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Transcript of Mathematics model exam_vol-2_ppt_design
CSEC Model Exam 1 Paper 2
CSEC Model Exam 2 Paper 1
CSEC Model Exam 2 Paper 2
CSEC Model Exam 3 Paper 1
CSEC Model Exam 3 Paper 2
CSEC Model Exam 4 Paper 1
CSEC Model Exam 4 Paper 2
CSEC Model Exam 5 Paper 1
CSEC Model Exam 5 Paper 2
CSEC Model Exam 1 Paper 1
CSEC MODEL EXAMINATIONS
Caribbean Educational Publishers Ltd.
CSEC MODEL EXAMINATION 1MATHEMATICS
Paper 190 minutes
Answer ALL the questions
NEXT
1. The decimal fraction 0.625 written as a common fraction, in its lowest terms, is
ANSWER
(C)(A) (D)(B)4
5
3
45
88
9
Model Exam 1 Paper 1
1. Multiply the decimal fraction with three
decimal places by , which is 1, to make
the decimal fraction a common fraction.
625 25
1 000 2525
4025 5
405
8
5
1 0000.625 0.625
1 000625
1 000
1000
1000
Divide both the numerator and the denominator by their common factor 25.
Divide both the numerator and the denominator by their common factor 5.
This is the common fraction written in its lowest terms.
Model Exam 1 Paper 1
2. The number 8.150 46 written correct to 3 decimal places is
ANSWER
(C) 8.151(A) 8.149 (D) 8.152(B) 8.150
Model Exam 1 Paper 1
2. 8.150 46 = 8.150 4 6 = 8.150
The digit in the 4th decimal place is 4, which is less than 5, so we do not add 1 to the digit in the 3rd decimal place. The digit in the 3rd decimal place remains unchanged.
Model Exam 1 Paper 1
3. The exact value of 0.615 × 0.07 is
ANSWER
(C) 4.305
(A) 0.043 05
(D) 43.05
(B) 0.430 5
Model Exam 1 Paper 1
3. 0.615 οr
0.07
0.043 05
0.615 × 0.07 = 0.043 05 3 dp + 2 dp = 5 dp
615
7
4 305
Model Exam 1 Paper 1
4. The exact value of 7 ÷ (0.01)2 is
ANSWER
(C) 7 000
(A) 0.000 7
(D) 70 000
(B) 0.000 07
Model Exam 1 Paper 1
4.
Invert the product of fractions which is the divisor and multiply instead of divide.
Model Exam 1 Paper 1
22
2
77 (0.01)
(0.01)
7(0.01) 0.01 0.01
0.01 0.017 1
0.011 1 100
100 1007 100 100
1 1 170 000
5. If $450 is divided into two portions in the ratio 4:5, then the smaller portion is
ANSWER
(C) $200(A) $50 (D) $250(B) $150
Model Exam 1 Paper 1
5. The number of equal parts = 4 + 5 = 9
the smaller portion= of $450
= × $450
= 4 × $50 = $200
4
9
4
9
Model Exam 1 Paper 1
6. If 40% of a number is $70, what is the number?
ANSWER
(C) $175(A) $110 (D) $200(B) $150
Model Exam 1 Paper 1
6.
Model Exam 1 Paper 1
3 2
40% of the number $70
100100% of the number $70
40
$
$ 5
70
7
0
14
7. What is the least number of cherries that can be shared equally among 5, 10 or 15 children?
ANSWER
(C) 60(A) 30 (D) 75(B) 45
Model Exam 1 Paper 1
7. 2 5, 10, 15 3 5, 5, 15 5 5, 5, 5
1, 1, 1
LCM = 2 × 3 × 5 = 30The answer to this question is the LCM of 5, 10 and 15.
Model Exam 1 Paper 1
8. What is the greatest number that can divide exactly into 12, 20 and 60?
ANSWER
(C) 6(A) 2 (D) 8(B) 4
Model Exam 1 Paper 1
8. 2 12, 20, 60 2 6, 10, 30 3, 5, 15
HCF = 2 × 2 = 4The answer to this question is the HCF of 12, 20 and 60.
2 is a common factor of thethree numbers.2 is a common factor of thethree numbers.
Model Exam 1 Paper 1
9. The exact value of 85 × 104 is
ANSWER
(C) (85 × 100) (85 × 4)
(A) (85 × 100) + 4
(D) (85 × 100) + (85 × 4)
(B) (85 × 100) – (85 × 4)
Model Exam 1 Paper 1
9. 85 × 104 = 85 × (100 + 4)= 85 × 100 + 85
× 4= (85 × 100) +
(85 × 4)
Using the distributivelaw.
Model Exam 1 Paper 1
10. The value of the digit 3 in 736.2 is
ANSWER
(C) 3 tens
(A) 3 tenths
(D) 3 hundreds
(B) 3 ones
Model Exam 1 Paper 1
10. Hundreds Tens Ones Tenths7 3 6 2
The value of the digit 3 in 736.2 is 3 tens.
Model Exam 1 Paper 1
11. The simple interest earned on $600 at 5% perannum for 3 years is given by
ANSWER
(C)
(A)
(D)
(B)600 5
$100 3
600 3$
100 5
600 100$
3 5
600 5 3$
100
Model Exam 1 Paper 1
11. The simple interest,
P = $600R = 5 %T = 3 years$60
1000 5 3
100
PRTI
Model Exam 1 Paper 1
12. A woman bought a sheep for $800 and sold it for $1200. Her gain as a percentage of the cost price is
ANSWER
(C) 45%(A) (D) 50%(B) 40%1
33 %3
Model Exam 1 Paper 1
12.
Model Exam 1 Paper 1
The gain $(1 200 800) $400
$ 400The percentage gain
1
$ 8002
100%
50%
13. An insurance salesman is paid 4% of his sales as commission. His sales for March were $5 025. How much commission was he paid?
ANSWER
(C) $201.00
(A) $50.25
(D) $402.00
(B) $100.50
Model Exam 1 Paper 1
13. The commission 4% of $5 025
4$5 025
1004 $50.25
$201.00
Model Exam 1 Paper 1
14. If the simple interest on $700 for 4 years is $168, then the rate of interest per annum is
ANSWER
(C) 7%(A) 5% (D) 8%(B) 6%
Model Exam 1 Paper 1
14. The rate of interest per annum,
100
100
IR
PT
168
42
7 00 41
%
6%
I = $168P = $700T = 4 years
Model Exam 1 Paper 1
15. The exchange rate for one United States dollar (US $1.00) is six dollars and thirty-four cents in Trinidad and Tobago currency (TT $6.34). What is the value of US $50 in TT currency?
ANSWER
(C) $264(A) $634 (D) $134(B) $317
Model Exam 1 Paper 1
15. US $1.00 TT $6.34
US $50 TT $6.34 50
TT $317
Model Exam 1 Paper 1
16. A sales tax of 10% is charged on an article. How much does a customer pay for an article marked as $75?
ANSWER
(C) $80.00
(A) $82.50
(D) $79.50
(B) $82.00
Model Exam 1 Paper 1
16. The amount the
customer paid (100 10)% of $75
110% of $75
1.10 $75
$82.50
Model Exam 1 Paper 1
17. Samuel invested $800 for 3 years at 5% per annum. Marina invested $600 at the same rate.If they both earned the same sum as simpleinterest, how many years did Marina invest hermoney?
ANSWER
(C) 4(A) 6 (D) 3(B) 5
Model Exam 1 Paper 1
17. Samuel’s interest,
P = $800R = 5%T = 3 years
The time,
I = $120P = $600R = 5%
Model Exam 1 Paper 1
18. A discount of of the marked price is offered
for cash. What is the discount on a dress with amarked price of $150?
ANSWER
(C) $37.50
(A) $25.00
(D) $40.50
(B) $30.00
1
4
Model Exam 1 Paper 1
18.1
The discount of $15041
$1504$37.50
Model Exam 1 Paper 1
19. If P = {2, 3, 5, 7, 9}, Q = {3, 7, 8} andS = {7, 8, 9}, then
ANSWER
(C) {7}
(A) { }
(D) {2, 3, 5, 7, 8, 9}
(B) {2}
Model Exam 1 Paper 1
19.
Model Exam 1 Paper 1
20. U = {integers} N = {natural numbers}Zn = {negative integers}
Which of the Venn diagrams given below illustrates the statement:
Model Exam 1 Paper 1
“No natural numbers are negative integers”?
ANSWER
(C)
(A)
(D)
(B)
Model Exam 1 Paper 1
20. No natural numbers are negative integers:
Model Exam 1 Paper 1
21.
ANSWER
(C) 6(A) 2 (D) 8(B) 4
In the Venn diagram shown above, n(L) = 8, n(M) = 10 and What is
Model Exam 1 Paper 1
21.
Model Exam 1 Paper 1
22.
ANSWER
The two circles above represent set X and set Y.If X = {factor of 8} and Y = {factor of 12}, thenthe shaded region represents
(C) {2, 4, 6, 8}
(A) { }
(D) {4, 6, 8, 12}
(B) {1, 2, 4}
Model Exam 1 Paper 1
22. 8 1 8
2 4
12 1 12
2 6
3 4
Model Exam 1 Paper 1
23. The scale on a map is stated as 1:500 000. Thedistance between two towns as measured onthe map is 1.8 cm. What is the actual distancebetween the two towns?
ANSWER
(C) 9.0 km
(A) 0.9 km
(D) 50 km
(B) 1.8 km
Model Exam 1 Paper 1
23. 1:500000 cm 500000 cm
5 00 000
100 000
km
5 km
1.8 cm 5 km×1.8
9.0 km
Model Exam 1 Paper 1
24. The number of kilometres travelled by a vehicle in t hours at a rate of s km per hour is
ANSWER
(C) (A) (D) (B) stst
60ts
ts
Model Exam 1 Paper 1
24. Speed,
Distance, d = st
ds
t Formula
Multiply both sides by t.d is the subject of the formula.
Model Exam 1 Paper 1
25. A cuboid with dimensions 12 cm, 10 cm and5 cm occupies space of volume
ANSWER
(C) 81 cm3
(A) 27 cm3
(D) 600 cm3
(B) 54 cm3
Model Exam 1 Paper 1
25. The volume of the cuboid,
The formula for the volume of acuboid.
3
V
12cm 10cm 5cm
600cm
lbh
Model Exam 1 Paper 1
26. A cylindrical block of cheese 8 cm thick has a volume of 500 cm3. A student cuts a uniform slice of 2 cm thickness. What volume of the cheese did the student take?
ANSWER
(C) 100 cm3
(A) 50 cm3
(D) 125 cm3
(B) 75 cm3
Model Exam 1 Paper 1
26.
Model Exam 1 Paper 1
3
3
3
The volume of
2the cheese taken 500 cm
81
500 cm4
125 cm
27.
ANSWER
The figure above, not drawn to scale, shows the sector of a circle with centre O. The length of the minor arc PQ is 7 cm. The length of the circumference of the circle is
(C) 56 cm
(A) 21 cm
(D) 63 cm
(B) 28 cm
Model Exam 1 Paper 1
27.
Model Exam 1 Paper 1
360The number of 45 in 360 8
45The length of the circumference 7 cm 8
56 cm
28. The distance around the edge of a circular table top is 352 cm. The radius of the table top, in centimetres, is
ANSWER
(C) (A) 88 (D) (B) 352176
352
Model Exam 1 Paper 1
28.
Model Exam 1 Paper 1
The circumference, 2
So 352 2
352 i.e.
1762
C r
r
r
29. A plane left Guyana at 21:00 h. The next day, the plane arrived at its destination in the same time zone at 02:30 h. How many hours did the flight take?
ANSWER
(C) (A) (D) (B) 31
22
15
2
123
2
Model Exam 1 Paper 1
29.
Model Exam 1 Paper 1
The number of hours to
midnight (24:00 21:00) h
3 hours
The number of hours after
1midnight 2 hours
2The number of hours taken for
1the flight 3
15 hours
2
2 hours2
30. An aircraft leaves airport A at 07:30 h and arrives at airport B at 12:30 h, the same day, in the same time zone. The distance between the two airports is 3 600 kilometres. What was the average speed of the aircraft for the flight?
ANSWER
(C) 480 km/h
(A) 180 km/h
(D) 720 km/h
(B) 288 km/h
Model Exam 1 Paper 1
30. The time taken, (12:30 07:30) hours
5 hours
The average speed,
3 600 km
5 h720 km/h
t
ds
t
Model Exam 1 Paper 1
31. Each of the letters of the word ‘PERFORM’ is written on a piece of paper. One piece of paper is drawn at random. What is the probability that a letter ‘R’ is drawn?
ANSWER
(C) (A) (D) (B) 1
6
1
3
1
7
2
7
Model Exam 1 Paper 1
31.
Model Exam 1 Paper 1
The number of Rs 2
The total number of letters 7
(2
7 )P R
32. A die is tossed twice. What is the probability that a ‘2’ followed by an odd number turns up?
ANSWER
(C) (A) (D) (B) 1
12
1
4
1
36
2
3
Model Exam 1 Paper 1
32.
Model Exam 1 Paper 1
(2 followed by
an odd number) (2) (odd number)
1 3
6 61 1
61
12
2
P
P P
33.
ANSWER
The bar chart shows the number of students who liked one of five stated colours. How many students took part in the survey?
(C) 80(A) 5 (D) 125(B) 45
Model Exam 1 Paper 1
33. The number of students = 20 + 10 + 5 + 20 + 25
= 80
Model Exam 1 Paper 1
34.
ANSWER
The pie-chart above represents the fruit a group of students ate. If 16 students ate mandarin, then the total number of students in the group is
(C) 128(A) 125 (D) 135(B) 45
Model Exam 1 Paper 1
34. The angle representing the number of students
who ate mandarin 180° 135°
45°
360The total number of student
12
s 1645
6 8
8
1
Model Exam 1 Paper 1
35. The lowest weekly wage of a group of employees is $520.60. What is the wage of the highest paid employee, if the range of the wages is $63.20?
ANSWER
(C) $520.60
(A) $63.20
(D) $583.80
(B) $457.40
Model Exam 1 Paper 1
35. The range = The highest weekly wage − The lowest weekly wage $520.60 = The highest weekly wage − $63.20The highest weekly wage = $520.60 + $63.20
= $583.80
Model Exam 1 Paper 1
36.
ANSWER(C) 8.5 and 14.5
(A) 0 and 2.5
(D) 14.5 and 20.5
(B) 2.5 and 8.5
The lengths of the pencils of 40 students were measured, to the nearest cm, and the information collected is shown in the frequency table above.
The least and greatest length of the class interval 15–20 are
Frequency 9 17 14
Length of pencil (cm) 3–8 9–14 15–20
Model Exam 1 Paper 1
36.
Model Exam 1 Paper 1
3 8 9 14 15 20
2.5 8.5 14.5 20.5
class interval
class boundary
37. (5a)2 =
ANSWER
(C) 10a2(A) 10a (D) 25a2(B) 25a
Model Exam 1 Paper 1
37. Meaning of a square.
Expanding the term.
Grouping like values.
Multiplying like values.
Simplifying.2
2
2
(5 ) 5 5
5 5
5 5
25
25
a a a
a a
a a
a
a
Model Exam 1 Paper 1
38. 2x3 × 3x2 =
ANSWER
(C) 6x6(A) 5x5 (D) 36x5(B) 6x5
Model Exam 1 Paper 1
38.
Expanding the term.
Grouping like values.
Multiplying like values.
Adding the indices.
Simplifying.
3 2
3 2
3 2
3+2
5
5
2 3
2 3
2
6
6
6
3
x x
x x
x x
x
x
x
Model Exam 1 Paper 1
39. (6a) × (3b) =
ANSWER
(C) 9ab(A) 9ab (D) 18ab(B) 18ab
Model Exam 1 Paper 1
39.
Expanding theterm.
Grouping likevalues.
Multiplying likevalues.
Simplifying.
( 6 ) ( 3 )
( 6) ( 3)
( 6)
18
( 3)
18
a b
a b
a
b
b
a
a
b
Model Exam 1 Paper 1
40. 4(3x y) − 2(5y 3x) =
ANSWER
(C) 6x 14y
(A) 18x 14y
(D) 18x 6y
(B) 8x 4y
Model Exam 1 Paper 1
40. 4(3x y) 2(5y 3x) = 4 × 3x + 4 × (y) 2 × 5y 2 × (3x) = 12x 4y 10y 6x = 12x 6x 4y 10y = 18x 14y
Using the distributivelaw.
Simplifying each term.Grouping like terms.Adding like terms.
Model Exam 1 Paper 1
41. If
ANSWER
(C) 10(A) (D) 3(B)
2 , then 5 2p q p pq
15 6
Model Exam 1 Paper 1
41. Substituting the value forp and for q in the formula.Simplifying the twoterms.Subtracting.
25 2 5 5 2
25 1
15
0
Model Exam 1 Paper 1
42. If 40 − 3x = x + 8, then x =
ANSWER
(C) 8(A) 4 (D) 29(B) 8
Model Exam 1 Paper 1
42.
Grouping like terms.
Adding like terms.
Dividing both sides by −4.
Simplifying.
Model Exam 1 Paper 1
40 3 8
3 8 40
So 4 32
2
48
3
x x
x x
x
x
43.
ANSWER
(C)
(A)
(D)
(B)
2 ( 3 ) (3 2 )a a b b a b 2 22 3a ab b 2 22 3 2a ab b
2 22 9 2a ab b 2 22 9 2a ab b
Model Exam 1 Paper 1
43. Use the distributive
law to remove the
brackets.
Adding the middle
terms.
2
2
2
2
2 ( 3 ) (3 2 )
2 6
2
3 2
3 2
a a b b a b
a ab a
a b
b b
ab
Model Exam 1 Paper 1
44. If
ANSWER
(C) (A) 5 (D) (B) 5
2
, when 5, then1
vK v K
v
1
64
16
4
Model Exam 1 Paper 1
44.
Substituting the value for v in theformula.
Squaring and subtracting values.
Dividing.
2
2
1
( 5)
5 125
41
64
vK
v
Model Exam 1 Paper 1
45. Yuri’s age is ten years less than twice that of Christine’s age. If Christine’s age is x years, then Yuri’s age, in years, is
ANSWER
(C) x 10
(A) 2(x 5)
(D) 2x 5
(B) 2(x 10)
Model Exam 1 Paper 1
45. Yuri’s age (2 10) years
(2 10) year
2( 5 ears
s
) yx
x
x
Model Exam 1 Paper 1
46. Which of the equations stated below represents the equation of a straight line?
ANSWER
(C) y = 5x2
(A) y = 3x
(D) y = 4x3
(B)2
yx
Model Exam 1 Paper 1
46. The equation of a straight line is y = mx + cIf c = 0, then y = mxSo y = −3x is the equation of a straight line.
Model Exam 1 Paper 1
47. The gradient of the straight line 2y = 4 5x is
ANSWER
(C) 4(A) 5 (D) 2(B) 5
2
Model Exam 1 Paper 1
47.
Writing the terms on the RHS
in the form mx + c.
Dividing each term by 2.
2 4 5
2 5 4
52
2
y x
y x
y x
It is in the form y = mx + c.
So the gradient,5
2.m
Model Exam 1 Paper 1
48. If
ANSWER
(C) 11(A) 1 (D) 17(B) 7
2( ) 5, then ( 3)g x x x g
Model Exam 1 Paper 1
48. Substitute −3 for x.
Simplifying.
Adding.
2( 3) ( 3) ( 3) 5
9 3 5
17
g
Model Exam 1 Paper 1
49.
ANSWER
The relation diagram shown above represents a function. Which of the following equations best describes the function?
(C) f (x) = x 2
(A) f (x) = x
(D) f (x) = 2(x 1)
(B) f (x) = y
Model Exam 1 Paper 1
49. (2) 2(2 1) 2(1) 2 2 2(3) 2(3 1) 2(2) 4 3 4(5) 2(5 1) 2(4) 8 5 8(7) 2(7 1) 2
( ) 2( 1)
(6) 12 7 12
f
f
f
f
ff xff
f x
Model Exam 1 Paper 1
50. Which of the following diagrams is not the graph of a function?
(A)
Model Exam 1 Paper 1
50. (B)
Model Exam 1 Paper 1
50. (C)
Model Exam 1 Paper 1
50. (D)
ANSWER
Model Exam 1 Paper 1
50.
Using the vertical line test for a function, it can be seen that The graph represents a one-to-many relation and it is therefore not a function.
Model Exam 1 Paper 1
51.
Model Exam 1 Paper 1
In the graph above, when y = 2, the values of x are:
ANSWER
(C) ±1.4(A) ±1.2 (D) ±1.5(B) ±1.3
Model Exam 1 Paper 1
51.
From the construction on the graph, when y = 2, then x = 1.4 and x = 1.4, that is x = ±1.4.
Model Exam 1 Paper 1
52.
ANSWER
The half-lines BA and CD are parallel. If angle BCD is 65°, then angle ABC is
(C) 130°(A) 65° (D) 145°(B) 115°
Model Exam 1 Paper 1
52. Interior angles are
supplementary.
Substitute the
value of angle
BCD.
Subtract 65° from
both sides.
Subtracting.
ˆˆ 180°
ˆ 65° 180°
ˆ 180° 65
115°
°
ABC BCD
ABC
ABC
Model Exam 1 Paper 1
53.
ANSWER
AC and DE are straight lines that intersects at B.Angle ABE = 127°The size of angle ABD is
(C) 127°(A) 53° (D) 233°(B) 74°
Model Exam 1 Paper 1
53. The sum ofangles on astraight line.Substitute thevalue for angleABE.Subtract 127°from both sides.Subtracting.
ˆ ˆ 180°
ˆ 127° 180°
ˆ 180° 127°
53°
ABD ABE
ABD
ABD
Model Exam 1 Paper 1
54.
ANSWER
The line segment PQ is mapped onto the line segment P′ Q′ by a translation. The matrix that represents this translation is
(C) (A) (D)(B)1
2
2
1
2
3
3
5
Model Exam 1 Paper 1
54.
5 3
2
3 2
5 3
3 2
1
T P P
T P P
or
8 6
6
1
5
2
8 6
6 5
T Q Q
T Q Q
Model Exam 1 Paper 1
or
Model Exam 1 Paper 1
55.
Model Exam 1 Paper 1
The shaded triangle is rotated through an angle of 90° in a counter-clockwise direction about thepoint P. Which of the four triangles represent theimage of the shaded triangle?
ANSWER
(C) C(A) A (D) D(B) B
Model Exam 1 Paper 1
55.
Model Exam 1 Paper 1
56.
Model Exam 1 Paper 1
In the diagram above, the line segment PQ is the image of LM after
ANSWER
(C) a reflection in the x-axis
(A) an enlargement of scale factor 1
(D) a rotation through with centre O
(B) a translation by vector2
5
90
Model Exam 1 Paper 1
Mx means a reflection in the x-axis.
56.
Model Exam 1 Paper 1
57.
