 cancels itself and 60, 12 times

Alternatively

1 minute = 5 tins
60 minutes = 5 tins x 60 = 300 tins

Either 288 or 300 tins is the answer

288 tins = 1 hour
1440 tins = $\frac{1440}{288}$ = 5 hours

Alternatively

300 tins = 1 hour
1440 tins = $\frac{1440}{300}$ = 4.8 hours

s = $\frac{n}{2}$ [2a + (n - 1)d]
a = 3, d = 4 and n = 10

Substitute the values into the equation

s = $\frac{10}{2}$ [2x3 + (10 - 1)4]
s = 5 x [6 + 9 x 4]
s = 5 x [6+36]
s = 5 x 42 = 210

Using a rule and a pair of compasses only, construct:

a triangle ABC with |BC| = 9 cm, |AC| = 8 cm and |AB| = 6 cm;

the perpendicular bisector of line BC;

iii

the bisector of angle ACB.

Label the point of intersection of the two bisectors as Y.

Draw a line to join B and Y.

Measure:

|BY|;

|YC|;

iii

the base angles of triangle BYC.

What type of triangle is BYC?

Show Solution

The table below shows the ages of students admitted in a hospital.

Age (years)	10	11	12	13	14	15
Number of students	5	1	7	10	3	4

Use the information to answer the following question:

What is the modal age?

Calculate, correct to two decimal places, the mean age of the students.

Rice is sold at GH₵56.00 per bag of 50 kg. A trader bought some bags of rice and paid GH₵1,344.00.00

How many bags of rice did the trader buy?

If the trader retailed the bags of rice at GH₵1.40 per kg, how much profit was made on 1 kg of rice?

Show Solution

Modal age = 13

Note: Modal or mode is the one which occurred the most. Thus the one with the highest frequency.

Age (years) (x)	Number of students (f)	fx
10	5	5 x 10 = 50
11	1	1 x 11 = 11
12	7	7 x 12 = 84
13	10	10 x 13 = 130
14	3	3 x 14 = 42
15	4	4 x 15 = 60
	∑f = 30	∑fx = 377

the mean age = $\frac{∑fx}{∑f}$ = $\frac{377}{30}$ = 12.57

	12.566
30	377 -30 77 -60 170 -150 200 -180 200 -180

Number of bags bought = $\frac{Amount Paid}{Cost per bag}$ = $\frac{1344}{56}$ = 24 bags

	24
56	1344 -112 224 -224 000

Cost price per kg = $\frac{GH₵56}{50}$ = GH₵1.12

	1.12
50	56 -50 60 -50 100 -100 0000

Profit per kg = Selling Price per kg - Cost Price per kg

Profit per kg = GH₵1.40 - GH₵1.12

Profit per kg = GH₵0.28 or 28GP

Alternatively

Total kg for the 24 bags = 24 x 50 kg = 1200 kg

Total amount sold for the 24 bags = GH₵1.40 x 1200
Total amount sold for the 24 bags = 14 x 10^-1 x 12 x 10²
Total amount sold for the 24 bags = 14 x 12 x 10^-1 x 10²
Total amount sold for the 24 bags = 168 x 10^-1 x 10²
Total amount sold for the 24 bags = 168 x 10^-1+2
Total amount sold for the 24 bags = 168 x 10¹
Total amount sold for the 24 bags = 168 x 10 = 1680

Profit = Selling Price - Cost Price

Cost Price = 1344

Profit = 1680 - 1344 = GH₵336

Profit per kg = $\frac{GH₵336}{1200}$ = GH₵0.28 or 28GP

	0.28
1200	3360 -2400 9600 -9600 0000

Using a scale of 2 cm to 1 unit on both axes, draw on a graph sheet two perpendicular axes 0x and 0y for -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5.

Plot, indicating the coordinates of all perpendicular points A(2,3) and B(-3,4). Draw a straight line passing through the points A and B.

Plot on the same graph sheet, indicating the coordinates of the points C(4,2) and D(-2,-3). Draw a straight line passing through the points to meet line AB.

Using the graphs in (a):

find the values of y when x = -2;

measure the angle between the lines AB and CD.

Show Solution

If m = $(\begin{matrix} 2x + 1 \\ 2 - 3y \end{matrix})$ , n = $(\begin{matrix} 6 \\ -8 \end{matrix})$ and (m + n) = $(\begin{matrix} 9 \\ -12 \end{matrix})$ , find the:

values of x and y;

components of m.

Solve the inequality:

$\frac{3}{4}$ (x + 1) + 1 ≤ $\frac{1}{2}$ (x -2) + 5.

Illustrate the answer in b(i) on a number line.

In the diagram, AB is parallel to CD.

Find the value of:

Show Solution

$(\begin{matrix} a \\ b \end{matrix})$ + $(\begin{matrix} c \\ d \end{matrix})$ = $(\begin{matrix} a + c \\ b + d \end{matrix})$

m + n = $(\begin{matrix} 2x + 1 \\ 2 - 3y \end{matrix})$ + $(\begin{matrix} 6 \\ -8 \end{matrix})$

m + n = $(\begin{matrix} 2x + 1 + 6 \\ 2 - 3y + -8 \end{matrix})$

m + n = $(\begin{matrix} 2x + 7 \\ - 3y -6 \end{matrix})$

(m + n) = $(\begin{matrix} 9 \\ -12 \end{matrix})$

$(\begin{matrix} 2x + 7 \\ -3y - 6 \end{matrix})$ = $(\begin{matrix} 9 \\ -12 \end{matrix})$

Corresponding components are the same. Thus the top at the left is the same as the top at the right and the bottom at the left is the same as the bottom at the right.

2x + 7 = 9

2x = 9 - 7

2x = 2

Divide both sides by 2

$\frac{2x}{2}$ = $\frac{2}{2}$

x = 1

-3y - 6 = -12

-3y = -12 + 6

-3y = -6

Divide both sides by -6

$\frac{-3y}{-3}$ = $\frac{-6}{-3}$

y = 2

m = $(\begin{matrix} 2x + 1 \\ 2 - 3y \end{matrix})$

x = 1, y = 2

2x + 1 = 2 x 1 + 1 = 2 + 1 = 3

2 - 3y = 2 - 3 x 2 = 2 - 6 = -4

m = $(\begin{matrix} 3 \\ -4 \end{matrix})$

$\frac{3}{4}$ (x + 1) + 1 ≤ $\frac{1}{2}$ (x -2) + 5

Multiply both sides of the equation by the L.C.M of the denominators (4 and 2). The L.C.M is 4.

4 x $\frac{3}{4}$ (x + 1) + 4 x 1 ≤ 4 x $\frac{1}{2}$ (x -2) + 4 x 5

3(x + 1) + 4 ≤ 2(x -2) + 20

Expand the brackets

3 x x + 3 x 1 + 4 ≤ 2x -4 + 20

3x + 3 + 4 ≤ 2x + 16

3x + 7 ≤ 2x + 16

3x - 2x ≤ 16 - 7

x ≤ 9

Note: The dot must be shaded since it's less than or equal to.

Vertically opposite angles are equal

x = 47 + 102

x = 149^o

Corresponding angles are equal

The angle on the same line AB with y = x

Sum of angles on a straight line = 180^o

y + x = 180

But x = 149^o

y + 149 = 180

y = 180 - 149

y = 31^o

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OBJECTIVE TEST

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