Model Exam 1 Paper 1
The point P shown in the graph above is reflected in the x-axis. What are the co-ordinates of the image of P?
ANSWER
(C) (2, 3)
(A) (3, 2)
(D) (2, 3)
(B) (3, 2)
Model Exam 1 Paper 1
57.
Model Exam 1 Paper 1
58. In a triangle ABC, if angle A = 2x° and angle B = 3x°, then angle C =
ANSWER
(C) (180 5x)°
(A) 36°
(D)
(B) 72°
180
5x
Model Exam 1 Paper 1
58. The sum ofthe angles of atriangle.Substitute thevalue for angle Aand for angle B.Add the xs.
Subtract 5x fromboth sides.
Simplifying.
Model Exam 1 Paper 1
ˆ ˆˆ 180°
ˆ2 ° 3 ° 180°
ˆSo 5 ° 180°
ˆ
(18
180°
0 5 )°
5 °
A B C
x x C
x
C
x
C
x
59.
ANSWER
In the right-angled triangle, tan θ =
(C) (A) (D)(B)5
13
12
13
5
12
12
5
Model Exam 1 Paper 1
59. Definition of the tangent of anangle.
Using the capital lettersnotation.
Substituting the length foreach side.
tanθ
12 cm
Opp
Adj
AB
BC
5 cm
12
5
Model Exam 1 Paper 1
60.
Model Exam 1 Paper 1
The diagram above, not drawn to scale, shows that the angle of depression of a point A on the ground from T, the top of a tower, is 40°. A is 25 m from B, the base of the tower. The height, TB, of the tower, in metres, is
ANSWER
(C) 25 tan 40°
(A) 25 sin 40°
(D) 25 sin 60°
(B) 25 cos 40°
Model Exam 1 Paper 1
60.
Model Exam 1 Paper 1
Alternateangles.Definition ofthe tangent ofan angle.Substitute thelength of AB.Multiply bothsides by 25 m.
Model Exam 1 Paper 1
ˆ 40°
tan 40°
25 m25 m tan 40°
or 25 m tan 4 0°
TAB
Opp
Adj
TB
ABTB
TB
TB
CSEC MODEL EXAMINATION 1
MATHEMATICS
Paper 2
2 hours 40 minutes
SECTION I
Answer ALL the questions in this section
All working must be clearly shown
NEXT
1. (a) Using a calculator, or otherwise, calculate
the EXACT value of
ANSWER
(i)
(3 marks)
73
151 3
2 43 5
giving your answer as a common fraction
Model Exam 1 Paper 2
1. (a) (i)
Model Exam 1 Paper 2
(ii)
ANSWER
(3 marks)giving your answer in standard form.
0.0225
36
Model Exam 1 Paper 2
1. (a) Using a calculator, or otherwise, calculate
the EXACT value of
1. (a)(ii)
2
4
2
0.0225 225 10
36 36
15 10
6
2.5 10
Standard form
Model Exam 1 Paper 2
The basic wage earned by a factory worker for a 40-hour week is $640.00.
(i) Calculate her basic hourly rate.
For overtime work, the factory worker is paid one and a half times the basic hourly rate.
1. (b)
ANSWER
(1 mark)
Model Exam 1 Paper 2
1. (b) (i) The basic hourly rateThe basic wage
The basic week$640
4
$1
0
6
Model Exam 1 Paper 2
The basic wage earned by a factory worker for a 40-hour week is $640.00.
(ii) Calculate her overtime wage for 15 hours of overtime.
1. (b)
ANSWER
(2 marks)
Model Exam 1 Paper 2
1. (b) (ii) The overtime hourly rate = The overtime rate The basic hourly rate
The overtime wage = The overtime hourly rate The number of hours worked overtime= $24 15= $360
Model Exam 1 Paper 2
1. (b) The basic wage earned by a factory worker for a 40-hour week is $640.00.
(iii) Calculate the total wages earned by the factory worker for a 60-hour week.
ANSWER
(3 marks)Total 12 marks
Model Exam 1 Paper 2
1. (b)
(iii)
The number of hours worked overtime
= (60 40) hours = 20 hours
The overtime wage = $24 20= $480
The total wages earned
= The basic wage The overtime wage= $(640 480)= $1120
Model Exam 1 Paper 2
Factorise completely:2. (a)
(i) 8px 5py 8qx 5qy
ANSWER
(2 marks)
Model Exam 1 Paper 2
2. (a)(i) 8px 5py 8qx 5qy
= p(8x 5y) q(8x 5y)= (8x 5y) (p q)
Factorise pairwise
Factoriseusing 8x 5yas a commonfactor.
Model Exam 1 Paper 2
(ii) 4x2 36
ANSWER
Model Exam 1 Paper 2
Factorise completely:2. (a)
(2 marks)
2. (a)(ii) 4x2 36
= 4(x2 9)= 4(x2 32)= 4(x 3)(x 3)
Factorise using 4 as the HCF.
Write as the differenceof two squares.Factorise as the difference of two squares.
Model Exam 1 Paper 2
(iii) 5x2 6x 8
ANSWER
Model Exam 1 Paper 2
Factorise completely:2. (a)
(2 marks)
2
2
610
5( 8)
( 2)(5 4
440
5 6 8
5 10 4 8
5 ( 2 ( 2)
)
) 4
x
x x
x x x
x x
m nm
m
x
n
x
n
2. (a)(iii)
Factorise pairwiseFactorise usingx 2 as a commonfactor.
Model Exam 1 Paper 2
2. (b)One cup of yogurt costs $x and one granolabar costs $y.
One cup of yogurt and three granola bars cost$32.00, while two cups of yogurt and two granola bars cost $30.00.
(i) Write a pair of simultaneous equations in x and y to represent the given information above.
ANSWER(2 marks)
Model Exam 1 Paper 2
2. (b) (i) The cost of one cup of yogurt = $x
The cost of one granola bar = $y
The first equation is:x 3y = 32 (in dollars)
The second equation is:2x 2y = 30 (in dollars)
The pair of simultaneous equations in x and y:
x 3y = 32 2x 2y = 30
Model Exam 1 Paper 2
2. (b)
ANSWER
One cup of yogurt costs $x and one granola bar costs $y.
One cup of yogurt and three granola bars cost $32.00, while two cups of yogurt and two granola bars cost $30.00.
Solve the equations to find the cost of one cup of yogurt and the cost of one granola bar. (4 marks)
Total 12 marks
Model Exam 1 Paper 2
(ii)
Hence, the cost of a yogurt is $6.50 andthe cost of a granola bar is $8.50.
2. (b) (ii)
Model Exam 1 Paper 2
So
In a survey of 85 students,
25 played drums
20 played tassa
x played drums and tassa
3x played neither.
Let D represent the set of students in the survey who played drums, and T the set of students who played tassa.
Copy and complete the Venn diagram
below to represent the information obtained from the survey.
(i) ANSWER
3. (a)
(2 marks)
3. (a)
(i)
The Venn diagram is shown above.
The students who played drums only,
The students who played tassa only,
Model Exam 1 Paper 2
( ) 25 xn D T ( ) 20 xn T D
In a survey of 85 students,
25 played drums
20 played tassa
x played drums and tassa
3x played neither.
Let D represent the set of students in
the survey who played drums, and T
the set of students who played tassa.
Write an expression in x for the totalnumber of students in the survey.
(ii) ANSWER
(1 mark)
Model Exam 1 Paper 2
3. (a)
The total number of students in thesurvey, n(U ) = 25 x + x + 20 x + 3x
= 2x + 45
3. (a)
(ii)
Model Exam 1 Paper 2
3. (a)
ANSWER(2 marks)Calculate the value of x.
Model Exam 1 Paper 2
In a survey of 85 students,
25 played drums
20 played tassa
x played drums and tassa
3x played neither.
Let D represent the set of students in the survey who played drums, and T the set of students who played tassa.
(iii)
3. (a)
(iii) n(U ) = 85 and n(U ) = 2x + 45
so we have the following equation:
Hence, the value of x is 20.
Subtract 45 fromboth sides.
Divide both sidesby 2.
Model Exam 1 Paper 2
2 45 85So 2 85 45i.e. 2 4
20
040
2
xxx
x
(i) Using a ruler, a pencil, and a pair of compasses, construct the kite PQRS accurately. ANSWER(4 marks)
Model Exam 1 Paper 2
The diagram below, not
drawn to scale, shows a
kite, PQRS, with the
diagonal PR = 6 cm,
3. (b)
3. (b) (i)
Draw a horizontal line greater than 6 cm. Mark a point P to the left of the line. Set your compasses to a separation of 6 cm using a ruler. Place the steel point of the compasses at point P and construct an arc to intersect the horizontal line at point R. PR = 6 cm.
Model Exam 1 Paper 2
Set your compasses to a separation of
that is, 3.25 cm. With point P as centre, construct an arc above PR. With point R as centre and the same compasses separation, construct another arc to intersect the previous arc at point Q.
13 cm,
4
13 cm.
4PQ RQ
Set your compasses to a separation of 5 cm. With centres P and R, construct two arcs below PR to intersect at point S.
PS = RS = 5 cm.
Use a ruler and pencil to draw the four sides of the kite PQRS.
Model Exam 1 Paper 2
(ii) Join QS. Measure and state, in centimetres, the length of QS.
ANSWER(2 marks)
Total 11 marks
Model Exam 1 Paper 2
The diagram below, not drawn to scale,
shows a kite, PQRS, with the diagonal
PR = 6 cm,
3. (b)
Draw a straight line from Q to S. Take a divider and open it from point Q to point S. Measure the separation of the divider using a ruler.
3. (b) (ii)
15 cm 5.
425 cmQS The length of
Model Exam 1 Paper 2
The table below shows two readings from the records of a train.
4. (a)
Town Time Distance travelled (km)
X 07:20 538
Y 09:50 773
Calculate(i) the number of hours taken for the
journey from town X to town YANSWER
(1 mark)
Model Exam 1 Paper 2
4. (a) (i) The number of hours taken for the journey from town X to town Y,
(09 : 50 07 : 20) h
2h 30min
12 h
2
t =
=
Model Exam 1 Paper 2
4. (a)
Calculate (ii) the distance travelled, in kilometres, between the two towns ANSWER(1 mark)
Model Exam 1 Paper 2
The table below shows two readings from the records of a train.
Town Time Distance travelled (km)
X 07:20 538
Y 09:50 773
4. (a) (ii) The distance travelled between the two towns, (773 538) km
235 kmd
Model Exam 1 Paper 2
4. (a)
ANSWER(2 marks)
Model Exam 1 Paper 2
Calculate (iii) the average speed of the train in km/h
The table below shows two readings from the records of a train.
Town Time Distance travelled (km)
X 07:20 538
Y 09:50 773
4. (a)
235 km1
2 h2
235km/h
52
2235 km/h
54794 km/h
2 km/h
ds
t
The average speed of the train,(iii)
Model Exam 1 Paper 2
4. (b) The map shown below is drawn to a scale of 1:500 000.
ANSWER(2 marks)
(i) Measure along a straight line and state, in centimetres, the distance on the map from P to Q.
Model Exam 1 Paper 2
Open your divider from P to Q, then measure the separation using a ruler.
4. (b) (i)
The distance on the map from P to Q = 5.8 cm
Model Exam 1 Paper 2
(ii) Calculate the actual distance, in kilometres, from P to Q ANSWER
(2 marks)
Model Exam 1 Paper 2
4. (b) The map shown below is drawn to a scale of 1:500 000.
The scale is 1: 500 0004. (b) (ii)
The actual distancefrom P to Q = 5.8 5 km = 29.0 km
Model Exam 1 Paper 2
That is 1cm 500 000 cm
5 00 000So 1cm
100 000
km
1cm 5 km
(iii) The actual distance between two places is 8.5 km. Calculate the number of centimetres that represent this distance on the map
ANSWER(3 marks)Total 11 marks
Model Exam 1 Paper 2
4. (b) The map shown below is drawn to a scale of 1:500 000.
4. (b) (iii) 5 km is represented by 1 cm
1 km is represented by1
cm5
8.5 km is represented by 8.5
= 1.7 cm
1cm
5
Model Exam 1 Paper 2
ANSWER
(1 mark)
5. (a) Given that f (x) = 4x − 7 and g(x) = x2 − 15, calculate the value of
(i) f (−3)
Model Exam 1 Paper 2
5. (a) (i) ( ) 4 7
( 3) 4( 3) 7
12 7
19
f x x
f
Substitute −3 for x.
Model Exam 1 Paper 2
ANSWER
(2 marks)
5. (a) Given that f (x) = 4x − 7 and g(x) = x2 − 15, calculate the value of
Model Exam 1 Paper 2
(ii) gf (2)
5. (a) (ii)
Substitute 2 for x.( ) 4 7
(2) 4(2) 7
8 7
1
f x x
f
Substitute 1 for x.
2
2
( ) 15
(1) (1) 15
1 15
(2) 14
g x x
g
gf
Model Exam 1 Paper 2
Substitute f (x)into g(x) for x.
Or
Model Exam 1 Paper 2
2
2
2
2
2
( ) 4 7
( ) 15
( ) (4 7) 15
(2) (4 2 7) 15
(8 7) 15
1 15
1 15
14
f x x
g x x
gf x x
gf
ANSWER
(2 marks)
5. (a) Given that f (x) = 4x − 7 and g(x) = x2 − 15, calculate the value of
Model Exam 1 Paper 2
(iii) f −1(−1)
5. (a) (iii)
Defining equation for f(x).
Interchanging x and y.
Adding 7 to both sides.
Dividing both sides by 4.
Defining equation for f −1(x).1
( ) 4 7
4 7
4 7
4 7
7
4
( )7
4
f x x
y x
x y
y x
y
fx
x
x
Model Exam 1 Paper 2
ANSWER
(2 marks)
5. (b) (i) Given that y = x2 + x − 6, copy and complete the table below.
x −4 −3 −2 −1 0 1 2 3y 6 −4 −6 −6 −4 6
Model Exam 1 Paper 2
5. (b) (i) 2
2
2
6
when 3, then
( 3) ( 3) 6
9 3 6
when 2, then
2 2
0
4 4
0
6
y x x
x
y
x
y
Model Exam 1 Paper 2
The completed table is shown below.
x −4 −3 −2 −1 0 1 2 3y 6 0 −4 −6 −6 −4 0 6
Model Exam 1 Paper 2
ANSWER
(5 marks)Total 12 marks
(ii) Using a scale of 2 cm to represent 1 unit
on the x-axis and 1 cm to represent
1 unit on the y-axis, draw the graph of
y = x2 + x − 6 for −4 ≤ x ≤ 3.
Model Exam 1 Paper 2
5. (b) Given that y = x2 + x − 6, copy and complete the table below.
x −4 −3 −2 −1 0 1 2 3y 6 −4 −6 −6 −4 6
5. (b) (ii)
Using the given scales, the graph of y = x2 + x − 6 for −4 ≤ x ≤ 3 was drawn on graph paper as shown above.
Model Exam 1 Paper 2
ANSWER
6. The diagram below shows trapeziums A, B and C. The line y = −x is also shown.
Model Exam 1 Paper 2
ANSWER
(3 marks)
6. (a) Describe, fully, the single transformation which maps trapezium A onto
(i) trapezium B
Model Exam 1 Paper 2
6. (a) (i) The single transformation which mapstrapezium A onto trapezium B is a
translation
with vector
3.
6
Each point on trapezium A is moved 3 units horizontally to the right, then 6units vertically downwards.
Model Exam 1 Paper 2
ANSWER
(3 marks)6. (a) (ii) trapezium C
Model Exam 1 Paper 2
6. (a) (ii) The single transformation which maps
trapezium A onto trapezium C is a
reflection in the line y = −x .
Model Exam 1 Paper 2
ANSWER
(4 marks)Total 10 marks
6. (b) State the coordinates of the vertices of trapezium D, the image of trapezium B after a reflection in the line y = −x.
Model Exam 1 Paper 2
6. (b) The coordinates of the vertices of trapezium D are:
(3, 0), (1, 2), (1, 4) and (3, 4).
Model Exam 1 Paper 2
7. The waiting time, to the nearest minute, experienced by 100 people to catch a bus is shown in the table below.
Waiting Time (minutes)
Number of Students
Cumulative Frequency
1 – 5 9 9 6 – 10 12 2111 – 15 15 3616 – 20 1921 – 25 2226 – 30 1631 – 35 436 – 40 3
Model Exam 1 Paper 2
ANSWER
(2 marks)7. (a) Use the table given above to construct a
cumulative frequency table.
Model Exam 1 Paper 2
7. (a) Interval(minutes)
CumulativeFrequency
< 5.5 9 < 10.5 9 + 12 = 21< 15.5 21 + 15 = 36< 20.5 36 + 19 = 55< 25.5 55 + 22 = 77< 30.5 77 + 16 = 93< 35.5 93 + 4 = 97< 40.5 97 + 3 = 100
The cumulative frequency table is shown above.
Model Exam 1 Paper 2
ANSWER
(4 marks)7. (b) Use the values from your table to draw a
cumulative frequency curve.
Model Exam 1 Paper 2
7. (b)
The completed cumulative frequency curve is shown above.
Model Exam 1 Paper 2
ANSWER
(2 marks)
7. (c) Use your graph to estimate
(i) the median for the data
Model Exam 1 Paper 2
7. (c) (i) Half of the total frequency,
From the graph, the waitingtime corresponding to a total
frequency of 50, Q2 = 19 minutes
Hence, the median for the data is
19 minutes.
1 1100
2 250
n
Model Exam 1 Paper 2
ANSWER
(2 marks)7. (c) (ii) the number of people who waited less
than 23 minutes
Model Exam 1 Paper 2
7. (c) (ii) From the graph, the number of people whowaited less than 23 minutes =
65
Model Exam 1 Paper 2
ANSWER
(2 marks)Total 12 marks
7. (c) (iii) the probability that a person, chosen at
random from the group, waited for at
least 18 minutes
Model Exam 1 Paper 2
7. (c) (iii) From the graph, the number of people whowaited less than 18 minutes =
45 The number of people whowaited for at least 18 minutes
P(x ≥ 18 minutes)
100 45
55
55
1011
20
0
Model Exam 1 Paper 2
8. The first three diagrams in a sequence are shown below. Diagram 1 has a single circle, which can be considered as a square pattern formed by a single circle.
Diagram 2 consists of a square of side two circles with two triangles formed at the ends as shown.
Diagram 3 consists of a square of side three circles with two triangles formed at the ends as shown.
Diagram 1 Diagram 2 Diagram 3
Model Exam 1 Paper 2
Diagram Number
Number of Circles Forming
the Square
Number of Additional Circles in Two
Triangles
Pattern for Calculating the Total Number of
Circles in the Diagram1 12 1(0) 12 + 1(0)2 22 2(1) 22 + 2(1)3 32 3(2) 32 + 3(2)
(i) 4 42 — —
(ii) — — 8(7) —
(iii) n — — —
Model Exam 1 Paper 2
ANSWER
(2 marks)8. (a) Draw Diagram 4 in the sequence.
Model Exam 1 Paper 2
8. (a)
Diagram 4
Diagram 4 in the sequence is shown above.
Model Exam 1 Paper 2
ANSWER
(8 marks)Total 10 marks
8. (b) Complete the table by inserting the appropriate values at the rows
marked (i), (ii) and (iii).
Model Exam 1 Paper 2
(b) Diagram Number
Number of Circles Forming
the Square
Number of Additional Circles in Two
Triangles
Pattern for Calculating the Total Number of
Circles in the Diagram1 12 1(0) 12 + 1(0)2 22 2(1) 22 + 2(1)3 32 3(2) 32 + 3(2)
(i) 4 42 4(3) 42 + 4(3)
(ii) 8 82 8(7) 82 + 8(7)
(iii) n n2 n(n 1) n2 + n(n 1)
8.
The completed table is shown above.
Model Exam 1 Paper 2
SECTION II
Answer TWO questions in this section
Model Exam 1 Paper 2
NEXT
ANSWER
(4 marks)
9. (a) Solve the pair of simultaneous equations
y = 1 − 2x
y = 2x2 + 5x − 3
Model Exam 1 Paper 2
9. (a)
Model Exam 1 Paper 2
2
2
2
2
2
1 2
2 5 3
: 2 5 3 1 2
Transfer all terms to the LHS:
2 5 2 3 1 0
Add like terms:
2 7 4 0
Factorise the expression on the LHS:
(2 1)( 4) 0 2 2
4 1 4
7 8
y x
y x x
x x x
x x x
x x
x x x x x
x x x
‚
‚
Either 2x – 1 = 0i.e. 2x = 1
Or x + 4 = 0 x = – 4
1
2x
1When , then
21 2
11 2
2
1 1
0
x
y x
Model Exam 1 Paper 2
Model Exam 1 Paper 2
When 4, then
1
9
2( 4)
1 8
x
y
10 Hence, , and .4
2 9,x y x y
ANSWER
(3 marks)
9. (b) Express in the form where a, h and k are
realnumbers
22 5 3x x 2( ) ,a x h k
Model Exam 1 Paper 2
9. (b) Factorise out thecoefficient of x2
i.e. 2.
Write as a perfect square.
The LCM of 2 and 16 is 16.
Adding the fractions.
2 2 21 1 5 5
coefficient of 2 2 2 4
x
Model Exam 1 Paper 2
2
2
2 22
2
2
2
2 5 3
5 32
2 2
5 5 3 52
2 4 2 4
5 3 252
4 2 16
5 3(8) 25(1)2
4 16
5 24 252
4 16
x x
x x
xx
x
x
x
22 5 5 3 5
22 4 2 4
x x
Simplifying the fraction.
Multiplying the fraction by 2.
It is in the form where a, h and k are 2, respectively.
2
2
5 492
4 16
5 492
4 8
x
x
2( )a x h k
Model Exam 1 Paper 2
and5
4
49
8
ANSWER
(1 mark)
9. (c) Using your answer from (b) above, or otherwise, calculate.
(i) the minimum value of22 5 3x x
Model Exam 1 Paper 2
9. (c)
(i) The minimum value of
2
2
2 5 3
5 492
4 8
x x
x
22 5 3
4
16
8
9
8
x x
Model Exam 1 Paper 2
ANSWER
(1 mark)
9. (c) Using your answer from (b) above, or otherwise, calculate.
Model Exam 1 Paper 2
(ii) the value of x where the minimum
occurs
9. (c) (ii) The minimum occurs where the value of x
5
41
14
Model Exam 1 Paper 2
4
50
45
x
x
ANSWER
(4 marks)
9. (d) Sketch the graph of y = 2x2 + 5x − 3, clearly showing
the coordinates of the minimum point.
the value of the y-intercept.
the values of x where the graph cuts the
x-axis.
Model Exam 1 Paper 2
9. (d) The coordinates of the minimumpoint are
y = 2x2 + 5x − 3 the value of the y intercept, c = 3
y = 2x2 + 5x − 3 and y = 0 on the x-axis,
so 0 = (2x − 1)(x + 3) by factorising the
expression.
1 11 , 6 .
4 8
Model Exam 1 Paper 2
Hence, x = and x = −3 are the values
of x where the graph cuts the x-axis.
A sketch of the graph of y = 2x2 + 5x − 3 is
shown below.
1
2
Model Exam 1 Paper 2
Model Exam 1 Paper 2
ANSWER
(2 marks)Total 15 marks
9. (e) Sketch on your graph of y = 2x2 + 5x − 3,
the line which intersects the curve at the values of x and y as calculated in (a) above.
Model Exam 1 Paper 2
9. (e) A sketch of the line y = 1 − 2x which intersects the curve
y = 2x2 + 5x − 3 at the points (−4, 9) and is shown below.
1,0
2
Model Exam 1 Paper 2
ANSWER(1 mark)(i) ABC
Model Exam 1 Paper 2
10. (a) The diagram following, not drawn to scale, shows a circle, centre O. The line DCE is a tangent to the circle. Angle ACE = 46° and angle OCB = 25°.
Calculate:
10. (a) (i) 46ABC ACE in alternate segment
Model Exam 1 Paper 2
ANSWER(1 mark) (ii) AOC
Model Exam 1 Paper 2
10. (a) The diagram following, not drawn to scale, shows a circle, centre O. The line DCE is a tangent to the circle. Angle ACE = 46° and angle OCB = 25°.
Calculate:
10. (a) (ii) at centre =2· at circumference
2
9
46
2
2
AOC ABC
Model Exam 1 Paper 2
ANSWER(1 mark)
Model Exam 1 Paper 2
(iii) BCD
10. (a) The diagram following, not drawn to scale, shows a circle, centre O. The line DCE is a tangent to the circle. Angle ACE = 46° and angle OCB = 25°.
Calculate:
10. (a) (iii) betweenradius andtangent atpoint oftangency.
Model Exam 1 Paper 2
90
So 2
6
5 90
90
5
25
BCD OCB
BCD
BCD
ANSWER(1 mark)
Model Exam 1 Paper 2
(iv) BAC
10. (a) The diagram following, not drawn to scale, shows a circle, centre O. The line DCE is a tangent to the circle. Angle ACE = 46° and angle OCB = 25°.
Calculate:
10. (a) (iv) in alternate segment.
65BAC BCD
Model Exam 1 Paper 2
ANSWER(1 mark)
Model Exam 1 Paper 2
(v) OAC
10. (a) The diagram following, not drawn to scale, shows a circle, centre O. The line DCE is a tangent to the circle. Angle ACE = 46° and angle OCB = 25°.
Calculate:
10. (a) (v)ΔOAC isisosceles,since OC =OA = r.
180
218
4
28
24
0 92
8
AOCOAC
Model Exam 1 Paper 2
ANSWER(1 mark) (vi) OAB
Model Exam 1 Paper 2
10. (a) The diagram following, not drawn to scale, shows a circle, centre O. The line DCE is a tangent to the circle. Angle ACE = 46° and angle OCB = 25°.
Calculate:
10. (a) (vi)
2
65 4
1
4
OAB BAC OAC
Model Exam 1 Paper 2
10. (b) The diagram below, not drawn to scale, shows the positions of two ships, P and Q, relative to a point O. P is on a bearing of 045° from O and the distance OP = 500 km.
Q is on a bearing of 080° from P and the distance PQ = 800 km.
Model Exam 1 Paper 2
ANSWER
(2 marks)
10. (b) (i) Copy the diagram above. On you diagram indicate the angles that represent the
bearings of 045° and 080°.
Model Exam 1 Paper 2
10. (b) (i)
A copy of the diagram is shown above. The angles that represent the bearings of 045° and 080° are indicated. The
distances are also indicated.
Model Exam 1 Paper 2
10. (b) The diagram below, not drawn to scale, shows the positions of two ships, P and Q, relative to a point O. P is on a bearing of 045° from O and the distance OP = 500 km.
Q is on a bearing of 080° from P and the distance PQ = 800 km.
Model Exam 1 Paper 2
ANSWER
(7 marks)Total 15 marks
10. (b) (ii) Calculate
a) OPQ
b) the distance OQ, to the nearest kilometre
c) the bearing of Q from O
Model Exam 1 Paper 2
10. (b) (ii) a)
Model Exam 1 Paper 2
Interior angles are supplementary,NO //NP.
Model Exam 1 Paper 2
180
45 180
180 45
135
NOP NPO
NPO
NPO
Sat apoint.
Hence, OPQ is 145°.
Model Exam 1 Paper 2
360
135 80 360
215 360
360 215
145
OPQ NPO NPQ
OPQ
OPQ
OPQ
10. (b) (ii) b)
Model Exam 1 Paper 2
Hence, the distance OQ is 1 243 km, to the
nearest kilometre.
Considering ΔOPQ and using the cosine rule:
Model Exam 1 Paper 2
2 2 2
2 2
ˆ2 cos
500 800 2 500 800 cos145
250 000 640 000 800 000 ( 0.819)
890 000 655 200
1545 200
1545 200 km
(to1 243km the nearest km)
P OPQOQ OP PQ OP Q
OQ
10. (b) (ii) c)
Model Exam 1 Paper 2
Hence, the bearing of Q from O is 066.7°.
Considering ΔOPQ and using the sine rule:
1
1 243 800ˆsin145 sin
800 sin145ˆsin1 243
0.369
ˆ sin 0.369
21.7
POQ
POQ
POQ
45 21.7
66.7
NOQ NOP POQ
Model Exam 1 Paper 2
ANSWER
(4 marks)
11. (a) The value of the determinant of
is −36.
Calculate the values of x.
4
5
xM
x
Model Exam 1 Paper 2
11. (a)
2
2
2
4
5
The determinant of , 4( 5)
20
And 36.
So 36 20
i.e. 36 20 16
16 4
xM
x
M M x x
x
M
x
x
x
Hence, the values of x are +4 and −4.
Model Exam 1 Paper 2
ANSWER(2 marks)
11. (b) The transformation R is represented by the
matrix1 0
.0 1
The transformation S is represented by the
matrix1 0
.0 1
(i) Write a single matrix, in the form
to represent the combined
transformation S followed by R.
a b
c d
Model Exam 1 Paper 2
11. (b) (i) The combined transformation S followed by R,
Hence, the single matrix that represents the combined transformation S followed by R is .
1 0
0 1
Model Exam 1 Paper 2
ANSWER
(3 marks)
Model Exam 1 Paper 2
(ii) Calculate the image of the point P(−7, 4) under the combined transformation S followed by R.
11. (b) The transformation R is represented by the
matrix1 0
.0 1
The transformation S is represented by the
matrix1 0
.0 1
11. (b) (ii) RS P P′
P′ (7, 4)
Hence, the image of the point P (−7, 4) under the combined transformation S followed by
R is P′ (7, 4).
1 0 7 1 ( 7) 0 4
0 1 4 0 ( 7) 1 4
7
4
P
Model Exam 1 Paper 2
(ii)Or
Model Exam 1 Paper 2
ANSWER
(2 marks)
11. (c)5 2
The matrix .3 1
N
(i) Determine the inverse matrix of N.
Model Exam 1 Paper 2
11. (c) (i) 5 2The matrix
3 1N
The determinant of , 5( 1) 2(3)
5 6
11
N N
The adjoint matrix of 1 2, adjoint
3 5N N
Model Exam 1 Paper 2
Model Exam 1 Paper 2
(4 marks)Total 15 marks
(ii) Hence, calculate the value of x and the
value of y for which5 2 4
3 9
x y
x y
Model Exam 1 Paper 2
11. (c)5 2
The matrix .3 1
N
ANSWER
11. (c) (ii)Given 5 2 4 then
3 9
5 2 4
3 1 9
x y
x y
x
y
1
It is in the form
So
NX B
X N B
Model Exam 1 Paper 2
Hence, x = 2 and y = −3.
Model Exam 1 Paper 2
CSEC MODEL EXAMINATION 2MATHEMATICS
Paper 190 minutes
Answer ALL the questions
NEXT
1. (1)3 + (3)2 =
ANSWER
(A) 4
(B) 9
(C) 8
(D) 10
Model Exam 2 Paper 1
1. (1)3 + (3)2 =
= (1) (1) (1) + (3) (3)
= 1 (1) + 9
Use the meaning of a square and a cube.
The product of two negative signs is a positive sign.
The product of a positive sign and a negative sign is negative.
= 1 + 9
= 8 Subtracting.
Model Exam 2 Paper 1
2. Express as a decimal correct to 3 significant figures.
ANSWER
(A) 5.27
(B) 5.28
(C) 5.29
(D) 5.30
25
7
Model Exam 2 Paper 1
20.285 . . .
7
25 5.28 5 . . .
5.297
(3 sf )
2. 0.2857 20
1460
56
40
355
The digit after the 3rd significant figure is 5, so we add 1 to the digit 8.
Model Exam 2 Paper 1
3. The decimal fraction 0.016 expressed as a commonfraction in its lowest terms is
ANSWER
(A)
(B)
(C)
(D)
16
100
4
2502
125
16
1000
Model Exam 2 Paper 1
Write the decimal fraction as an equivalent common fraction.
Divide both the numerator and the denominator by their common factor 8.
This is the common fraction written in its lowest terms.
3. 160.016
1000
16 8
1000 8
2
125
Model Exam 2 Paper 1
4. In standard form, 8 504 is
ANSWER
(A) 8.504 102
(B) 8.504 103
(C) 8.504 102
(D) 8.504 103
Model Exam 2 Paper 1
4. 8 504 = 8.504 1 000
= 8.504 103
The first number must have a value between 1 and 10.
That is, 1 < first number < 10
Model Exam 2 Paper 1
5.
ANSWER
(A)
(B) 12
(C)
(D)
41
3
1
12
1
81
4
3
Model Exam 2 Paper 1
4
1
1 1 1 1 1
3 3 3 3 3
1 1 1 1
3 3 3 3
81
5.
Model Exam 2 Paper 1
6. If 70% of a number is 80, then the number is
ANSWER
(A) 10
(B) 56
(C) 80
(D) 2114
7
Model Exam 2 Paper 1
6. The number 8 0100
7 0
800
72
1147
Model Exam 2 Paper 1
7. The multiplicative inverse of –5 is
ANSWER
(A) 5
(B) 5
(C)
(D)
1
51
5
Model Exam 2 Paper 1
5 1
1
5
51
x
x
7. Definition
Divide both sides by –5.
A positive value divided by a negative value is a negative value.
Model Exam 2 Paper 1
8. The HCF of 15, 30 and 60 is
ANSWER
(A) 3
(B) 5
(C) 15
(D) 45
Model Exam 2 Paper 1
8. 3 15, 30, 60
5 5, 10, 20
1, 2, 4
Each of the numbers 15, 30 and 60 is divisible by 15.
The HCF = 3 5 = 15
Model Exam 2 Paper 1
9. If 2n is an even number, which of the following is an odd number?
ANSWER
(A) 2n 1
(B) 2(n + 1)
(C) 2n 2
(D) 2(n + 3)
Model Exam 2 Paper 1
9. Even number = 2n
Odd number = 2n 1
Model Exam 2 Paper 1
10. The next term in the sequence 5, 2, 1, 4 is
ANSWER
(A) 5
(B) 6
(C) 7
(D) 8
Model Exam 2 Paper 1
10. 5, 5 3 = 2,
2 3 = 1,
1 3 = 4,
4 3 = 7
A term in the sequence is obtained by subtracting 3 from the term just to its left (the preceding term).
Model Exam 2 Paper 1
11. A butcher bought a car for $2 500 and sold it for $3 000. His profit as a percentage of the cost price is
ANSWER
(A) 5%
(B) 10%
(C) 15%
(D) 20%
Model Exam 2 Paper 1
11. The profit = $(3 000 2 500)
= $500
The percentage profit =
1
$ 500
5
$ 2500100%
= 20%
Model Exam 2 Paper 1
12. A boutique gives 10% discount for cash. What is the cash price of a dress with a marked price of $350?
ANSWER
(A) $35
(B) $315
(C) $340
(D) $360
Model Exam 2 Paper 1
12. (100 10)% of $350 = 90% of $350
90$350
100
= $315
Model Exam 2 Paper 1
13. If J $90.00 is equivalent to US $1.00, then J $5 400.00 equivalent to
ANSWER
(A) US $6.00
(B) US $60.00
(C) US $600.00
(D) US $540
Model Exam 2 Paper 1
13. J $90.00 US $1.00
1.00J $1.00 US $
90.001.00
J $5 400.00 US $ 5 400.0090.005 400
US $90
US $60
Model Exam 2 Paper 1
14. The freight charges on a parcel is $150 plus custom duties of 20%. What amount of money was paid to collect the parcel?
ANSWER
(A) $160
(B) $170
(C) $180
(D) $190
Model Exam 2 Paper 1
14. (100 20)% of $150 120% of $150
120$150
1
$ 0
0
18
0
Model Exam 2 Paper 1
15. A man pays $0.25 for each unit of electricity used up to 400 units and $0.31 for each unit of electricity used in excess of 400 units. How much does he pay for consuming 1 200 units of electricity?
ANSWER
(A) $56
(B) $324
(C) $348
(D) $672
Model Exam 2 Paper 1
15. The cost for the first 400 units = $0.25 400
= $25 4 = $100
The cost for the remaining = $0.31 800
800 units = $31 8 = $248
The electricity bill = $(100 + 248)
= $348
Model Exam 2 Paper 1
16. The table below shows the rates charged by an insurance company for home insurance.
ANSWER
(A) $2 100 (B) $4 500
(C) $4 020 (D) $6 600
House $4.50 per $1 000
Contents $2.10 per $1 000
A house is valued at $800 000 and the contents at $200 000. How much will the owner pay for home insurance?
Model Exam 2 Paper 1
16. The cost for insuring the house = $4.50
= $4.50 800
= $450 8
= $3 600
$800 000
$1 000
The cost for insuring the contents = $2.10
= $2.10 200
= $210 2
= $420
$200 000
$1 000
Model Exam 2 Paper 1
The cost for the home insurance = $(3 600 + 420)= $4 020
Model Exam 2 Paper 1
17. A student bought 12 blue pens at $15 each and 13 green pens at $10 each. What is the mean cost per pen?
ANSWER
(A) $12.40
(B) $12.50
(C) $12.60
(D) $12.70
Model Exam 2 Paper 1
17. The cost for the 12 blue pens = $15 12
= $180
The cost for the 13 green pens = $10 13
= $130
The total cost for the 25 pens = $(180 + 130)
= $310
The mean cost per pen =
= $12.40
$310
25
Model Exam 2 Paper 1
18. A woman invested a sum of money at 6% per annum for 2 years. If she collected $ 300 as simple interest, what was the sum of money that she invested?
ANSWER
(A) $2 500
(B) $2 700
(C) $2 800
(D) $10 000
Model Exam 2 Paper 1
18. I = $300
R = 6%
T = 2 years
100The principal,
100 300$
IP
RT
50
6 2
100 50$
25
12
$100
50
25
$2 0
Model Exam 2 Paper 1
19.
ANSWER
(A) (P Q)
(B) (P Q)
(C) P Q
(D) P Q
In the Venn diagram above, the shaded region represents
Model Exam 2 Paper 1
19. The unshaded region represents
P Q P or Q
The shaded region represents
(P Q) Not P or Q
Model Exam 2 Paper 1
20. If U = {2, 3, 5, 7, 11, 13} and A = {5, 11}, then n(A) =
ANSWER
(A) 2
(B) 4
(C) 6
(D) 8
Model Exam 2 Paper 1
20. A = {2, 3, 7, 13}
n(A) = 4
Model Exam 2 Paper 1
21.
ANSWER
(A) {3, 9, 15, 18, 24}
(B) {3, 6, 9, 12, 15}(C) {6, 12, 18, 24}(D) {6, 12}
In the Venn diagram, set L and set M are represented by two intersecting circles. If L = {multiples of 3 less than 16} and M = {multiples of 6 less than 25}, then the shaded region represents
Model Exam 2 Paper 1
21.
Model Exam 2 Paper 1
{3, 6, 9, 12, 15}
{6, 12, 18, 24}
which are common elements.{6, 12}
L
M
L M
22. Which of the following pairs of sets are equivalent?
ANSWER
(A) {2, 3} and {a, b, c}
(B) { } and {1, 2, 3}
(C) {a, b, c} and {2, 4}
(D) {1, 2, 3} and {a, b, c}
Model Exam 2 Paper 1
22. {1, 2, 3} {a, b, c}
n{1, 2, 3} = 3 n{a, b, c} = 3
The number of elements in each of the sets is 3, therefore the sets are equivalent.
Or
1 a
2 b
3 c
There is a 1 1 correspondence between the elements of the two sets, therefore the sets are equivalent.
Model Exam 2 Paper 1
23. The volume of a cube with edges of length 1 cm is
ANSWER
(A) 1 cm3
(B) 12 cm3
(C) 16 cm3
(D) 24 cm3
Model Exam 2 Paper 1
23. The volume of the cube,
V = l3
= (1 cm)3
= 1 cm3
The formula for thevolume of a cube.
Model Exam 2 Paper 1
24. Expressed in millimetres, 470 centimetres is
ANSWER
(A) 4.7
(B) 47
(C) 4 700
(D) 47 000
Model Exam 2 Paper 1
24. 1 cm = 10 mm
470 cm = 10 470 mm
= 4 700 mm
Model Exam 2 Paper 1
25. The lengths of the sides of a triangle are x, 2x and 3x centimetres. The perimeter of the triangle is 30 centimetres. What is the value of x?
ANSWER
(A)
(B) 5
(C) 10
(D) 15
3 5
Model Exam 2 Paper 1
25. The perimeter = (x + 2x + 3x) cm
= 6x cm
Equating the values for the perimeter:
6x = 30
= 5
30
6x
Model Exam 2 Paper 1
26. If Usain Bolt runs the 100 metres race in 9.6 seconds, what was his average speed in metres per second?
ANSWER
(A)
(B)
(C)
(D) 96
24
254
45
510
12
Model Exam 2 Paper 1
26. The average speed,d = 100 m and t = 9.6 s
100 m
9.6 s
1000m/s
96250
m/s24
510 m/s
12
ds
t
Model Exam 2 Paper 1
27. Forty students each drank 2 bottles of sweet drink. Each bottle held 250 millilitres of sweet drink. How many litres of sweet drink were used?
ANSWER
(A) 20
(B) 80
(C) 500
(D) 20 000
Model Exam 2 Paper 1
27. The number of bottles used = 40 2
= 80
The number of millilitres used = 250 80
= 20 000
The number of litres used =
= 20
20 000
1000
Model Exam 2 Paper 1
28. The length of a rectangle is three times that of its width. If the area of the rectangle is 108 cm2, then its width, in cm, is
ANSWER
(A) 6
(B) 26
(C) 27
(D) 36
Model Exam 2 Paper 1
l = 3w cm28.
A = 108 cm2 b = w cm
The area of the rectangle,A = lb
= (3w w) cm2 Substitute 3w for l and w for b.
= 3w2 cm2
Equating the values for the area:2
2
3 108
108
6 cm
363
36
w
w
w
Model Exam 2 Paper 1
29. A student leaves home at 06:25 h and arrives at school at 07:45 h. The student travels non-stop at an average speed of 60 km/h. What distance, in kilometres, is the student’s home from school?
ANSWER
(A) 40
(B) 50
(C) 70
(D) 80
Model Exam 2 Paper 1
29. The time taken,
t = (07:45 – 06:25) h
= 1 h 20 min 1
1 h3
The distance,
160 1 km
34
60 km3
80 k20 4 km
m
d st
s = 60 km/h
t 1
1 h3
Model Exam 2 Paper 1
30.
ANSWER
(A) (B)
(C) (D)
The diagram above shows a sector POQ withsector angle POQ = 45° and radius OQ = r units.The area of the sector POQ is
21π
2r 21
π4
r
21π
6r 21
π8
r
Model Exam 2 Paper 1
30. 2
2
2
2
θThe area of the sector, π
36045
π3601
π8
1π
8
A r
r
r
r
Model Exam 2 Paper 1
Items 31 – 34 refer to the following frequency distribution.The distribution shows the mass of parcels, in kilograms, sent to a skybox by an individual.
Mass of parcel (kg) Number of parcel
2 3
3 7
4 2
5 1
Model Exam 2 Paper 1
31. The mode, in kilograms, of the distribution is.
ANSWER
(A) 2
(B) 3
(C) 4
(D) 5
Model Exam 2 Paper 1
31. Mode = 3 kg 7 (highest frequency)
Model Exam 2 Paper 1
32. What is the median, in kilograms, of the distribution?
ANSWER
(A) 4
(B) 3.5
(C) 3
(D) 2
Model Exam 2 Paper 1
32. The total frequency = 3 + 7 + 2 + 1
= 13
So the middle value is in the 7th ordered position.
The 7th parcel in ascending or descending order has a
mass of 3 kg.
So the median of the distribution has a mass of 3 kg.
Model Exam 2 Paper 1
33. The total mass, in kilograms, of all the parcels sent to the skybox by the individual is
ANSWER
(A) 13
(B) 14
(C) 40
(D) 182
Model Exam 2 Paper 1
x(kg) f fx
2 3 63 7 214 2 85 1 5
fx = 40
33.
Model Exam 2 Paper 1
34. The mean, in kilograms, of the distribution is
ANSWER
(A)
(B)
(C)
(D)
13
131
321
4131
42
Model Exam 2 Paper 1
34.
13 kg
The mean,
40 kg
13
13
fxx
f
Model Exam 2 Paper 1
35.
ANSWER
(A) 400
(B) 300
(C) 200
(D) 100
The pie chart shown above represents the ways in which a school of 600 children watched a movie. The number of children who watched the movie at a cinema is approximately
Model Exam 2 Paper 1
35. 1 1The number of children of 600
3 21
3003100
Model Exam 2 Paper 1
36. The volume, in millilitres, of five sizes of bottled orange juice are 500, 250, 2 000, 750, 1 000. The range, in millilitres, is
ANSWER
(A) 250
(B) 500
(C) 1 000
(D) 1 750
Model Exam 2 Paper 1
36. The range
= The greatest volume The least volume
= (2 000 250) ml
= 1 750 ml
Model Exam 2 Paper 1
37. 5(x 2) =
ANSWER
(A) 5x 2
(B) 5x + 2
(C) 5x 10
(D) 5x + 10
Model Exam 2 Paper 1
37. 5(x 2) = 5 x5 (2)
= 5x + 10
Use the distributivelaw to remove thebrackets.
The product of apositive and a negativesign is a negativesign. The product oftwo negative signs is apositive sign.
Model Exam 2 Paper 1
38. 4(2x 1) 3(x 5) =
(A) 5x 11
(B) 5x +11
(C) 5x – 6
(D) 5x + 6
ANSWER
Model Exam 2 Paper 1
38. 4(2x 1) 3(x 5)
= 8x 4 3x + 15
= 8x 3x + 15 4
= 5x + 11
Use the distributive law on the terms in each pair of brackets.Group like terms.Add like terms.
Model Exam 2 Paper 1
39. For all x, 4x(x + 3) 2x(5x 1) =
ANSWER
(A) 6x2 + 14x
(B) 6x2 14x
(C) 4x2 10x + 4
(D) 4x2 10x 4
Model Exam 2 Paper 1
39. 4x(x + 3) –2x(5x 1)
≡ 4x2 + 12x 10x2 + 2x
≡ 4x2 10x2 + 12x + 2x
≡ 6x2 + 14x
Use the distributive law twice toremove the two pairs of brackets.
Group like terms.
Add like terms.
Model Exam 2 Paper 1
40.
ANSWER
(A) 1
(B) 1
(C)
(D)
If 1 , then 10*25*p
p qq
251
10
251
10
Model Exam 2 Paper 1
40. 1*
1010 25 1*
2510
15
1 2
1
pp q
q
State the given formula.
Substitute the value for p and for q.
Use the meaning of a square root.
Dividing.
Subtracting.
Model Exam 2 Paper 1
41. If a = 2 and ab = 10, then (a + b)2 (a2 + b2) =
ANSWER
(A) 20
(B) 20
(C) 78
(D)
Model Exam 2 Paper 1
2 2 2
2 2 2
2
So
2 5 2 5
7 4 25
49 29
20
a b a b
41. If 2 and 10
10then
10
25
a ab
ba
Substitute 2 for a and 5 for b.
Model Exam 2 Paper 1
42.
ANSWER
(A)
(B)
(C) 3x
(D)
8 5
9 9x x
13
9x13
6x
1
3x
Model Exam 2 Paper 1
42.
The common denominator is 9x.
Simplify the values in the numerator by subtracting.
Divide both the numerator and denominator by their common factor 3.
Model Exam 2 Paper 1
43. The statement “When 2 is added to five times a number n, the result is 40.” May be represented by the equation
ANSWER
(A) 2(5n) = 40
(B) 2 5n = 40
(C) 5n + 2 = 40
(D) 5n 40 = 2
Model Exam 2 Paper 1
43. Five times a number n = 5n
2 added to five times a number n = 5n + 2
The equation is: 5n + 2 = 40
Model Exam 2 Paper 1
44. If x and y are numbers with x greater than y, then the statement. “The square of the difference of two numbers is always positive.” May be represented as
ANSWER
(A) (x y)2 > 0
(B) x2 y2 > 0
(C) 2(x y) > 0
(D) (x + y)2 > 0
Model Exam 2 Paper 1
44. The difference of the two numbers = x y
The square of the difference of the two numbers = (x y)2
The statement is:
(x y)2 > 0
A positive number is greater than zero.
Model Exam 2 Paper 1
45. Given that 3x + 8 29, then the range of values of x is
ANSWER
(A) x 7
(B) x > 7
(C)
(D)
37
3x
37
3x
Model Exam 2 Paper 1
45. Subtract 8 from both sides.
Subtracting.
Divide both sides by 3.
Dividing.
Model Exam 2 Paper 1
3 8 29
3 29 8
3 21
2
7
1
3
x
x
x
x
x
So
46.
ANSWER
(A) y is greater than x
(B) x is a factor of y
(C) x is less than y
(D) x is a multiple of y
The arrow diagram above describes the relation
Model Exam 2 Paper 1
46. 2 4 = 8
4 2 = 8
3 3 = 9
2 5 = 10
Hence, x is a multiple of y.
Model Exam 2 Paper 1
47. Which of the following relation diagrams illustrates a function?
ANSWER
(A) (B)
(C) (D)
Model Exam 2 Paper 1
47.
Each element in the domain is mapped onto one
and only one element in the range.
This relation diagram represents a function.
Model Exam 2 Paper 1
48. If f(x) = x2 + x 1, then f(3) =
ANSWER
(A) 5
(B) 5
(C) 7
(D) 13
Model Exam 2 Paper 1
2
2
( ) 1
( 3) ( 3) ( 3) 1
9 3 1
13
f x x x
f
48. Substitute 3 for x.
Simplify each term.
Subtracting.
Model Exam 2 Paper 1
49. Which of the following sets is represented by the relation f: x x2 3?
ANSWER
(A) {(0, 3), (1, 2), (2, 1), (3, 6)}
(B) {(0, 3), (1, 2), (2, 1), (3, 0)}
(C) {(0, 3), (1, 6), (2, 9), (3, 12)}
(D) {(0, 3), (1, 3), (2, 3), (3, 4)}
Model Exam 2 Paper 1
49. f(x) = x2 3
f(0) = 02 3 = 3 (0, 3)
f(1) = 12 3 = 1 3 = 2 (1, 2)
f(2) = 22 3 = 4 3 = 1 (2, 1)
f(3) = 32 3 = 9 3 = 6 (3, 6)
The set is {(0, –3), (1, 2), (2, 1), (3, 6)}
Model Exam 2 Paper 1
50.
ANSWER
(A) y = ax2 + bx
(B) y = bx ax2
(C) y = ax2 + bx + c
(D) y = c + bx ax2
If a, b and c are constants with a > 0, then theequation of the graph could be
Model Exam 2 Paper 1
a A maximum turning point
c 0 (y-intercept)
Equation is: y = c + bx ax2
50.
Model Exam 2 Paper 1
51. Which of the following diagrams is the graph of a function?
ANSWER
(A) (B)
(C) (D)
Model Exam 2 Paper 1
51.
Using the vertical line test for a function:x1 y1
x2 y2
The graph represents a 1 1 relation which is a function.
Model Exam 2 Paper 1
52.
ANSWER
(A) x = y
(B) x < y
(C) x + y = 180
(D) x + y > 180
In the figure above, AB and CD are parallel. Therelation between x and y is
Model Exam 2 Paper 1
52. x + y = 180 The interior angles are supplementary.
Model Exam 2 Paper 1
53. Which of the following plane shapes has no line of symmetry?
ANSWER
(A) (B)
(C) (D)
Model Exam 2 Paper 1
53. Each of these three plane figures has a line of symmetry.
This figure has no line of symmetry.
Model Exam 2 Paper 1
54.
ANSWER
(A) 6 8 (B) 6 10
(C) 8 10 (D) 6 16
The area of PQR, in cm2, is given by
1
2
1
2
1
2
1
2
Model Exam 2 Paper 1
54.
The area of PQR, A = bh
= 6 cm 8 cm
= 6 8 cm2
1
21
21
2
Model Exam 2 Paper 1
55.
ANSWER
(A) 28
(B) 56
(C) 102
(D) 124
In ABC, angle ABC = x and angle BAC = 28.What is the value of x?
Model Exam 2 Paper 1
55.
Δ ABC is isosceles since AB = CB.
Also angle BCA = angle BAC = 28°
So x + 28 + 28 = 180i.e. x + 56 = 180 x = 180 – 56 = 124 x = 124
Sum of the angles of a triangle.
Model Exam 2 Paper 1
56.
ANSWER
(A) 640 m(B) 160 m
(C)
(D)
In the diagram above, not drawn to scale, TB represents a hill which is 320 m high, and S is the position of a ship. The angle of elevation of S from T is 30°. The distance of the ship from the top of the hill is
160 3 m320
m2
Model Exam 2 Paper 1
56.
320 msin 30
1 320 m
22 320
6 0 m
m
4
ST
STST
Model Exam 2 Paper 1
57.
ANSWER
(A) DAB = 90(B) ADB = ACB
(C) CAB = ACB
(D) ACB + ABD = 90
In the diagram above, not drawn to scale, BODis a diameter of the circle centre O. Which of thefour statements below is false?
Model Exam 2 Paper 1
57. DAB = 90ADB = ACB
CAB + ADB = 90CAB = ACB
DAB = 90
ADB = ACB
Each of these three statements is true.
ADB + ABD = 90ACB + ABD = 90 since ADB = ACB
This statement is false.
The angle in a semicircle is 90º
Angles at the circumference standing on the same arc.
Model Exam 2 Paper 1
58.
ANSWER
(A) (B)
(C) (D)
In the triangle shown above, tan M is
3
5
4
3
3
4
4
5
Model Exam 2 Paper 1
58.
Opptan
A
4
3
dj
4 cm
3 cm
M
KL
ML
Definition of the tangent of an angle.
Using the capital letters notation.
Substitute the length of each side.
Model Exam 2 Paper 1
59. A ship sailed 75 km due east from A to B. It then sailed 50 km due south to C. Which of the diagrams below best represents the path of the ship?
ANSWER
(A) (B)
(C) (D)
Model Exam 2 Paper 1
59. This diagram bestdescribes the pathof the ship.
Model Exam 2 Paper 1
60.
ANSWER
(A) x = 0
(B) y = 0
(C) y = x
(D) x = –y
In the diagram shown, if the line y = –x is rotated about 0 through a clockwise angle of 90°, then its image is
Model Exam 2 Paper 1
60.
The image is the line y = x.
Model Exam 2 Paper 1
CSEC MODEL EXAMINATION 2
MATHEMATICS
Paper 2
2 hours 40 minutes
SECTION I
Answer ALL the questions in this section
All working must be clearly shown
NEXT
1. (a) (i) Using a calculator, or otherwise, determine the exact value of
(2 marks)
ANSWER
5.8 0.39 0.5625
Model Exam 2 Paper 2
(ii) Express as a single fraction
(3 marks)
1 35
2 51
32
1. (a) (i) 5.8 0.39 0.5625
2.262 0.75
(exact value3 ).012
Model Exam 2 Paper 2
1. (a) (ii)1 35
2 51
32
94
101
32
57
5
21
Use the mixed number
Function, , to simplify
the numbers in the numerator. Use the mixed
number function, , to
divide the mixed number in the numerator by the mixed number in the denominator. (single fraction)
ba
c
ba
c
Model Exam 2 Paper 2
1. (b) In this question, use CAN $1.00 = GUY
$164.00.
(2 marks) ANSWER
(i) While vacationing in Canada, Robert used his credit card to buy a camcorderfor CAN $450.00.
How many Guyanese dollars is Robert owing on his credit card for this transaction?
Model Exam 2 Paper 2
1. (b) (i) CAN $1.00 = GUY $164.00CAN $450.00 = GUY $164.00 × 450
= GUY $73 800.00
Hence, Robert is owing GUY $73 800.00 on his credit card for the transaction.
Model Exam 2 Paper 2
(3 marks)Total 10 marks
ANSWER
(ii) Robert’s credit card balance is GUY $102 500.00. After buying the camcorder, how many canadian dollars does he have left on his credit card for spending?
Model Exam 2 Paper 2
1. (b) In this question, use CAN $1.00 = GUY
$164.00.
1. (b) (ii) The credit card balance after the transaction
GUY$ 102 500.00 73 800.00
GUY$ 28 700.00
Now GUY $164.00 = CAN $1.00
So GUY $1.00 = CAN
GUY $28 700.00 = CAN × 28 700.00
= CAN $175.00
Hence, Robert has CAN $175.00 left onhis credit card for spending.
1.00$
164.001.00
$164.00
Model Exam 2 Paper 2
2. (a) Find the value of each of the following algebraic expressions when a 3, b
1and c 2
(1 mark)
ANSWER
(i) a (b c)
Model Exam 2 Paper 2
2. (a) (i) ( ) 3( 1 2
3( 3)
9
)a b c
Substitute the values for a,b and c intothe algebraicexpression, thensimplify.
Model Exam 2 Paper 2
(2 marks)
ANSWER
(ii)25 3b ac
a b c
Model Exam 2 Paper 2
2. (a) Find the value of each of the following algebraic expressions when a 3, b
1and c 2
2. (a) (ii) 2
2
5 3
5( 1) 3( 3)2
3 ( 1) 2
5(1) 18
3 1 25 18
6
53
3
6
6
2
b ac
a b c
Substitute thevalues for a,b and c intothe algebraicexpression,then simplifyaccording tothe arithmeticrules
Model Exam 2 Paper 2
2. (b) Change the following statements into algebraic expressions:
(1 mark)
ANSWER
(i) Seven times the sum of x and 3.
Model Exam 2 Paper 2
2. (b) (i) The sum of and 3 3
Seven times the
sum of and 3 7 ( 3)7( 3)
x x
x xx
Model Exam 2 Paper 2
(2 marks)
ANSWER
(ii) Fifteen more than the product of p and q.
Model Exam 2 Paper 2
2. (b) Change the following statements into algebraic expressions:
2. (b) (ii) The product of p and q p × q pqFifteen more than theproduct of p and q pq+15
Model Exam 2 Paper 2
2. (c) Solve the equation3(2x + 1) 4x 1 (2 marks)
ANSWER
Model Exam 2 Paper 2
2. (c) 3(2 1) 4 1
6 3 4 1
6 4 3 1
2 4
4
22
x x
x x
x x
x
x
Use the distributive law Group like terms
Add like termsDivide both sides by 2
Simplify
Model Exam 2 Paper 2
So
2. (d) Factorise completely
(2 marks)
ANSWER
(i) 8a3b4 − 16a6b2
Model Exam 2 Paper 2
2. (d) (i) Factorise using8a3b2 as the HCF
6
3
3
2
4
3 2
2
8 ( 2
6
)
8 1
a b
a
b a
a b b
Model Exam 2 Paper 2
ANSWER
(ii) 3m2 + 11m − 4 (2 marks)Total 12 marks
Model Exam 2 Paper 2
2. (d) Factorise completely
2. (d) (ii)
Factorise pairwise.
Factorise using thecommon factor (3m – 1).
2
2
3 11 4
3 12 4
(3 1) 4(3
(3 1)( 4)
1)
m m
m m m
m m m
m m
11
3( 4) 12
1 12 11
( 1) 12 12
p q
pq
Model Exam 2 Paper 2
3. Students taking part in a community project were surveyed to find out the type of movies that they were most likely to view. Each student choose only one type of movie and 1 260 students were surveyed. The results are shown in the table below.
MovieNumber of Students
Horror 168
Detective 210
Romance r
War 182
Musical 462
Model Exam 2 Paper 2
3. (a) Calculate the value of r, the number of students who were most likely to view romance movies.
ANSWER
(2 marks)
Model Exam 2 Paper 2
3. (a) 168 210 182 462 1 260
1022 1 260
1 26
2 8
1
3
0 022
r
r
r
Hence, 238 students were most likely to view romance movies.
Model Exam 2 Paper 2
So
(i) The data collected in the table are to be represented on a pie chart. Calculate the size of the angle in each of the five sectors of the pie chart.
ANSWER(4 marks)
Model Exam 2 Paper 2
MovieNumber of Students
Horror 168
Detective 210
Romance r
War 182
Musical 462
3. (b)
3. (b) (i)
The sector angle
representing168
horror movies 3601 260
2168
7
The sector angle
representing2
detec
360
1
tive
260
6
212
48
movies 2107
7
60
Model Exam 2 Paper 2
The sector anglerepresenting romance movies
The sector angle representingromance movies
The sector angle representingmusical movies
238
768
2
282
752
1
2462
7132
Model Exam 2 Paper 2
(ii) Using a circle of radius 4.5 cm, construct a pie chart to represent the data.
ANSWER
(4 marks)Total 10 marks
Model Exam 2 Paper 2
MovieNumber of Students
Horror 168
Detective 210
Romance r
War 182
Musical 462
3. (b)
3. (b) (ii)
The constructed pie chart with radius 4.5 cm is shown above.
Model Exam 2 Paper 2
4. (a) A universal set, U, is defined asU = {25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38}.Sets P and E are subsets of U such thatP = {Prime Numbers} and E = {EvenNumbers}.
(5 marks)
ANSWER
(i) Draw a Venn diagram to represent the sets P, E and U.
Model Exam 2 Paper 2
4. (a) (i) U = { 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38}, P = {29, 31, 37} and
E = {26, 28, 30, 32, 34, 36, 38}.
The Venn diagram representing the sets P, E and U is shown above.
Model Exam 2 Paper 2
(ii) List the elements of the set (1 mark)
ANSWER
Model Exam 2 Paper 2
4. (a) A universal set, U, is defined asU = {25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38}.Sets P and E are subsets of U such thatP = {Prime Numbers} and E = {EvenNumbers}.
4. (a) (ii) The elements of the set = {25, 27, 33, 35}.
Model Exam 2 Paper 2
4. (b) (i) Using only a pair of compasses and a
pencil, construct parallelogram ABCD in
which AB = 5 cm, AD = 8 cm and the
angle BAD is 60º.
(5 marks)
ANSWER
Model Exam 2 Paper 2
4. (b) (i)
The constructed parallelogram ABCD
with AD = BC = 8 cm, AB = DC =
5 cm and BAD = 60°.
Model Exam 2 Paper 2
4. (b) (ii) Measure and state the length of the
diagonal AC.
ANSWER
(1 mark)Total 12 marks
Model Exam 2 Paper 2
4. (b) (ii) The length of the diagonal AC = 11.4 cm.
Model Exam 2 Paper 2
5. The diagram below, not drawn to scale, represents the floor plan of a house. The broken line PS, divides the floor plan into a semi-circle, A, and a rectangle, B.
Use as 22
.7
Model Exam 2 Paper 2
(a) Calculate the radius of the semi-circle PST. ANSWER(1 mark)
5. (a) The radius of the semi-cir
7m
cle2
27m
2
2
PSPST
QR
r
Model Exam 2 Paper 2
(b) Calculate the perimeter of the entire floor plan.ANSWER
(3 marks)
Model Exam 2 Paper 2
5. The diagram below, not drawn to scale, represents the floor plan of a house. The broken line PS, divides the floor plan into a semi-circle, A, and a rectangle, B.
Use as 22
.7
5. (b) The length of the arc PTS 2
2
22 7m
7
11 m
2
r
r
The perimeter of the entire floor plan PQRST = (12 + 7 + 12 +11) m
= 42 m
Model Exam 2 Paper 2
(c) Evaluate the area of the entire floor plan.ANSWER
(4 marks)
Model Exam 2 Paper 2
5. The diagram below, not drawn to scale, represents the floor plan of a house. The broken line PS, divides the floor plan into a semi-circle, A, and a rectangle, B.
Use as 22
.7
5. (c) The area of thesemi-circle PST
2
11
2
1 22
2
r
2
2
27 7m
7 2 2
77
119
4
4
m
m
The area of the rectangle PQRS = lb= 12 × 7 m2
= 84 m2
Model Exam 2 Paper 2
The area of the entire floor plan 2
21103 m
4
119 84 m
4PQRST
Model Exam 2 Paper 2
(d) Section B of the floor is to be covered with floor tiles measuring 1 m by 50 cm. How many floor tiles are needed to just completely cover Section B?
ANSWER
(4 marks)Total 12 marks
Model Exam 2 Paper 2
5. The diagram below, not drawn to scale, represents the floor plan of a house. The broken line PS, divides the floor plan into a semi-circle, A, and a rectangle, B.
Use as 22
.7
5. (d) The area of a floor tile
2
1m 50cm
11m m
11 m
2
2
lb
The number of floor tiles needed to just
completely cover Section B 2
2
12 7 m1
1 m2
12
6
7 2
1 8
Model Exam 2 Paper 2
6. (a) In the diagram below, not drawn to scale, TB is a vertical lantern post standing on a horizontal plane. B, P and Q are points on the horizontal plane.
TB = 10 metres and the angles of depression from the top of the pole T to P and Q are 35º and 29º respectively.
Model Exam 2 Paper 2
(i) Copy the diagram and insert the angles of depression. (1 mark)
ANSWER
6. (a) (i)
The angles can be seen inserted in the diagram above.
TBP = TBQ = 90°Vertical post standing on horizontal ground.
Model Exam 2 Paper 2
(ii) Calculate to one decimal placea) the length of BPb) the length of PQ
ANSWER(5 marks)
Model Exam 2 Paper 2
6. (a) In the diagram below, not drawn to scale, TB is a vertical lantern post standing on a horizontal plane. B, P and Q are points on the horizontal plane.
TB = 10 metres and the angles of depression from the top of the pole T to P and Q are 35º and 29º respectively.
6. (a) (ii)
In the diagram above:
of depression =
of elevation
35
29
PTU BPT
QTU BQT
Model Exam 2 Paper 2
(a)
Model Exam 2 Paper 2
Considering ΔTBP:
tan35
10m
10m
tan3510m
0. 047
1 .3m0
TB
BP
BP
BP
(to one decimal place)
Model Exam 2 Paper 2
(b)
Model Exam 2 Paper 2
Considering ΔTBQ:
(to one decimal place)
tan 29
10m
10m
tan 2910m
0.55418.1m
TB
BQ
BQ
BP
The length of
(18.1 14
3 m
.3
8
)m
.
PQ BQ BP
Model Exam 2 Paper 2
6. (b)
ANSWER
Model Exam 2 Paper 2
(i) The figure labelled P undergoes a transformation, such that its image is Q. Completely describe this
transformation. (2 marks)
6. (b)
Model Exam 2 Paper 2
(i) The transformation is a translation represented by
the column vector .
3
8T
(ii) On graph paper, draw and labela) the line y = −xb) S, the image of P under a
reflection in the line y = −x.
ANSWER(4 marks)Total 12 marks
Model Exam 2 Paper 2
6. (b)
(ii) a) The line y = −x can be seen drawn and labelled on graph paper.
b) S, the image of P under a reflection in the line y = −x can be seen drawn and labelled on graph paper.
Model Exam 2 Paper 2
6. (b)
(a) The equation of the line above is y = mx + c.ANSWER(1 mark) (i) State the value of c.
Model Exam 2 Paper 2
7. The diagram below shows the graph of a straight line passing through the points A and B.
7. (a)
(i) From the graph, theintercept on
the y-axis = 4.The value of
c = 4.
Model Exam 2 Paper 2
(ii) Determine the value of m. ANSWER(2 marks)
Model Exam 2 Paper 2
7. (a) The diagram below shows the graph of a straight line passing through the points A and B.
7. (a) (ii)
The gradient of theline segment AB
The vertical rise
The horizontal shift4
34
34
So the value of 3
m
Model Exam 2 Paper 2
(ii) From the graph, the slope of AB indicates a negative gradient.
The gradient of theline segment AB
4
3
So the value of 4
3
AO
BO
m
Or
Model Exam 2 Paper 2
(a) (iii) Determine the coordinates of the midpoint of the line
segment AB.
ANSWER(2 marks)
Model Exam 2 Paper 2
7. The diagram below shows the graph of a straight line passing through the points A and B.
7. (a) (iii) Let the mid-point of the line segmentAB be M (x, y).
The x-coordinate of M2
3
21
12
BO
Model Exam 2 Paper 2
The -coordinate of 2
2
4
2
AOy M
So the coordinates of the mid-point of
the line segment AB is .
11 ,2
2M
Model Exam 2 Paper 2
(iii) From the construction on the graph, thecoordinates of the mid-point of the line
Or
11se ,2
2gment is .MAB
Model Exam 2 Paper 2
(iii) Using A(0, 4) andB(3, 0), the midpointof the linesegment AB,
Or
1 2 1 2,2 2
0 3 4 0,
2 2
11 , 2
3 4,
2
2
2
x x y yM
Model Exam 2 Paper 2
7.
ANSWER(3 marks)
14 ,
2p
Model Exam 2 Paper 2
(b) The point lies on the line. State
the value of p.
7. (b) From the construction on the graph, when
, then y = p = −2.
So the value of p is –2.
14
2x
Model Exam 2 Paper 2
(b) The equation of AB is
When and y = p, then
Hence, the value of p is –2.
44
3y x
14
2x
4 14 4
3 2
4 94
3 2
2(3) 4
6
2
4
p
Or
Model Exam 2 Paper 2
7.
ANSWER
(4 marks)Total 12 marks
Model Exam 2 Paper 2
(c) Determine the coordinates of the point of intersection of the line y = x − 3 and the line shown previously.
7. (c) 44
33
:=
y x
y x
‚
‚
Model Exam 2 Paper 2
43 4
34
4 334
1 737
73
7
3
3
7
x x
x x
x
x
x
x
Group like terms
Model Exam 2 Paper 2
When x = 3, then
y = x − 3
= 3 − 3
= 0
Hence, the coordinates of the point of
intersection of the line y = x − 3 and the line
shown is (3, 0) 4
43
y x
Model Exam 2 Paper 2
(c) Given y = x − 3, then m = 1 and c = −3.
Using c = −3 and , the graph of the
line y = x − 3 was drawn on the same graph paper as shown above.
The graph of the lines and
y = x − 3 intersect at B (3, 0).
1
1m
44
3y x
Or
Model Exam 2 Paper 2
8. The first three diagrams in a sequence are shown below. Diagram 1 has a single dot, which can be considered as a triangular pattern formed by a single dot. Diagram 2 consists of a triangle formed by three dots.Diagram 3 consists of a triangle formed by six dots.
Model Exam 2 Paper 2
8. (a) Draw Diagram 4 in the sequence.
ANSWER
(2 marks)
Model Exam 2 Paper 2
8. (a)
Diagram 4Diagram 4 in the sequence can be seen above.
Model Exam 2 Paper 2
8. (b) Complete the table by inserting the appropriate values at the row 2 marked (i), (ii) and (iii). (6 marks)
Model Exam 2 Paper 2
ANSWER
Diagram Number
Number of Dots Forming
the triangle
Pattern for Calculating the Total Number of Dots in the Diagram
1 1 1 (1 + 1) ÷ 22 3 2 (2 + 1) ÷ 23 6 3 (3 + 1) ÷ 2
(i) 4 — —
(ii) — 21 —
(iii) n — —
Model Exam 2 Paper 2
Diagram Number
Number of Dots Forming the
triangle
Pattern for Calculating the Total Number of Dots
in the Diagram1 1 1 (1 + 1) ÷ 22 3 2 (2 + 1) ÷ 23 6 3 (3 + 1) ÷ 24 10 4 (4 + 1) ÷ 26 21 6 (6 + 1) ÷ 2n n (n + 1) ÷ 2
8. (b)
The completed table can be seen above.
2
2
n n
Model Exam 2 Paper 2
8. (c) How many dots will be needed to form the triangle in Diagram 100?
ANSWER
(2 marks)Total 10 marks
Model Exam 2 Paper 2
8. (c) The total number ofdots in the diagram 100(100 1) 2
100(101) 2
50(101)
5 050
Model Exam 2 Paper 2
(c) The total number ofdots in the diagram
Or
2100 100
210 000 100
210 1
25 050
00
Model Exam 2 Paper 2
SECTION II
Answer TWO questions in this section
Model Exam 2 Paper 2
9. (a) Simplify
ANSWER
(1 mark)(i) x3 × x4 ÷ x6
Model Exam 2 Paper 2
9. (a) (i) 3 4 6
3 4 6
7 6
7 6
1
x x x
x x
x x
x
x
x
Model Exam 2 Paper 2
ANSWER
(2 marks)(ii)5 7
2 2a b ab
Model Exam 2 Paper 2
9. (a) Simplify
9. (a) (ii)5 7
2 2
5 7 1
2 2 2
5 7 1 1
2 2 2 2
5 1 7 1
2 2 2 2
5 1 7 1
2 2 2 2
6 8
2 2
3 4
3 4
( )
a b ab
a b ab
a b a b
a a b b
a b
a b
a
a b
b
Model Exam 2 Paper 2
ANSWER
(1 mark)
9. (b) If f(x) = 4x − 1, find the value of
(i) f (3)
Model Exam 2 Paper 2
9. (b) (i) ( ) 4 1
(3) 4(3) 1
11
12 1
f x x
f
Model Exam 2 Paper 2
(ii) f –1(0)
ANSWER
(2 marks)
Model Exam 2 Paper 2
9. (b) If f(x) = 4x − 1, find the value of
9. (b) (ii) Given
then
So
i.e.
1
1
( ) 4 1
4 1
4 1
1 4
1
41
41
( )4
0 1 (0)
41
4
f x x
y x
x y
x y
xy
xy
xf x
f
is the defining equation for f (x)
Interchanging x and y
Adding 1 to both sides
Dividing both sides by 4
is the defining equation for f–1(x)
Model Exam 2 Paper 2
(iii) f –1 f (3)
ANSWER
(2 marks)
Model Exam 2 Paper 2
9. (b) If f(x) = 4x − 1, find the value of
9. (b) (iii) 1
1
1(3) 11 and ( )
411 1
S o (34
12
3
)
4
xf f x
f f
Model Exam 2 Paper 2
Or
1 1( ) 4 1 and ( )
4
xf x x f x
1
1
4 1 1So ( )
44
4
(3) 3
xf f x
x
x
f f
Model Exam 2 Paper 2
(i) Using a scale of 8 cm to represent 100 years on the horizontal axis and a scale of 4 cm to represent 100 kg on the vertical axis, construct a mass-time graph to show
how the solid decays in the 168 years interval.
ANSWER(4 marks)
Draw a smooth curve through all the plotted points.
Model Exam 2 Paper 2
9. (c) The mass, in kg, of strontium, a radioactivematerial, after a number of years is given inthe table below.
t (time in years)
0 28 56 84 112 140 168
m (mass in kg)
400 200 100 50 25 12.5 6.25
9. (c) (i)
The points were plotted on graph paper and a smooth curve drawn as shown above.
Model Exam 2 Paper 2
(ii) Use your graph to estimatea) the mass of the solid after 50
yearsb) the rate of decay of the solid at t
= 75 years.ANSWER
(3 marks)Total 15 marks
Model Exam 2 Paper 2
9. (c) The mass, in kg, of strontium, a radioactivematerial, after a number of years is given inthe table below.
t (time in years)
0 28 56 84 112 140 168
m (mass in kg)
400 200 100 50 25 12.5 6.25
9. (c) (ii) (a) From the construction on the graph:
The mass of the solid after 50 years
= 116 years
Model Exam 2 Paper 2
(b) Draw a tangent to the curve at t = 75 years.
Using two points on the tangent, (0, 180) and (112.5, 0), the gradient of the tangent
2 1
2 1
(0 180)kg
(112.5 0) years
180kg year
112.51.6 kg year
m m
t t
Hence, the rate of decay of the solid at t = 75 years is −1.6 kg/year.
Model Exam 2 Paper 2
10. (a) In the diagram below, not drawn to scale,
PQ is a tangent to the circle, centre O.
PS is parallel to OR and angle RPS = 32º.
ANSWER(2 marks)
Calculate, giving reasons for your answer, the size of
(i) angle PQR
Model Exam 2 Paper 2
10. (a)
Model Exam 2 Paper 2
(i) 32
32
180 (32 32 )
180 64
116
258
2116
ORP RPS
OPR ORP
POR
PORPQR
Alternate S. ΔOPR is isosceles sinceOP = OR = r (radius)
Sum of theangles of a Δ
at centre = 2. at circumference
Hence, the size of angle PQR is 58º.
Model Exam 2 Paper 2
ANSWER(2 marks)
Model Exam 2 Paper 2
10. (a) In the diagram below, not drawn to scale,
PQ is a tangent to the circle, centre O.
PS is parallel to OR and angle RPS = 32º.
Calculate, giving reasons for your answer, the size of
(ii) angle SPT
10. (a) (ii)
Hence, the size of angle SPT is 26º.
90
( )
90 (32 32 )
90 64
26
OPT
SPT OPT OPR RPS
between radius and tangentat point of tangency.
Model Exam 2 Paper 2
10. (b) In the diagram below, not drawn to scale, O is centre of the circle of radius 9.5 cm and AB is a chord of length 16.5 cm.
ANSWER(3 marks)
(i) Calculate the value of θ to the nearest degree.
Model Exam 2 Paper 2
10. (b) (i)
Considering Δ AOB and using the cosine rule:
Model Exam 2 Paper 2
Model Exam 2 Paper 2
(b) (i) Considering Δ AOB and using the cosine
rule:
Or
2 2 2
2 2 2
1
2 2 cos
16.5 2(9.5) 2(9.5) cos
272.25 180.5 180.5cos
So 180.5cos 180.5 272.25
180.5cos 91.75
91.75i.e. cos
180.50.5083
cos ( 0.5083)
121 (to the nearest
degree)
AB r r
Model Exam 2 Paper 2
10. (b) (i)
Δ AOB is isosceles since AO = BO = r (radius)
Model Exam 2 Paper 2
Considering AOB:
ˆsin
8.25 cm
ADAOB
AO
9.5 cm
1ˆSo sin 0.868 4
60.27
And 2
2(60.27
121 (to the nearest
)
1
deg
20.5
r
4
ee)
AOB
AOB AÔD
16.5cmSo 8.25cm
2AD BD
Model Exam 2 Paper 2
ANSWER(2 marks)
(ii) Calculate the area of triangle AOB.
Model Exam 2 Paper 2
10. (b) In the diagram below, not drawn to scale, O is centre of the circle of radius 9.5 cm and AB is a chord of length 16.5 cm.
10. (b) (ii)
Model Exam 2 Paper 2
2
1
2
2
The area of ,
1sin
21
sin21
9.5 9.5 sin cm2
0.5 90.25 0.8572 cm
(correct to 3 s.f.)38.7 cm
AOB A
ab C
AO BO
(ii)
Or
1
2
2 2
2
2
The area of , :
1sin
21
(9.5) sin cm2
45.125 0.8572 cm
(correct 38.7 to 3 s.f.)cm
AOB A
r
Model Exam 2 Paper 2
(ii) The semi-perimeter of ΔAOB,
Or
2(9.5 9.5 16.5)cm
235.5
cm2
17.75cm
a b cs
Model Exam 2 Paper 2
The area of ΔAOB, A1
2
2
2
( )( )( )
17.75(17.75 9.5)(17.75 9.5)
(17.75 16.5) cm
17.75(8.25)(8.25)(1
38.7c
.25) cm
(correct to 3 s )m .f.
s s a s b s c
Model Exam 2 Paper 2
ANSWER(3 marks) (iii)Hence, calculate the area of the
shaded region. [Use = 3.14]
Model Exam 2 Paper 2
10. (b) In the diagram below, not drawn to scale, O is centre of the circle of radius 9.5 cm and AB is a chord of length 16.5 cm.
10. (b) (iii)
2
2 2
295
3
.
60
2 c
1213.14 9.5 cm
360
(correct to 3 s.f.m )
r
The area of the minor sector AOB, A2
Model Exam 2 Paper 2
The area of the shaded region,
1
2
2
2(95.2 38.7)cm
56.5cm
A A A
Model Exam 2 Paper 2
ANSWER (iv) Calculate the length of the major arc AB.
(3 marks)Total 15 marks
Model Exam 2 Paper 2
10. (b) In the diagram below, not drawn to scale, O is centre of the circle of radius 9.5 cm and AB is a chord of length 16.5 cm.
10. (b) (iv)
Model Exam 2 Paper 2
The major sector angle, reflex
360 121
239
AOB
2360
2392 3.14 9.5 cm360
(correct to 3 s.39 ..6 cm f )
l r
The length of the major arc AB,
Model Exam 2 Paper 2
ANSWER(2 marks)
(a) Copy the diagram and complete it to show the points of P and M.
Model Exam 2 Paper 2
11.
In the diagram above, the position vectors of A and B relative to the origin are a and b respectively.
The point P is on OA such that OP = 3 PA.
The point M is on BA such that BM = MA.
11. (a)
The diagram was copied and completed as shown above. The points P and M are learly shown.
Model Exam 2 Paper 2
ANSWER
(1 mark)(b) OB is produced to N such that OB = 2 BN
(i) Show the position of N on your diagram.
Model Exam 2 Paper 2
11.
11. (b) (i)
The position of N is shown in the diagram.
Model Exam 2 Paper 2
ANSWER
(5 marks)
(ii) Express in terms of a and b the vectors.
Model Exam 2 Paper 2
(b) OB is produced to N such that OB = 2 BN
, and .AB PA PM���������������������������� ��������������
11.
11. (b) (ii)
AB AO OB
OA OB
������������������������������������������
����������������������������
a b
b a
Model Exam 2 Paper 2
3
1
3
PA OP
PA OP
given
3
43
4
OP OA
����������������������������
a
1
3
4
1
1 3
3
4
PA OP
����������������������������
a
a
Model Exam 2 Paper 2
given
1
21
21
( )2
1( )
2
BM MA
BM MA BA
AB
������������������������������������������
��������������
b a
a b
Model Exam 2 Paper 2
1 1( )
4 21 1 1
4 2
1 1
2
4
1 1
4
2
2
PM PA AM
PA MA
������������������������������������������
����������������������������
a a b
a a b
a b
b a
Hence,1 1 1
4, an
2 4dAB PA PM
������������������������������������������b a a b a
Model Exam 2 Paper 2
ANSWER
(4 marks)11. (c) Use a vector method to prove that P, M
and N are collinear.
Model Exam 2 Paper 2
11. (c) 2
1
21
21
2
BN OB
BN OB
BN OB
����������������������������
b
Model Exam 2 Paper 2
1 1( )
2 21 1 1
2
2
2 21
21
21 1
22 4
MN M
PM
B BN
BM BN
MN
������������������������������������������
����������������������������
a b b
a b b
a b
b a
b a
Since , then the vectors are either parallel or coincident. Since the vectorshave a common point M, then P, M and N are collinear.
2MN PM����������������������������
Model Exam 2 Paper 2
ANSWER
(3 marks)Total 15 marks
11. (d) Calculate the length of AN if.
8 2 and
4 4
a b
Model Exam 2 Paper 2
11. (d)
Model Exam 2 Paper 2
1
23
22 83
4 42
32 82
3 44
2
OA AN ON
AN ON OA
OB BN OA
=
������������������������������������������
������������������������������������������
������������������������������������������
b b a
b a
Considering ΔNOA:
Model Exam 2 Paper 2
1
23
22 83
4 42
32 82
3 44
2
OA AN ON
AN ON OA
OB BN OA
=
������������������������������������������
������������������������������������������
������������������������������������������
b b a
b a
Model Exam 2 Paper 2
3 8
6 4
3 8
6 4
5
2
Model Exam 2 Paper 2
Or
Model Exam 2 Paper 2
Considering ΔMAN:
1 1( )
2 21 1 1
2 2 21 1 1
2 2 23
28 23
4 42
AN AM MN
������������������������������������������
b a b a
b a b a
a a b b
a b
Model Exam 2 Paper 2
328 2
4 34
2
8 3
4 6
8 3
4
2
6
5
Model Exam 2 Paper 2
2 2
(exact value)
5.39 (c
( 5) 2
orrect to 3 s.f
25
29
.)
4
AN
The length of
Hence, the length of AN is 5.39 units.
Model Exam 2 Paper 2
CSEC MODEL EXAMINATION 3MATHEMATICS
Paper 190 minutes
Answer ALL the questions
NEXT
1. The decimal fraction 0.85 written as a common fraction, in its simplest form, is
ANSWER
(A)
(B)
(C)
(D) 10
17
17
2020
1717
10
Model Exam 3 Paper 1
1000.85 0.85
10085
10085 5
100 517
20
1. Multiply the decimal fraction with two
decimal places by , which is 1, to make
the decimal fraction a common fraction.
100
100
Divide both the numerator and thedenominator by their common factor 5.
This is the common fraction written in its simplest form.
Model Exam 3 Paper 1
2. The number 75 836 written correct to 4 significant figures is
ANSWER
(A) 80 000
(B) 76 000
(C) 75 800
(D) 75 840
Model Exam 3 Paper 1
2. 75 836
= 75 840 (4 s.f.)
The digit after the 4th significant figure is 6, so we add 1 to the digit 3. 0 is needed as a place holder.
Model Exam 3 Paper 1
3. Given that 768 51.2 = 39 321.6, then 76.8 0.512 =
ANSWER
(A) 3 932.16
(B) 393.216
(C) 39.321 6
(D) 3.932 16
Model Exam 3 Paper 1
3.
4 dp
768 51.2 39 321.6
1dp 1dp
76.8 0.512
1dp 3 dp 4dp
39.3216
Model Exam 3 Paper 1
4.
ANSWER
(A) 0.018
(B) 0.18
(C) 1.8
(D) 18
If 144 225 180, then 1.44 2.25
Model Exam 3 Paper 1
4.
2 dp 2 dp 2 dp 2 dp
144 225 12 12 15 15
12 15
180
1.44 2.25 1.2 1.2 1.5 1.5
1.2 1.5
1.80
1.8
Model Exam 3 Paper 1
5. y is inversely proportional to the square root of 7 may be expressed as
ANSWER
(A)
(B)
(C) y 72
(D)
7y 1
7y
2
1
7y
Model Exam 3 Paper 1
5. y is inversely proportional to means 71
7y
Model Exam 3 Paper 1
6. One hundred thousand written as a power of 10 is
ANSWER
(A) 104
(B) 105
(C) 106
(D) 107
Model Exam 3 Paper 1
6. One hundred thousand = 100 000
= 105
Model Exam 3 Paper 1
7. By the distributive law, 74 13 + 74 12 =
ANSWER
(A) 86 87
(B) 74 25
(C) 86 + 87
(D) 74 + 25
Model Exam 3 Paper 1
7. 74 13 + 74 12
= 74 (13 + 12)
= 74 25
The common
factor is 74.
Adding.
Model Exam 3 Paper 1
8. The highest common factor of 12, 24 and 30 is
ANSWER
(A) 2
(B) 4
(C) 5
(D) 6
Model Exam 3 Paper 1
8.
The HCF = 2 3 = 6
2 is a common factor of thethree numbers.3 is a common factor of thethree numbers.
2 12, 24, 30
3 6, 12, 15
2, 4, 5
Model Exam 3 Paper 1
9. The lowest common multiple of 5, 8 and 20 is
ANSWER
(A) 1
(B) 10
(C) 20
(D) 40
Model Exam 3 Paper 1
9. 2 5, 8, 20
2 5, 4, 10
2 5, 2, 5
5 5, 1, 5
1, 1, 1
The LCM = 2 2 2 5 = 40
Model Exam 3 Paper 1
10. The next two terms in the sequence 7, 6, 8 . . . is
ANSWER
(A) 7, 9
(B) 7, 7
(C) 7, 8
(D) 7, 6
Model Exam 3 Paper 1
10. 7, 6, 8, 7, 9, . . . 1 + 2 –1 + 2
Model Exam 3 Paper 1
11. A man’s annual income is $60 000. His non-taxable allowances is $15 000. If he pays a tax of 25% on his taxable income, then the tax payable is
ANSWER
(A) $3 750
(B) $11 250
(C) $15 000
(D) $33 750
Model Exam 3 Paper 1
11.
Model Exam 3 Paper 1
The taxable income $(60 000 15 000)
$45 000
the tax payable 25% of $45 000
25
100
$45 000
$11 250
12. The basic rate of pay is $28.00 per hour. What is the overtime rate of pay if it is one-and-a-half times the basic rate?
ANSWER
(A) $32.00
(B) $35.00
(C) $36.00
(D) $42.00
Model Exam 3 Paper 1
12. 1The overtime rate 1 $28
23
$282
4
1
2
$
$
3 4
Model Exam 3 Paper 1
13. Alfred saved $74 when he bought a cell phone at a sale which gave a discount of 20% on the marked price. What was the marked price of the cellphone?
ANSWER
(A) $370
(B) $296
(C) $222
(D) $148
Model Exam 3 Paper 1
13. The discount of 20% $74
100the marked price $74
20(which is 100%) $74 5
$370
Model Exam 3 Paper 1
14. A store offers a discount of 10% off the marked price for cash. If the cash price of a calculator is $135, what is the marked price?
ANSWER
(A) $13.50
(B) $121.50
(C) $148.50
(D) $150.00
Model Exam 3 Paper 1
14. 90% of the marked price $135
100the marked price $135
90
(which is 100%) $15
$150
10
Model Exam 3 Paper 1
15. The charge per kWh of electricity used is 35 cents. There is also a fixed charge of $27.00. What amount is the electricity bill if 80 kWh of electricity is consumed?
ANSWER
(A) $55
(B) $62
(C) $142
(D) $307
Model Exam 3 Paper 1
15. The cost for the electricity = 35¢ 80
= 2 800¢
= $28.00
The fixed charge = $27.00
\ the amount of the bill = $(28.00 + 27.00)
= $55.00
Model Exam 3 Paper 1
16. The exchange rate for US $1.00 is GUY $200. What amount of Guyanese dollars will a tourist receive for changing US $75.00?
ANSWER
(A) $150
(B) $1 500
(C) $15 000
(D) $150 000
Model Exam 3 Paper 1
16. US $1.00 = GUY $200
US $75.00 = GUY $200 75
= GUY $15 000
Model Exam 3 Paper 1
17. Calculate the book value of a computer valued at $3 000, after two years, if it depreciates by 10% each year.
ANSWER
(A) $300
(B) $2 400
(C) $2 430
(D) $2 920
Model Exam 3 Paper 1
17. The book value after 1 year 90% of $3 000
90$3 000
100
$2 700
The book value after 2 years 90% of $2 700
90$2 700
100
$2 430
Model Exam 3 Paper 1
18. A man pays $540 as income tax. If income tax is charged at 20% of the taxable income, what was his taxable income?
ANSWER
(A) $1 800
(B) $2 160
(C) $2 700
(D) $3 100
Model Exam 3 Paper 1
18. 100The taxable income $540
20
(which is 100%) $270 1
$2 70
0
0
Model Exam 3 Paper 1
19. X = {a, p, e}. How many subsets has the set X?
ANSWER
(A) 3
(B) 6
(C) 8
(D) 10
Model Exam 3 Paper 1
19. { }, {a}, {p}, {e}
{a, p}, {a, e}, {p, e}
{a, p, e}
The number of subsets = 8.
or
The number of subsets, N = 2n X = {a, p, e}
= 23 n(X) = 3
= 8 n = 3
Model Exam 3 Paper 1
20. A school has 200 students. 108 students play both soccer and basketball, 52 students play soccer only, and 15 students play neither sport. How many students play basketball only?
ANSWER
(A) 25
(B) 40
(C) 50
(D) 77
Model Exam 3 Paper 1
20.
Hence, 25 students play basketball only.
Model Exam 3 Paper 1
108 52 15 200
So 175 200
200 175
25
x
x
x
21. All students in a class play chess or scrabble or both. 15% of the students play chess only, and 37% of the students play scrabble only. What percentage of students play both games?
ANSWER
(A) 22
(B) 48
(C) 52
(D) 78
Model Exam 3 Paper 1
21.
Hence, 48% of the students play both games.
Model Exam 3 Paper 1
15 37 100
So 52 100
100 5
8
2
4
x
x
x
22.
ANSWER
(A) X Y
(B) Y X
(C) X Y = { }
(D) X Y { }
The Venn diagram above is best represented by the statement
Model Exam 3 Paper 1
22. Sets X and Y have no common elements, so X Y = { }.
Model Exam 3 Paper 1
23. 5:30 p.m. may be represented as.
ANSWER
(A) 05:30 h
(B) 17:30 h
(C) 15:30 h
(D) 18:30 h
Model Exam 3 Paper 1
23. 5 : 30 p.m. = (12 + 5) : 30 h
= 17 : 30 h
Model Exam 3 Paper 1
24.
ANSWER
(A) 8 (B) 16
(C) 24 (D) 32
The diagram above shows a circle with centre O and diameter 8 cm. The area of the circle, in cm2, is
Model Exam 3 Paper 1
24. The diameter of the circle, 8 cm
8 cmt he radius of the circle,
24 cm
d
r
2
2
2
2
The area of the circle, A π
π(4 cm)
π(16 cm )
c6 m1 π
r
Formula for the area of a circle.Substitute r = 4 cm.
Squaring.
Model Exam 3 Paper 1
25.
ANSWER
(A) (B)
(C) (D)
In the diagram above, POQ is a minor sector of a circle with angle POQ = 60° and OQ = r cm.The area, in cm2, of the minor sector POQ is
1π
3r
21π
6r 21
π3
r
1π
6r
Model Exam 3 Paper 1
25.
Model Exam 3 Paper 1
26. Mark takes 35 minutes to drive to university which is 45 km away from his apartment. His speed in km per hour is
ANSWER
(A)
(B)
(C)
(D) 45 35
60
35 60
45
35 45
60
45 60
35
Model Exam 3 Paper 1
26.
Model Exam 3 Paper 1
27.
ANSWER
(A) (B)
(C) (D)
The diagram above, not drawn to scale, shows a cone of radius r cm and height r cm. The volume of the cone, in cm3, is
21π
3r
34π
3r 24
π3
r
31π
3r
Model Exam 3 Paper 1
27.
Model Exam 3 Paper 1
28. The length of the edge of a cube is 20 cm. Thevolume of the cube is
ANSWER
(A) 8 000 cm3
(B) 400 cm3
(C) 240 cm3
(D) 200 cm3
Model Exam 3 Paper 1
28. 3
3
3
38 00
The volume of the cube,
(20 cm)
20 20 20 cm
cm0
V l
Model Exam 3 Paper 1
29. The mass of one tonne of sugar in kilograms is
ANSWER
(A) 100
(B) 1 000
(C) 10 000
(D) 100 000
Model Exam 3 Paper 1
29. 1 tonne = 1 000 kg
Model Exam 3 Paper 1
30. Robert has 0.75 kg of sweets. He has bags which can each hold 15 g of sweets. How many bags of sweets can he fill?
ANSWER
(A) 0.5
(B) 5
(C) 50
(D) 500
Model Exam 3 Paper 1
30.
Model Exam 3 Paper 1
0.75kg 0.75×1000g
750gThe number of bags that can 750gbe filled with s
5
we
0
ets15g
31. A bowl contains 6 green marbles and 7 yellow marbles. A marble is picked at random from the bowl. The marble is found to be green and it is not replaced. What is the probability that the next ball picked at random from the bowl will be yellow?
ANSWER
(A) (B)
(C) (D) 7
13
1
2
7
12
6
13
Model Exam 3 Paper 1
The number of green marblesremaining in the bowl = 6 1 = 5
The number of yellow marblesin the bowl = 7
The total number of marblesremaining in the bowl = 5 + 7 = 12
P(second marble is yellow) =
31.
7
12
Model Exam 3 Paper 1
32. The mode of the numbers 1, 2, 3, 4, 4, 5, 5, 5, 6, 7, 7, 8 is
ANSWER
(A) 4
(B) 5
(C) 6
(D) 7
Model Exam 3 Paper 1
32. The mode is 5, since it occurs the most number of times.
Model Exam 3 Paper 1
33. The median of the numbers 1, 2, 3, 4, 4, 5, 5, 5, 6, 7, 7, 8 is
ANSWER
(A) 6
(B) 5.5
(C) 5
(D) 4
Model Exam 3 Paper 1
1, 2, 3, 4, 4, 5, 5, 5, 6, 7, 7, 8
Two middle values
The median of the numbers,2
5 5 10
2 25Q
Q2 = 533.
Model Exam 3 Paper 1
34. The mean of the numbers 1, 2, 3, 4, 4, 5, 5, 5, 6, 7, 7, 8 is
ANSWER
(A) 4
(B)
(C) 5
(D)
34
4
15
2
Model Exam 3 Paper 1
34. The sum of the numbers, x = 1 + 2 + 3 + 4 +4 + 5 + 5 + 5 + 6 + 7 + 7 + 8
= 57The total frequency of the number, n = 12
The mean of the numbers,
57
1
4
23
4
xx
n
Model Exam 3 Paper 1
35. The scores of 100 students who took part in a shooting competition at a May Fair is recorded in the table shown below.
ANSWER
(A) (B)
(C) (D)
Score 0 1 2 3 4 5 6 7 8 9 10
Frequency 2 4 5 7 10 31 20 12 5 3 1
The probability that a student chosen at randomfrom these students scored exactly 6 is
1
1003
101
53
50
Model Exam 3 Paper 1
35. The frequency of the score 6 20
The total frequency, 100
20P(score 6)
1
1
0
5
0
n
Model Exam 3 Paper 1
36. The mean of the five numbers 7, p, 5, 9 and 18 is 12. The number p is
ANSWER
(A) 15
(B) 17
(C) 19
(D) 21
Model Exam 3 Paper 1
The sum of the numbers, p = 7 + p + 5 + 9 + 18
= p + 39
The total frequency, f = 5
The mean of the numbers, 12x
36.
Model Exam 3 Paper 1
37. (7a) (+2b) =
ANSWER
(A) 14ab
(B) –14ab
(C)
(D) 2
7ab
2
7ab
Model Exam 3 Paper 1
37. (7a) (+2b)
= 7 a 2 b
= 7 2 a b
= 14ab
Expand each expression.
Group like values.
The product of a negative
sign times a positive sign
is a negative sign.
Model Exam 3 Paper 1
38. a(a 3b) + b(a 3b) =
ANSWER
(A) (a 3b) (a + b)
(B) (a + 3b) (a b)
(C) (a 3b) (a b)
(D) (a + 3b) (a + b)
Model Exam 3 Paper 1
38. a(a 3b) + b(a 3b)
= (a 3b)(a + b)
Factorize using (a 3b)
as a common factor.
Model Exam 3 Paper 1
39.
ANSWER
(A) 2.6
(B) 26
(C) 260
(D) 2 600
5If 13, then
100
xx
Model Exam 3 Paper 1
39.5
13100
1320
13
260
20
x
x
x
Divide both the numerator and the denominator of the fraction on the LHS by the common factor 5.
Multiply both sides by 20.
Multiply the numbers on the RHS.Multiplying.
Model Exam 3 Paper 1
40. Given that p q means (p q)2, the value of 5 3 is
ANSWER
(A) 2
(B) 8
(C) 2
(D) 8
1
2
Model Exam 3 Paper 1
p q = (p q)2
5 3 = (5 3)2
= (2)2
= (4)
= 2
Substitute 5 for p and 3 for q in the formula.
Simplify the numbers in the brackets by subtracting.
Square the number in the brackets.
Multiply.
40.1
2
1
2
1
2
1
2
Model Exam 3 Paper 1
41. The statement ‘9 is subtracted from four times a certain number and the result is 15’ is represented by the equation
ANSWER
(A) 4x 15 = 9
(B) 9x 4 = 15
(C) 4x 15 = 9
(D) 4x 9 = 15
Model Exam 3 Paper 1
Four times a certain number = 4x
9 subtracted from four timesa certain number = 4x 9
The equation is:4x 9 = 15
41.
Model Exam 3 Paper 1
42. If 40 5x = x 14, then x =
ANSWER
(A) 6
(B) 6
(C) 9
(D) 9
Model Exam 3 Paper 1
42. 40 5 14
5 14 40
6 54
5
9
4
6
x x
x x
x
x
-
Subtract x and 40 from each side.
Add like terms.
Divide both sides by 6.
A negative sign divided by anegative sign is a positive sign.
or
Model Exam 3 Paper 1
So
Add 14 and 5x to each side.
Add like terms.
Divide both sides by 6.A positive sign divided by a positive sign is a positive sign.
40 5 14
40 14 5
54 6
5
9
9
4
6
x x
x x
x
x
x
x
Model Exam 3 Paper 1
So
43. 2x(4x + 5) 4x(3x + 2) =
ANSWER
(A) 4x2 2x
(B) 2x 4x2
(C) 20x2 + 18
(D) 20x 4x2
Model Exam 3 Paper 1
43.2 2
2 2
2
2
2 (4 5) 4 (3 2)
8 10 1
2 4
2 8
8 12 10 8
4 2
x x x x
x x x x
x x x x
x x
x x
Use the distributivelaw twice.
Group like terms.
Add like terms.
Rearrange terms.
Model Exam 3 Paper 1
44.
ANSWER
(A) 2k2
(B) 4k2
(C) 6k2
(D) 8k2
The figure above, consists of a triangle resting on a square of length 2k cm. The height of the triangle is 2k cm. The area, in cm2, of the figure is
Model Exam 3 Paper 1
44.
1
2
2 2
1The area of the triangle,
21
( 2 ) 2 cm2
2 cm
A bh
k k
k
Model Exam 3 Paper 1
Model Exam 3 Paper 1
45. The width of a rectangular glass block is w centimeters. Its height is four-fifths its width and its length is 5 times its height. The volume of the rectangular glass block, in centimeters, is
14
13w
316
5w
316
25w
14
3w(A)
(B)
(C)
(D)
ANSWER
Model Exam 3 Paper 1
45.
Model Exam 3 Paper 1
46.
ANSWER
(A) x is a multiple of y (B) x is a factor of y
(C) x is greater than y (D) x is divisible by y
The arrow diagram above represents the relation
Model Exam 3 Paper 1
46. 8 = 1 8 = 2 4 2 and 4 are factor of 8.10 = 1 10 = 2 5 2 and 5 are factor of 10.12 = 1 15 = 2 6 = 3 4 2, 3, 4 and 6 are factors of 12.Hence, the diagram represents the relation, ‘x is a factor of y’.
Model Exam 3 Paper 1
47.
ANSWER
(A) 3 x < 2
(B) 3 < x 2
(C) 3 < x < 2
(D) 3 x 2
The diagram above is the number line of the inequality
Model Exam 3 Paper 1
47.
(3 is included) (2 is not included)
3 x < 2
Model Exam 3 Paper 1
48.
ANSWER
(A) {(x, y) : 2 y 1} (B) {(x, y) : 2 < y < 1}
(C) {(x, y) : 2 < y 1} (D) {(x, y) : 2 < y 1}
The shaded area in the graph above can be represented by
Model Exam 3 Paper 1
48.
(–2 is not included)
(1 is included)
Model Exam 3 Paper 1
49.
(A)
(B)
(C) 5
(D) 5 ANSWER
4 1If ( ) , then ( 9)
7
xg x g
37
7-
37
7
Model Exam 3 Paper 1
49.
Model Exam 3 Paper 1
4 1If ( )
74( 9) 1
then ( 9)7
36 1
735
75
xg x
g
50. Which of the following sets is represented by the relation f : x → x3 + 1?
(A) {(0, 0), (1, 1), (2, 8), (3, 27)}
(B) {(0, 1), (1, 2), (2, 9), (3, 28)}
(C) {(0, 0), (1, 1), (2, 4), (3, 9)}
(D) {(0, 1), (1, 2), (2, 5), (3, 10)}
ANSWER
Model Exam 3 Paper 1
50.
3
3
3
3
3
( ) 1
(0) 0 1 0 1 1 (0,1)
(1) 1 1 1 1 2 (1, 2)
(2) 2 1 8 1
(0,1), (1, 2),
9 (2, 9)
(3) 3 1 27 1 28 (
(2, 9), (3, 2
3
8
, )
)
28
f x x
f
f
f
f
Model Exam 3 Paper 1
51. Which of the following diagrams is the graph of a function?
(A) (B)
(C) (D)
ANSWER
Model Exam 3 Paper 1
51. Using the vertical line test for a function, the graph below is the graph of a function.
x1 y1
x2 y2
1 1 relation
Model Exam 3 Paper 1
52.
ANSWER
(A) x + y = 180 (B) x + y < 180
(C) x = y (D) x > y
In the figure above AB and CD are parallel. The relation between x and y is
Model Exam 3 Paper 1
52. x = y because they are alternate angles.
Model Exam 3 Paper 1
53.
ANSWER
(A) (B) 50 sin
(C) 50 cos (D) 50 tan
The triangle ABC is right-angled at B. AC = 50 cm and angle ACB = degrees. An expression for the length of AB, in cm, is
Model Exam 3 Paper 1
53.
Model Exam 3 Paper 1
54.
ANSWER
(A) (B)
(C) (D)
In the right-angled triangle PQR, angle Q = 90, PR = 90 cm, PQ = 50 cm and RQ = y cm. cos PRQ =
9
5
50
y
90
y
5
9
Model Exam 3 Paper 1
54.
Model Exam 3 Paper 1
55.
ANSWER
(A) 10 (B) 12
(C) 14 (D) 16
How many triangles congruent to BCE are needed to completely cover the rectangle ABCD?
Model Exam 3 Paper 1
55.
The number of triangles needed = 14
Model Exam 3 Paper 1
56.
ANSWER
(A) 40 (B) 50
(C) 80 (D) 90
The diagram shows ΔABC with AD = BD = CD.The magnitude of angle ABC is
Model Exam 3 Paper 1
56.
Model Exam 3 Paper 1
57.
ANSWER
(A) an enlargement about the origin of scale factor 2
(B) a rotation through 180° about the origin
(C) an enlargement about the origin of scale factor 2
(D) a rotation through 180° about the point
The transformation that maps KLM onto PQR is
Model Exam 3 Paper 1
57.
22
1
PQ QR PR
KL LM KM
The image is on the other side of the centre of enlargement, which is the origin, so the scale factor is 2.
The transformation is an enlargement about the origin of scale factor 2.
Model Exam 3 Paper 1
58.
(A) (3, 2) (B) (2, 3)
(C) (3, 2) (D) (3, 2)
The point P(−2, 3) is rotated about the origin through an angle of 270° in an anticlockwise direction. The coordinates of the image of P is
ANSWER
Model Exam 3 Paper 1
58.
The image of P(2, 3) under the rotation is P1(3, 2)
Model Exam 3 Paper 1
59. PQR is an isosceles triangle with angle P = 70°. The possible values of angle Q are
ANSWER
(A) 50 or 70 (B) 40 or 70
(C) 40 or 55 or 70 (D) 50 or 60 or 70
Model Exam 3 Paper 1
59.
or
or
Model Exam 3 Paper 1
60. The image of the point P(5, 8) under the translation is
ANSWER
(A) (3, 5)
(B) (3, 5)
(C) (5, 3)
(D) (5, 3)
23
Model Exam 3 Paper 1
60.
2 5 2 5
3 83 8
3
5
T: 5, 8 3, 5
T P P
P
Model Exam 3 Paper 1
CSEC MODEL EXAMINATION 3MATHEMATICS
Paper 22 hours 40 minutes
SECTION IAnswer ALL the questions in this section
All working must be clearly shown
NEXT
1. (a) Using a calculator, or otherwise, calculate the exact value of
ANSWER
(4 marks)
1 23 1
5 33
25
giving your answer as a fraction in its lowest terms.
Model Exam 3 Paper 2
1. (a)
Model Exam 3 Paper 2
(i) Calculate his annual salary.
ANSWER
(1 mark)
Model Exam 3 Paper 2
1. (b) Mr. John has a wife and two children. His monthly salary during the year 2008 was $9 500.
Information on calculating income tax for 2008 is shown in the table below.
Allowances Tax RateWorker $12 000
25% of the taxable incomeSpouse $7 000Each child $3 500Taxable Income = Annual Salary Total Allowances
1. (b) (i) Mr. John’s annual salary = $9 500 12
= $114 000
Model Exam 3 Paper 2
(ii) Calculate Mr. John’s total allowances for 2008.
ANSWER
(2 marks)
Model Exam 3 Paper 2
1. (b) Mr. John has a wife and two children. His monthly salary during the year 2008 was $9 500.
Information on calculating income tax for 2008 is shown in the table below.
Allowances Tax RateWorker $12 000
25% of the taxable incomeSpouse $7 000Each child $3 500Taxable Income = Annual Salary Total Allowances
1. (b) (ii) The worker allowance $12 000
The spouse allowance $7 000
The children allowances $3 500 2
$7 000
Mr. John's totalallowances for 2008 $(12 000 7 000 7 000)
$26 000
Model Exam 3 Paper 2
(iii) Calculate Mr. John’s Income tax for 2008.
ANSWER
(3 marks)
Model Exam 3 Paper 2
1. (b) Mr. John has a wife and two children. His monthly salary during the year 2008 was $9 500.
Information on calculating income tax for 2008 is shown in the table below.
Allowances Tax RateWorker $12 000
25% of the taxable incomeSpouse $7 000Each child $3 500Taxable Income = Annual Salary Total Allowances
1. (b) (iii)
Model Exam 3 Paper 2
Mr. John’s taxable
income $(114 000 26 000)
$88 000
Mr. John’s income
tax for 2008 25% of $88 000
25
1
10014
$ 8822
$2
0
0
00
2 00
(iv) What percentage of Mr. John’s annual salary was paid in income tax? ANSWER
(2 marks)Total 12 marks
Model Exam 3 Paper 2
1. (b) Mr. John has a wife and two children. His monthly salary during the year 2008 was $9 500.
Information on calculating income tax for 2008 is shown in the table below.
Allowances Tax RateWorker $12 000
25% of the taxable incomeSpouse $7 000Each child $3 500Taxable Income = Annual Salary Total Allowances
1. (b) (iv) The percentage of Mr. John’s annual salary that was paid in income tax
22 000
114 000
19.3%
100%
11 100%
57(correct to 3 s.f.)
Model Exam 3 Paper 2
2. (a) Simplify completely: (5a 1)2
ANSWER
(2 marks)
Model Exam 3 Paper 2
2. (a) (5a 1)2 = (5a 1)(5a 1)
= 5a(5a 1) 1(5a 1)
= 25a2 5a 5a + 1
= 25a2 10a + 1
Model Exam 3 Paper 2
2. (b) Make p the subject of the formula
ANSWER
(3 marks)8
.5
qp
Model Exam 3 Paper 2
2. (b)
Model Exam 3 Paper 2
2. (c) Factorize completely
ANSWER
(1 mark)(i) 8mn 6n2
Model Exam 3 Paper 2
(ii) 49p2 q2 (2 marks)
2. (c) (i) 8mn 6n2
= 2n(4m 3n)
Factorize using theHCF of the two terms.
Model Exam 3 Paper 2
(ii) 49p2 q2
= (7p)2 q2
= (7p + q)(7p q)
2. (d) Solve the following pair of simultaneous
equations:
ANSWER
(4 marks)Total 12 marks
5x 2y = 164x + 3y = 1
Model Exam 3 Paper 2
2. (d) 5x 2y = 16 4x + 3y = 1
3 and 2:15x 6y = 48 8x + 6y = 2
+ :15 8 48 2
23 46
4
2
6
23
x x
x
x
Model Exam 3 Paper 2
Substitute x = 2 in :
4(2) 3 1
8 3 1
3 1 8
3 9
9
33
y
y
y
y
y
Hence, x = 2, y = 3.
Model Exam 3 Paper 2
(i) List the elements of the set
ANSWER(2 marks)a) J Kb) J K.
Model Exam 3 Paper 2
3. (a) The figure below is a Venn diagram showing a Universal set, U, and two subsets, J and K. The numerals in the diagram represent elements of the sets.
3. (a) (i) a) J K = {10, 20, 30}
b) J K = {5, 15, 25, 35}
Model Exam 3 Paper 2
(ii) Determine the value of n (J K).ANSWER
(1 mark)
Model Exam 3 Paper 2
3. (a) The figure below is a Venn diagram showing a Universal set, U, and two subsets, J and K. The numerals in the diagram represent elements of the sets.
3. (a) (ii) J K = {5, 10, 15, 20, 25, 30, 35, 40, 50, 60, 70, 80}
n (J K) = 12
Model Exam 3 Paper 2
(iii) Describe in words
ANSWER(3 marks)a) the Universal set Ub) the set J
Model Exam 3 Paper 2
3. (a) The figure below is a Venn diagram showing a Universal set, U, and two subsets, J and K. The numerals in the diagram represent elements of the sets.
3. (a) (iii) a) The Universal set U = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100}
The Universal set U is the set of multiples of 5 less than 101 (or no more than 100, or between 1 and 100 inclusive).
b) The set K = {10, 20, 30, 40, 50, 60, 70, 80}
The set K is the set of multiples of 10 less than 81(or no more than 80, or between 1 and 80 inclusive).
Model Exam 3 Paper 2
3. (b) (i)
ANSWER(5 marks)
Use a ruler and a protractor to draw accurately the quadrilateral PQRS shown below. PQ = 9 cm, QR = 5 cm, PS = 7.5 cm, angle PQR = 135° and angle QPS = 60°.
Model Exam 3 Paper 2
3. (b) (i)
Model Exam 3 Paper 2
(ii) Measure and state the length of RS.
ANSWER
(1 mark)Total 12 marks
Model Exam 3 Paper 2
3. (b)
3. (b) (ii) By measurement, the length of RS = 9.2 cm
Model Exam 3 Paper 2
(a) Calculate the area of triangle ABC. ANSWER(2 marks)
Model Exam 3 Paper 2
4. The diagram below, not drawn to scale, shows a triangular prism with identical right angled isosceles triangles at both ends. Angle ABC = 90° and AB = BC = 6 cm.
4. (a)
2
2
1The area of triangle ,
21
6 cm
18 cm
6 cm2
3 6 cm
ABC A bh
Model Exam 3 Paper 2
(b) Calculate the length of the edge CD.ANSWER
(3 marks)
Model Exam 3 Paper 2
The volume of the prism is 270 cm3.
4.
4. (b) The volume of the prism, V = 270 cm3
The volume of the prism, V = Al18
18 270So
270i.e. cm
1830
cm215 cm
l
l
l
CD
Hence, the length of the edge CD is 15 cm.
Model Exam 3 Paper 2
(c) Calculate, to one decimal place, the length of the edge AC.
ANSWER
(2 marks)
Model Exam 3 Paper 2
4.
4. (c)
Considering the right-angled ABC and using Pythagoras’ theorem:2 2 2
2 2
8
6 6
36 36
72
5 c
7
.
2 c
m
m
AC AB BC
AC
Hence, the length of the edge AC is 8.5 cm.
Model Exam 3 Paper 2
4. (d) State the number of faces, edges, and vertices of the prism by completing the table
below.
ANSWER
(3 marks)Total 10 marks
Face Edge Vertices
Model Exam 3 Paper 2
4. (d)
Face Edge Vertices
5 9 6
The completed table is shown above.
Model Exam 3 Paper 2
(a) Draw the line x = 1 ANSWER(1 mark)
Model Exam 3 Paper 2
5. Triangle JKL has coordinates J(2, 1), K(4, 3) and L(2, 6) respectively.
5. (a) The vertical line x = 1 was drawn on the graph paper.
Model Exam 3 Paper 2
5. (b)
ANSWER
(3 marks)Draw the image of triangle JKL after a reflectionin the line x = 1. Label the image JKL.
Model Exam 3 Paper 2
The image triangle JKL was drawn or the graph paper. The vertices of triangle JK L are J(0, 1), K(2, 3), and L(0, 6).
5. (b)
Model Exam 3 Paper 2
(c)
ANSWER(2 marks)
A second transformation maps triangle JKL onto triangle JKL which has vertices J(4, 2), K(8, 6) and L(4, 12).(i) Draw the triangle JKL.
Model Exam 3 Paper 2
5.
5. (c) (i) The triangle JKL was drawn on the graph paper.The vertices of triangle JKL are J(4, 2), K(8, 6), and L(4, 12) respectively.
Model Exam 3 Paper 2
ANSWER(4 marks)
(ii) Describe completely the transformation which maps triangle JKL onto triangle JKL.
Model Exam 3 Paper 2
(c) A second transformation maps triangle JKL onto triangle JKL which has vertices J(4, 2), K(8, 6) and L(4, 12).
5.
5. (c) (ii) 2 ( 12)
6 12 12
510
52
J L
JL
The object and the image are on opposite sides of the centre of enlargement, which is the origin, 0.
Model Exam 3 Paper 2
∴ the scale factor, k = 2.
So E(0, 2) : JKL J K L.
Hence, the transformation which maps triangle JKLonto triangle JKL is an enlargement with the origin as the centre of enlargement and scale factor 2.
Model Exam 3 Paper 2
5. (d) Calculate the value of
ANSWER
(2 marks)Total 12 marks
Area of triangle
Area of triangle
J K L
JKL
Model Exam 3 Paper 2
5. (d)
Model Exam 3 Paper 2
6. The table below gives the birth rate per 1 000 people in Trinidad and Tobago during a six-year period.
ANSWER(6 marks)
Year 2003 2004 2005 2006 2007 2008Birth Rate 12.74 12.81 12.72 12.90 13.07 13.22
(a) Using a scale of 10 cm to represent a birth rate of 1 per thousand on the vertical axis and 2 cm to represent 1 year on the horizontal axis, draw a line-graph to represent the birth rate in Trinidad and Tobago from 2003 to 2008.
Model Exam 3 Paper 2
6. Using the given scales and data the graph was drawn as shown below.
(a)
Model Exam 3 Paper 2
6. (b) In which year was there
ANSWER
(1 mark)(i) the lowest birth rate?
Model Exam 3 Paper 2
(ii) the highest birth rate? (1 mark)
6. (b) (i) The lowest birth rate = 12.72 per 1 000 people
Model Exam 3 Paper 2
(ii) The highest birth rate = 13.22 per 1 000 people
6. (c) During which period was there
ANSWER
(1 mark)(i) the greatest increase in the birth rate?
Model Exam 3 Paper 2
(ii) the smallest increase in the birth rate?
(1 mark)
6. (c) (i) The period when there was the greatest increase in the birth rate = 2005 to 2006
Model Exam 3 Paper 2
(ii) The period when there was the smallest increase in the birth rate = 2003 to 2004
6. (d) During which period was there a decrease in the birth rate?
ANSWER
(1 mark)Total 11 marks
Model Exam 3 Paper 2
6. (d) The period when there was a decrease in the birth rate = 2004 to 2005
Model Exam 3 Paper 2
7. (a) A line segment connects the points A(3, 10) and B(f, g). If the mid-point of AB is (1, 2), calculate the values of f and g.
ANSWER
(4 marks)
Model Exam 3 Paper 2
7. (a) Calculate the x-coordinate of B:
1 2
23
12
3 1(2) 2
2 53
x xx
f
f
f
Model Exam 3 Paper 2
Hence, f = 5 and g = 6.
21
210
22
6
10 2(2) 4
4 10
y yy
g
g
g
Model Exam 3 Paper 2
Calculate the y-coordinate of B:
7. (b) Two functions, f and g, are defined as follows:
ANSWER
(1 mark)
: 7 4
5:
1
f x x
g xx
Calculate
(i) f(0)
Model Exam 3 Paper 2
7. (b) (i) ( ) 7 4
(0) 7(0) 4
0
4
4
f x x
f
Model Exam 3 Paper 2
(ii) g(2)
ANSWER
(1 mark)
Model Exam 3 Paper 2
7. (b) Two functions, f and g, are defined as follows:
: 7 4
5:
1
f x x
g xx
Calculate
7. (b) (ii)5
( )1
5(2)
2
21
3
15
3
g xx
g
Model Exam 3 Paper 2
(iii) f1(x)
ANSWER
(2 marks)
Model Exam 3 Paper 2
7. (b) Two functions, f and g, are defined as follows:
: 7 4
5:
1
f x x
g xx
Calculate
7. (b) (iii)
1
( ) 7 4
7 4
7 4
4 7
4
74
7
( )4
7
f x x
y x
x y
x y
xy
xy
xf x
Defining equation for f(x).
Interchanging x and y.
Adding 4 to both sides.
Dividing both sides by 7.
Defining equation for f1(x)
Model Exam 3 Paper 2
(iv) the value of x if fg(x) = 1
ANSWER
(3 marks)Total 11 marks
Model Exam 3 Paper 2
7. (b) Two functions, f and g, are defined as follows:
: 7 4
5:
1
f x x
g xx
Calculate
7. (b) (iv) 5( ) 7 4
1
354
1( ) 1
354 1
1
fg xx
xfg x
x
Equating
Model Exam 3 Paper 2
351 4 5
135
157 1
7 1
6
6
x
x
x
x
x
x
Adding 4 to both sides
Dividing both sides by 5, and multiplying by x + 1.
Subtracting 1 from both sides.
Hence, the value of x is 6.
Model Exam 3 Paper 2
The first three diagrams in a sequence are shownbelow. Diagram 1 has three coins, which can beconsidered as a square pattern with a missing coin.
8.
Diagram 2 consists of a square pattern with amissing coin formed by eight coins.Diagram 3 consists of a square pattern with amissing coin formed by fifteen coins.
Diagram 1 Diagram 2 Diagram 3
Model Exam 3 Paper 2
ANSWER
(2 marks)Draw Diagram 4 in the sequence.8. (a)
Model Exam 3 Paper 2
8. (a)
Diagram 4
Diagram 4 in the sequence can be seen above.
Model Exam 3 Paper 2
ANSWER
(6 marks)
Complete the table by inserting the appropriatevalues at the rows marked (i), (ii) and (iii).
8. (b)
Diagram Number
Number of Coins Forming the Square
Pattern for calculating the Total Number of Coins in the Diagram
1 3 22 12 8 32 13 15 42 14 — —
8 80 —
n — —
Model Exam 3 Paper 2
8. (b) Diagram Number
Number of Coins Forming the Square
Pattern for calculating the Total Number of Coins in the Diagram
1 3 22 12 8 32 13 15 42 14 24 52 18 80 92 1n n2 + 2n (n + 1)2 1
The completed table can be seen above.
Model Exam 3 Paper 2
ANSWER
(2 marks)Total 10 marks
Hence, determine a formula consisting of asingle term in n for calculating the total number of coins, N, in a diagram.
8. (c)
Model Exam 3 Paper 2
8. (c) The total number of coins in a diagram,
N = n2 + 2n
N = n(n + 2)
Hence, the formula is N = n(n + 2).
Model Exam 3 Paper 2
SECTION II
Answer TWO questions in this section
Model Exam 3 Paper 2
ANSWER
(3 marks)
Copy and complete the table below for the function9. (a)
2
10for 2 7.y x
x
x 2 3 4 5 6 7y 1.1 0.6 0.3
Model Exam 3 Paper 2
9. (a) When x = 2, then
When x = 5, then
2
10 10 5
2 4 22.5y
2
10 10 2
5 25 50.4y
When x = 7, then
2
10 10(correct to 1 d. p.)
7 40.2
9y
x 2 3 4 5 6 7y 2.5 1.1 0.6 0.4 0.3 0.2
The completed table is shown above.
Model Exam 3 Paper 2
ANSWER(4 marks)
Using a scale of 2 cm to represent 1 uniton both axes, plot the points whose x andy values are given in the table above.
(i)
Model Exam 3 Paper 2
Copy and complete the table below for the function9. (b)
2
10for 2 7.y x
x
x 2 3 4 5 6 7y 1.1 0.6 0.3
9. (b) (i) Using the given scale the points were plotted as shown.
Model Exam 3 Paper 2
ANSWER
(1 mark)Draw a smooth curve through the points.(ii)
Model Exam 3 Paper 2
Copy and complete the table below for the function9. (b)
2
10for 2 7.y x
x
x 2 3 4 5 6 7y 1.1 0.6 0.3
9. (b) (ii) A smooth curve was drawn through the points as shown.
Model Exam 3 Paper 2
ANSWER
(1 mark)
Use your graph to estimate9. (c)
(i) the value of y when x = 3.5
Model Exam 3 Paper 2
9. (c) From the construction on the graph:
(i) when x = 3.5, the value of y = 0.8
Model Exam 3 Paper 2
ANSWER
(2 marks)the value of x when y = 4.09. (c) (ii)
Model Exam 3 Paper 2
9. (c) (ii) when y = 4.0, the value of x = 1.6
Model Exam 3 Paper 2
ANSWER
(4 marks)Total 15 marks
Draw a tangent to the curve at the point (3, 1.1) as accurately as possible. Hence, estimate the gradient of the curve at the point (3, 1.1).
9. (d)
Model Exam 3 Paper 2
The tangent to the curve at the point (3, 1.1) was drawn as shown:9. (d)Choose two points on the tangent, for example, A(0, 3.3) and B(4.4, 0).The gradient of the tangent AB,
Hence, the gradient of the curve at the point (3, 1.1) is 0.75.
1
2
2 11
2 1
2
0 3.3
4.4 00.
(0,3.3) ( , )
(4.43.3
4.4
7
,
5
0)
0.75 ( , )
A A xy y
mx
y
B
B x y
x
m
Model Exam 3 Paper 2
The diagram below, not drawn to scale, shows three points P, Q and R on a straight line. Q and R are both due East of P such that PQ = 5 km, and QR = 12 km.
10.
Model Exam 3 Paper 2
ANSWER
(1 mark)Copy and label the diagram.10. (a)
Model Exam 3 Paper 2
The diagram was copied and labelled as shown above.10. (a)
Model Exam 3 Paper 2
ANSWER
(4 marks)
On your diagram, show the point, S, such that– the bearing of S from R is 035°– the bearing of Q from S is 121°
10. (b)
Model Exam 3 Paper 2
The bearing of S from R is 035° and the bearing of Q from S is 121° are shown in the diagram.
10. (b)
Model Exam 3 Paper 2
ANSWER
(1 mark)
State the size of
(i) SRQ
10. (c)
Model Exam 3 Paper 2
(ii) RSQ(iii) SQP
(1 mark)(1 mark)
(i) SRQ = 90° 35° = 55°10. (c)
Model Exam 3 Paper 2
(ii) NSR = 180 35 = 145
RSQ = 360 (145 + 121)
= 360 266
= 94
(iii) SQP = 55° + 94° = 149°
ANSWER
(2 marks)
Calculate, correct to the nearest kilometre, the distance.
(i) SQ
10. (d)
Model Exam 3 Paper 2
10. (d) (i)
Considering RSQ and using the sine rule:12 km
sin 55 sin 9412 km sin 55
sin 9412 km 0.819
0.9989.8
0
k
1 km
m
SQ
SQ
Hence, the distance SQ is 10 km.
(correct to the nearest kilometre).
Model Exam 3 Paper 2
ANSWER
(2 marks)
Calculate, correct to the nearest kilometre, the distance.
(ii) SP
10. (d)
Model Exam 3 Paper 2
10. (d) (ii)
Considering PSQ and using the cosine rule:
Hence, the distance SP is 14 km.
(correct to the nearest kilometre).
205.026 km14.3 km14 km
SP
2 2 29.8 5 2 9.8 5 cos14996.04 25 98 ( 0.857)121.04 83.986205.026
SP
Model Exam 3 Paper 2
ANSWER
(3 marks)Total 15 marks
Calculate the bearing of S from Q.10. (e)
Model Exam 3 Paper 2
10. (e)
SQR = 180 149 = 31
NQS reflex = 270 + 31 = 301
Hence, the bearing of S from Q is 301.
Model Exam 3 Paper 2
The points O(0, 0), A(4, 2), B(5, 7) andC(3, r) are plotted on graph paper.
11.
ANSWER
Model Exam 3 Paper 2
(a)
(i) OA��������������
(1 mark)
Write the following position vectors in the
form :x
y
11.
The points O, A and B can be seen plotted on graph paper.
Model Exam 3 Paper 2
(a) (i)4
2OA
��������������
ANSWER
(1 mark)(ii) OB��������������
Model Exam 3 Paper 2
The points O(0, 0), A(4, 2), B(5, 7) andC(3, r) are plotted on graph paper.
11.
(a) Write the following position vectors in the
form :x
y
(a) (ii)5
7OB
��������������
Model Exam 3 Paper 2
The points O, A and B can be seen plotted on graph paper.
11.
ANSWER
(1 mark)(iii)OC��������������
Model Exam 3 Paper 2
The points O(0, 0), A(4, 2), B(5, 7) andC(3, r) are plotted on graph paper.
11.
(a) Write the following position vectors in the
form :x
y
(a) (iii) 3O
rC
��������������
Model Exam 3 Paper 2
The points O, A and B can be seen plotted on graph paper.
11.
ANSWER
(2 marks)
Write as a column vector, in the form (b) :x
y
(i) BA��������������
Model Exam 3 Paper 2
The points O(0, 0), A(4, 2), B(5, 7) andC(3, r) are plotted on graph paper.
11.
11. (b) (i)
Model Exam 3 Paper 2
ANSWER
(2 marks)(ii) in terms of r. AC��������������
Model Exam 3 Paper 2
Write as a column vector, in the form (b) :x
y
The points O(0, 0), A(4, 2), B(5, 7) andC(3, r) are plotted on graph paper.
11.
11. (b) (ii)
Model Exam 3 Paper 2
ANSWER
(3 marks)
11. (c) Calculate the values ofr for which 37
The point D is such that7
.17
OD
��������������
AC��������������
Model Exam 3 Paper 2
Squaring both sides:
11. (c)
Model Exam 3 Paper 2
Factorizing as the difference of two squares:
Model Exam 3 Paper 2
ANSWER
(5 marks)Total 15 marks
11. (d) Using a vector method, prove that the pointsA, B and D are collinear.
Model Exam 3 Paper 2
11. (d)
DB and BA have the same direction and a common point B, so A, B, and D are collinear.
Model Exam 3 Paper 2
CSEC MODEL EXAMINATION 4MATHEMATICS
Paper 190 minutes
Answer ALL the questions
NEXT
1. (−4)2 + (−3)2 =
ANSWER
(C) −25(A) −7 (D) 25(B) 7
Model Exam 4 Paper 1
1. Expand usingthe definition ofa square.
The product oftwo negativesigns is apositive sign.
2 2( 4) ( 3)
( 4) ( 4) ( 3) ( 3)
16 9
25
Model Exam 4 Paper 1
2. 25.3 ÷ 0.01 =
ANSWER
(C) 253(A) 0.253 (D) 2 530(B) 2.53
Model Exam 4 Paper 1
2.Multiply both thenumerator andthe denominatorby 100, to makethe number in thedenominator awhole number.
25.325.3 0.01
0.0125.3 100
0.01 100
2530
1 2 3 5 0
Model Exam 4 Paper 1
3. The number 8 705 written in standard form is
ANSWER
(C) 8.705 × 103
(A) 8.705 × 10−3
(D) 8.705 × 102
(B) 8.705 × 10−2
Model Exam 4 Paper 1
3. 8 705 = 8.705 × 1000 = 8.705 × 103
Model Exam 4 Paper 1
4. The number 0.057 04 written in standard form is
ANSWER
(C) 5.704 × 103
(A) 5.704 × 10−3
(D) 5.704 × 102
(B) 5.704 × 10−2
Model Exam 4 Paper 1
4.
25.704 10
The first number must be between 1 and 10.
10.057 04 5.704
100
Model Exam 4 Paper 1
5. In a school, the ratio of the number of boys to the number of girls is 3:5. If the school has 864 students, how many are girls?
ANSWER
(C) 324(A) 172 (D) 540(B) 288
Model Exam 4 Paper 1
5.
Model Exam 4 Paper 1
6. A class has 40 students. 75% of the class are girls. 40% of the girls have a calculator. How many girls in the class have a calculator?
ANSWER
(C) 10(A) 30 (D) 4(B) 12
Model Exam 4 Paper 1
6.
Model Exam 4 Paper 1
7. Using the distributive law
ANSWER
(C) 70 × 20
(A) 39 × 41
(D) 35 × 10
(B) 35 × 20
35 16 4 35× + × =
Model Exam 4 Paper 1
7. Using thedistributivelaw.
Adding thenumbers inthe brackets.
35 16 4 35 35 (16 4)
35 20
× + × = × +
= ×
Model Exam 4 Paper 1
8. 8 05110 =
ANSWER
(C) 8 × 102 + 5 × 10 + 1
(A) 8 × 102 + 5 × 1
(D) 8 × 103 + 5 × 10 + 1
(B) 8 × 103 + 5 × 1
Model Exam 4 Paper 1
8. 8 05110 = 8 × 103 + 0 × 102 + 5 × 101 + 1 × 100
= 8 × 103 + 5 × 10 + 1 × 1
= 8 × 103 + 5 × 10 + 1
Expandingas a baseten number.
Model Exam 4 Paper 1
9. The next term in the sequence 1, 5, 25, 125, . . . is
ANSWER
(C) 500(A) 250 (D) 625(B) 450
Model Exam 4 Paper 1
9.
Model Exam 4 Paper 1
0 1 2 3
4
1, 5, 25, 125, . . . 5 , 5 , 5 , 5 , . . .
The next term 5
62 5
=
=
=
10. A stick is used to measure cloth of lengths 2.8 m, 4.2 m and 5.6 m. If the rod fitted each length of cloth an exact number of times, what is its greatest length?
ANSWER
(C) 2.1 m(A) 0.7 m (D) 2.8 m(B) 1.4 m
Model Exam 4 Paper 1
10. 2 28, 42, 56 7 14, 21, 28 2, 3, 4
The HCF of 28, 42 and 56 = 2 × 7 = 14
So the HCF of 2.8, 4.2 and 5.6 = = 1.4
Hence, the greatest length of the rod is 1.4 m.
14
10
Model Exam 4 Paper 1
11. The rateable value of a house is $1 850. If the rates charged for that area are 25¢ in the $1, then the amount payable per annum for rates is
ANSWER
(C) $1 825
(A) $462.50
(D) $1 875
(B) $925.00
Model Exam 4 Paper 1
11. The rates 25 ¢ in the $1
25%
0.25
=
=
=
The annual amount payable = 0.25 × $1 850= $462.50
Model Exam 4 Paper 1
12. The basic hourly rate of pay is $9.00. Overtime is paid for at double the basic rate. How much will Rita receive for overtime if she worked 20 hours overtime?
ANSWER
(C) $180.00
(A) $58.00
(D) $360.00
(B) $90.00
Model Exam 4 Paper 1
12.
$18.00
The overtime rate of pay $9.00 2
the overtime pay $18.00 20
$360.00
Model Exam 4 Paper 1
13. If $9 000 is borrowed for 3 years at the rate of 4% per annum, the simple interest is
ANSWER
(C) $750.00
(A) $120.00
(D) $1 080.00
(B) $167.50
Model Exam 4 Paper 1
13.
Model Exam 4 Paper 1
14. A woman’s taxable income is stated as $20 500. She pays tax at the marginal rate of 25%. The amount of income tax payable is
ANSWER
(C) $1 640
(A) $5 125
(D) $820
(B) $2 050
Model Exam 4 Paper 1
14. The amount of income tax payable = 25% of $20 500
Model Exam 4 Paper 1
1 2
25= $20 500
1001
$20 5004$5125
15. The cash price of a cellphone is $2 500. On a hire-purchase plan, a deposit of $250 is required, followed by 18 monthly payments of $137.50. How much is saved by paying cash?
ANSWER
(C) $200(A) $250 (D) $175(B) $225
Model Exam 4 Paper 1
15. The deposit = $250The amount of the monthlypayments = $137.50 × 18
= $2 475The hire-purchase price = $(250 + 2 475)
= $2 725The cash price = $2 500The difference between thehire-purchase price andthe cash price = $(2 725 − 2 500)
= $225
Hence, the amount saved by paying cash is $225.
Model Exam 4 Paper 1
16. After a discount of 40% is given on a stove, Angela saved $720. What was the marked price of the stove?
ANSWER
(C) $960(A) $2 160 (D) $760(B) $1 800
Model Exam 4 Paper 1
16.
Model Exam 4 Paper 1
17. A house costing $250 000 can be bought by making a 5% deposit and taking a mortgage loan for the remaining amount. What is the amount of the deposit?
ANSWER
(C) $12 500
(A) $2 500
(D) $25 000
(B) $5 000
Model Exam 4 Paper 1
17.
Model Exam 4 Paper 1
18. Orange juice is sold in packets of 250 ml, 500 ml and 750 ml. The size of the smallest container that can be completely filled by a whole number of packets of juice of either size is
ANSWER
(C) 1 500(A) 150 (D) 2 000(B) 750
Model Exam 4 Paper 1
18. 2 25, 50, 75
3 25, 25, 75
5 25, 25, 25
5 5, 5, 5
1, 1, 1
Hence, the smallest container that can be completely filled by a whole number of packets of juice of either size is 1 500 ml.
The LCM of 25,50 and 75 = 2 × 3 × 5 × 5 = 150So the LCM of 250,500 and 750 = 150 × 10 = 1 500
Model Exam 4 Paper 1
19.
ANSWER
In the Venn diagram above, the shaded regionrepresents
(C) Q P′
(A) Q′
(D) P Q′
(B) P Q′
Model Exam 4 Paper 1
19. The shaded region represents P only = P and not Q
= P Q′
Model Exam 4 Paper 1
20. If A = {1, 2, 3, 4, 5, 6}, then the number ofsubsets of A is
ANSWER
(C) 32(A) 5 (D) 64(B) 25
Model Exam 4 Paper 1
20. The number ofsubsets of A, N = 2n n(A) = 6
= 26 n = 6= 2 × 2 × 2 × 2 × 2 × 2= 64
Model Exam 4 Paper 1
21.
ANSWER
The two circles above represent P = {factors of 8} and Q = {factors of 12}. The shaded region represents
(C) {1, 2, 4, 8}
(A) {1, 2, 3, 4, 6, 8, 12}
(D) {1, 2, 3, 4, 6, 12}
(B) {3, 6, 8, 12}
Model Exam 4 Paper 1
21. 1, 2, 4, 8
1, 2, 3, 4, 6, 12
P
Q
Factors of 8 1 8
2 4
Factors of 12 1 12
2 6
3 4
P Q = P or Q These are the = {1, 2, 3, 4, 6, 8, 12} elements in the
shaded region.
Model Exam 4 Paper 1
22. If U = {2, 3, 5, 7, 11, 15, 17, 19}, A = {2, 3, 5, 7, 11} and B = {3, 7, 11, 15}, then (A B)′ =
ANSWER
(C) {3, 7, 11, 15}
(A) {2}
(D) {2, 3, 17, 19}
(B) {17, 19}
Model Exam 4 Paper 1
22.
15,17,19
2, 5,17,19
17,19
or
2, 3, 5, 7,11
3, 7,11,15
2, 3, 5, 7,11,15
2, 3, 5, 7,11,15,17,19
17,19
A
B
A B A B
A
B
A B
U
A B
Ç È
È
È
Model Exam 4 Paper 1
or
The shaded region in the Venn diagram represents (A B)′.
(A B)′ = {17, 19}
Model Exam 4 Paper 1
23. The distance around the boundary of a circular lake is 132 m. The radius of the lake is
ANSWER
(C)
(A) 66π
(D) 264π
(B) 132π m
66m
π
Model Exam 4 Paper 1
23.
Model Exam 4 Paper 1
The Circumference of the lake,
132 m
2π
2π 132
132
F
2π66
π
Hence, the radius of the lake
ormula for the circumference
of a circle.
66m .is
π
C
C r
r
r
24.
ANSWER
The diagram shows the sector of a circle with centre O, radius 6 cm and angle AOB = 150°. The length of the minor arc AB, in cm, is
(C) 5π(A) (D) 10π(B) 2
π5
5π
2
Model Exam 4 Paper 1
24.
Model Exam 4 Paper 1
25.
ANSWER
The area of the trapezium shown above is
(C) 44 cm2
(A) 31.5 cm2
(D) 45 cm2
(B) 34 cm2
Model Exam 4 Paper 1
25.
Model Exam 4 Paper 1
26. The volume of a cuboid with edges of lengths 10 cm, 100 cm and 1 000 cm is
ANSWER
(C) 1 110 cm3
(A) 1 000 000 cm3
(D) 60 cm3
(B) 2 000 cm3
Model Exam 4 Paper 1
26.
The volume of the cuboid, V = lbh Formula= 1 000 × 100 × 10 cm3 = 1 000 000
cm3
Model Exam 4 Paper 1
27. The length of a rectangle is tripled. By what value must the width of the rectangle be multiplied for its area to remain the same.
ANSWER
(C) (A) 3 (D) (B) 1
2
1
3
1
6
Model Exam 4 Paper 1
27.
Model Exam 4 Paper 1
28. The area of a triangle is 31.5 cm2. If the base of the triangle is multiplied by four and the altitude is halved, then the area would be
ANSWER
(C) 63 cm2
(A) 252 cm2
(D) 15.75 cm2
(B) 126 cm2
Model Exam 4 Paper 1
28.
Model Exam 4 Paper 1
29.
ANSWER
The diagram shows a cylinder with radius 4 cm and height 25 cm. The volume of the cylinder is
(C) 300π cm3
(A) 100π cm3
(D) 400π cm3
(B) 200π cm3
Model Exam 4 Paper 1
29. The volume of thecylinder, V = π2h Formula
= π(4)2 × 25 cm3 r = 4 cm and
= π(16) × 25 cm3 h = 25 cm
= 400π cm3
Model Exam 4 Paper 1
30. If a sailing boat travels a distance of 2 040 km in24 hours, what was its average speed?
ANSWER
(C) 255 km/h
(A) 85 km/h
(D) 340 km/h
(B) 170 km/h
Model Exam 4 Paper 1
30.
Model Exam 4 Paper 1
31.
ANSWER
The pie-chart shows the preference in fruits of the students in a school. If 78 students prefer melon, then the total number of students in the school is
(C) 540(A) 312 (D) 624(B) 468
Model Exam 4 Paper 1
31. 45° represents 78 students
360° represent 78 students ×
= 78 students × 8
= 624 students
Hence, the total number of students in the school
is 624 students.
360
45
Model Exam 4 Paper 1
32. If the mode of the numbers 2, 3, 4, 5, 5, 6, 6, x, 7, 7, 8 is 6, then x =
ANSWER
(C) 7(A) 5 (D) 8(B) 6
Model Exam 4 Paper 1
32. The mode is the number that occurs the most,so x = 6.The number 6 occurs three times.
Model Exam 4 Paper 1
33. An urn contains 5 blue balls and 6 green balls. A ball is picked at random from the urn and it is found to be blue. It is not replaced. What is the probability that the next ball to be taken randomly from the urn will also be blue?
ANSWER
(C) (A) (D) (B) 4
11
2
5
3
5
6
11
Model Exam 4 Paper 1
33.
P (second ball is blue) 24
10 5
B G Total 5 6 11 4 6 10
Model Exam 4 Paper 1
34. The ages of five students are: 15, 14, q, 16, 12 (years)If the mean age is 15 years, then q is
ANSWER
(C) 17(A) 19 (D) 16(B) 18
Model Exam 4 Paper 1
34.
Model Exam 4 Paper 1
35. If the lower quartile of the distribution of the masses of a class of students is 28 kg and the upper quartile is 65 kg, then the semi-interquartile range is
ANSWER
(C) 55.5 kg
(A) 18.5 kg
(D) 74 kg
(B) 37 kg
Model Exam 4 Paper 1
35.
Model Exam 4 Paper 1
36. In a survey to determine the number of children per household, the following table was obtained.
ANSWER
If a house is visited at random, the probability that it contains exactly 2 children is
(C) (A) (D)(B)
Number of children
0 1 2 3 4 5
Frequency 8 7 6 4 2 3
2
15
1
5
1
3
2
3
Model Exam 4 Paper 1
36.6
(exactly 2 children)8 7 6 4 2 36
5
301
P
Model Exam 4 Paper 1
37. 4x − 3(x + 6) =
ANSWER
(C) 7x − 18
(A) x − 18
(D) −7x + 18
(B) x + 18
Model Exam 4 Paper 1
37.
1
4 3( 6 3 1
8
) 4 8x x x x
x
Using the distributivelaw.
Model Exam 4 Paper 1
38. 3(x + y) − 4(x − y) =
ANSWER
(C) −7x + 7y
(A) − x
(D) −7x − 7y
(B) − x + 7y
Model Exam 4 Paper 1
38. Use the distributive law twice.
3( ) 4( )
3 3 4 4
3 4 3
7
4
x y x y
x y x
x y
y
x x y y
Group like terms.Add like terms.
Model Exam 4 Paper 1
39. For all a and b,
ANSWER
(C)
(A)
(D)
(B)
( ) ( )a a b b a b 2( )ab a b 2 2a b
2 2a b 2 22a ab b
Model Exam 4 Paper 1
39. Factorise using (a − b) as a common factor.
This is the form of thedifference of two squares.2 2
2 2
2 2
( ) ( )
( )( )
or
( ) ( )
a a b b a b
a b a b
a b
a a b b a b
a ab ab b
a b
Use the distributive lawtwice.
Add like terms.
Model Exam 4 Paper 1
40. 5x × 5y =
ANSWER
(C) 25xy(A) 25x+y (D) 5xy(B) 5x+y
Model Exam 4 Paper 1
40. 5x × 5y = 5x+y The bases are the same, so weadd the powers.
Model Exam 4 Paper 1
41.
ANSWER
(C)
(A)
(D)
(B)
2 3
5 4
x x
y y
8 15
20
xy xy
xy
8 15
20
x y
xy
223
20
x
y
23
20
x
y
Model Exam 4 Paper 1
41. 2 3
5 4
2 (4) 3 (5) 20 204 5
20 5 4
8 15
20
23
20
x x
y y
x x y y
y y y
x x
y
x
y
The LCM of the denominators 5yand 4y is 20y.
Add the like terms in thenumerator.
Model Exam 4 Paper 1
42.
ANSWER
(C) −20(A) −10 (D) 10(B)
If 5 , then 30 2k
k ll
112
2
Model Exam 4 Paper 1
42. * 5
3030*2 5
230,
5
2
15
10
k
k
k
l
l
l
Model Exam 4 Paper 1
43.
(C) (A) (D) (B)
If 24 16 8(5 ), thenx x x 7
4
7
2
3
4
4
3
ANSWER
Model Exam 4 Paper 1
43. Use the distributive law.
Group like terms.
Add like terms.
Divide both sides by 32.
Reduce the fraction to lowest terms.
24 16 8(5 )
24 16 40 8
24 8 40 16
32 24
24
323
4
x x
x x
x x
x
x
Model Exam 4 Paper 1
So
Use the distributive law.
Divide both sides by 8.
Group like terms.
Add like terms.
Divide both sides by 4.
or
24 16 8(5 )
8(3 2) 8(5 )
3 2 5
3 5 2
3
4
4
3
x x
x x
x x
x x
x
x
Model Exam 4 Paper 1
So
44. If x is an integer which satisfies the inequalities5 < x − 3 < 7, then the value of x is
ANSWER
(C) 8(A) 5 (D) 9(B) 7
Model Exam 4 Paper 1
44. Add 3 throughout the
inequality.
Add the numbers.
The only possible integeris 9.
5 3 7
5 3 7 3
8
9
10
x
x
x
x
Model Exam 4 Paper 1
45. Three times the square of the sum of two numbers, x and y, is 21. Which equation below describes the given statement?
ANSWER
(C)
(A)
(D)
(B)23( ) 21x y 2(3 3 ) 21x y
2[3( )] 21x y 2( ) 3(21)x y
Model Exam 4 Paper 1
45. The sum of two numbers, x and y = x + y
The square of the sum of two
numbers, x and y = (x + y)2
Three times the square of the
sum of two numbers, x and y = 3(x + y)2
The equation is:
3(x + y)2 = 21
Model Exam 4 Paper 1
46.
ANSWER
The graph of the inequality shown is defined by
(C) −3 < x ≤ 4
(A) −3 < x < 4
(D) −3 ≤ x < 4
(B) −3 ≤ x ≤ 4
Model Exam 4 Paper 1
46.
All values from −3 to 4 are included.
Model Exam 4 Paper 1
47.
ANSWER
In the graph above, the shaded region is represented by
(C) {(x, y) : 2 ≤ y ≤ 4}
(A) {(x, y) : 2 ≤ x ≤ 4}
(D) {(x, y) : 2 < y < 4}
(B) {(x, y) : 2 < x < 4}
Model Exam 4 Paper 1
47.
{(x, y) : 2 ≤ y ≤ 4} represents the shaded region.
Model Exam 4 Paper 1
48. If the real value of x which
cannot be in the domain of x is
ANSWER
(C) (A) −4 (D) 3 (B)
4 1( ) ,
3 1
xh x
x
1
4 1
3
Model Exam 4 Paper 1
48.4 1
If ( ) , th
1
en3 1
3 1 0
1
3
3
xh x
xx
x
x
since division by 0 is undefined.
Model Exam 4 Paper 1
So
49.
ANSWER
(C)
(A) −19
(D)
(B)
5 4If ( ) , then ( 9)
3
xf x f
116
3
116
3
213
3
Model Exam 4 Paper 1
49. 5 4( )
35( 9) 4
( 9)3
45 4
349
31
163
xf x
f
Model Exam 4 Paper 1
50. Which of the following sets is represented by therelation g: x x2 − 1?
ANSWER
(C)
(A)
(D)
(B)
{(0, 1), (1, 0), (2, 3), (3, 8)}{(0, 1), (1, 0), (2, 5), (3, 10)}{(0, 1), (1, 0), (2, 4), (3, 9)}{(0, 1), (1, 0), (2, 3), (3, 8)}
Model Exam 4 Paper 1
50. 2
2
2
2
2
( ) 1
(0) 0 1 0 1 1 (0, 1)
(1) 1 1 1 1 0 (1, 0)
(2) 2 1 4 1 3 (2, 3)
(3) 3 1 9 1 8 (3, 8)
g x x
g
g
g
g
The required set = {(0, 1), (1,0), (2,3), (3,8)}
Model Exam 4 Paper 1
51.
ANSWER
The relation diagram shown represents a
(C) many–1relation
(A) 1–1 relation
(D) many-to-many relation
(B) 1–many relation
Model Exam 4 Paper 1
51.
The relation diagram represents a 1– many relation
Model Exam 4 Paper 1
52.
ANSWER
In the figure above, AB and CD are parallel. The relation between x and y is
(C) x + y < 180
(A) x + y = 180
(D) x > y
(B) x = y
Model Exam 4 Paper 1
52.
x + y = 180
Model Exam 4 Paper 1
53.
ANSWER
The triangle ABC is right-angled at C. Angle CAB = 25° and AC = 18 m. If BC represents the height of a tower, then its height, in m, is
(C)
(A) 18 cos 25°
(D) 18 tan 25°
(B) 18 sin 25°
18
cos25
Model Exam 4 Paper 1
53.
Model Exam 4 Paper 1
54.
ANSWER
In the right-angled triangle above, not drawn to scale, angle R = 90°, PQ = 100 cm, PR = 90 cm and QR = h cm. tan
(C) (A) (D) (B) 9
10
10
9
90
h
h
90
Model Exam 4 Paper 1
54.
Model Exam 4 Paper 1
55.
ANSWER
In the circle ABCD above, O is the centre. Angle CAD = 37° and angle CDE = 105°. Angle ABD =
(C) 71° (A) 37° (D) 105° (B) 68°
Model Exam 4 Paper 1
55.
Model Exam 4 Paper 1
So
And
56. Which of the following shapes does not have a line of symmetry?
ANSWER
(C)
(A)
(D)
(B)
Model Exam 4 Paper 1
56.
These shapes each have a line of symmetry
Model Exam 4 Paper 1
57.
Model Exam 4 Paper 1
In the diagram above, the image of B was obtained by a
ANSWER
(C) translation parallel to the y-axis
(A) reflection in the x-axis
(D) translation parallel to the x-axis
(B) reflection in the y-axis
Model Exam 4 Paper 1
57.
The image of B is:
• congruent to its object• laterally inverted• the same distance away from the y-axis.
Hence, the image of B was obtained by a reflectionin the y-axis.
Model Exam 4 Paper 1
58. The image of the point P(5, −8) under the
translation is
ANSWER
(C) (−3, 4)
(A) (7, −12)
(D) (3, −4)
(B) (−7, 12)
2
4
Model Exam 4 Paper 1
58.
Model Exam 4 Paper 1
59. P1 (−7, 10) is the image of P(x, y) after a reflection in the line y = x. P (x, y) is
ANSWER
(C) P(−10, −7)
(A) P(−10, 7)
(D) P(10, 7)
(B) P(10, −7)
Model Exam 4 Paper 1
59. 1
1
1
: ( , ) ( , )
: ( , ) ( , )
: ( 7, 1
Reflection
Inverse reflection
(10, 7)0)
y x
y x
y x
M P x y P y x
M P y x P x y
M P P
- -
Model Exam 4 Paper 1
60.
ANSWER
In the diagram above, not drawn to scale, angle CAB = 29° and AOC is a diameter of the circle with centre O. Angle ACB =
(C) 45° (A) 61° (D) 29° (B) 58°
Model Exam 4 Paper 1
60.
in a semi-circle.ˆ 90
ˆ ˆSo 9
6
90 9
1
0
2
ABC
ACB CAB
Model Exam 4 Paper 1
CSEC MODEL EXAMINATION 4
MATHEMATICS
Paper 2
2 hours 40 minutes
SECTION I
Answer ALL the questions in this section
All working must be clearly shown
NEXT
1. (a) Using a calculator, or otherwise, determine the exact value of (4.7)2 − (7.65 ÷ 1.7). (3 marks)
ANSWER
Model Exam 4 Paper 2
1. (a) 2(4.7) (7.65 1.7)
22.09
17.59 (exa
4
ct value
.5
)
Model Exam 4 Paper 2
1. (b) A total of 1 600 students attend Central High School. The ratio of teachers to students
is 1:40
(2 marks)
ANSWER
(i) How many teachers are there at the school?
Three-eights of the students own a laptop
computer.
Model Exam 4 Paper 2
1. (b) (i) The number of teachers at the school
Model Exam 4 Paper 2
(2 marks)
ANSWER
1. (b) A total of 1 600 students attend Central High School. The ratio of teachers to students
is 1:40
Model Exam 4 Paper 2
(ii) How many students do not own a laptop
computer?
Thirty-six percent of the students who own a laptop computer also own a video game.
1. (b) (ii)
Model Exam 4 Paper 2
The number of students who do not own
a laptop computer
Hence, 1 000 students do not own a laptop computer.
Or
Model Exam 4 Paper 2
1. (b) (ii) The fraction of students who do not
own a laptop computer
The number of students who do not own a laptop computer = 1 600 − 600
= 1 000
Model Exam 4 Paper 2
(4 marks)Total 11 marks
ANSWER
1. (b) A total of 1 600 students attend Central High School. The ratio of teachers to students
is 1:40
Model Exam 4 Paper 2
(iii) What fraction of the students in theschool own a video game?
Express your answer in its lowest terms.
1. (b) (iii) The number of students who own a
video game = 36% of 600
Hence, 216 students own a video game.
Model Exam 4 Paper 2
2. (a) Given that evaluate
(4 marks)
ANSWER
(i) 2 * 8
Model Exam 4 Paper 2
(ii) 4 * (2 * 8)
2. (a) (i)
Model Exam 4 Paper 2
(ii)
Model Exam 4 Paper 2
2. (b) Simplify, expressing your answer in its
simplest form(2 marks)
ANSWER
Model Exam 4 Paper 2
2. (b)
Model Exam 4 Paper 2
2. (c) A theatre has two sections, balcony (B) andhouse (H)
Tickets for section B cost $b each.Tickets for section H cost $h each.
Paula paid $515 for 4 section B tickets and3 section H tickets.
Zuri paid $520 for 5 section B tickets and1 section H ticket.
(5 marks)Total 11 marks
Model Exam 4 Paper 2
(i) Write two equations in b and h to represent the information above.
ANSWER(ii) Calculate the values of b and h.
Model Exam 4 Paper 2
2. (c) (i) The equation for the tickets bought byPaula:4b + 3h = 515 (in dollars)
The equation for the tickets bought byZuri:5b + h = 520 (in dollars)
(ii)
Hence, b = 95 and h = 45.
Model Exam 4 Paper 2
Model Exam 4 Paper 2
3. (a) The Venn diagram below represents information on the type of games played by members of a youth club. All members of the youth club play at least one game.
(5 marks)
F represents the set of members who play football.
C represents the set of members who play cricket.
S represents the set of members who play squash.
Ada, Obi and Roy are three members of the youth club.
(i) State what game(s) is/are played bya) Adab) Obic) Roy
ANSWER(ii) Describe in words the members of the setF′ S.
Model Exam 4 Paper 2
3. (a) (i) a) Ada played all three games. or
Ada played football, cricket and squash. b) Obi played cricket and squash. c) Roy played football.
or Roy played only football.
(ii) F′ S represents the set of members who play squash (and cricket) but
not football.
or
S C means that all members who play squash also play cricket.
So F′ S represents the set of members who play squash and cricket but not football.
3. (b) (i) Use only a pencil, a ruler and a pair of
compasses to perform the followingconstruction.a) Construct a triangle PQR in
which QR = 9.5 cm, QP = 8 cm and PQR = 60°.
(7 marks)Total 12 marks
b) Construct a line PT such that PT is perpendicular to QR and meets QR at T.
Model Exam 4 Paper 2
(ii) a) Measure and state the size of angle PRQ.
b) Measure and state the length of PT.
ANSWER
3. (b) (i)
Model Exam 4 Paper 2
a) Draw a horizontal line greater than 9.5 cm. Mark off a point Q. Set your compasses to a separation of 9.5 cm. With Q as centre mark off the point R to the right of the horizontal line. QR = 9.5 cm.
With Q as centre and a suitable separation, drawn an arc to the right of Q, to intersect the horizontal line at A. With A as centre and the same separation, drawn an arc to intersect the previous arc at B. Draw a straight line passing through the points Q and B. Angle PQR = 60°.
With a compasses separation of 8 cm and centre Q, mark off point P on QB produced. QP = 8 cm.
Draw a straight line from P to R to complete the triangle PQR.
Model Exam 4 Paper 2
b) With P as centre and a suitable compasses separation, draw an arc to intersect QR at C and D. With C and D as centres, construct two arcs to intersect at E. Draw a straight line through the points P and E to intersect QR at T. PT is perpendicular to QR.
(ii) a) By measurement, the size of angle PRQ = 52°
b) By measurement, the length of PT = 7 cm.
Model Exam 4 Paper 2
4. (a) The diagram below shows a map of a recreation park drawn on a grid of 1 cm squares. The scale of the map is 1 : 5 000.
(6 marks)
Model Exam 4 Paper 2
Use the map of the recreation park, find
(i) the distance, to the nearest metre, from St. Maria to St. Albert on the ground.
(ii) the distance, to the nearest metre, from St. Albert to St. Raymond on the ground.
(iii) the area on the ground represented by 1 cm2 on the map.
(iv) the approximate area of the recreation park on the ground, stating the answer in square metres.
ANSWER
Model Exam 4 Paper 2
4. (a) The diagram below shows a map of a recreation park drawn on a grid of 1
cm squares. The scale of the map is 1 : 5 000.
(6 marks)
4. (a)
Model Exam 4 Paper 2
The scale of 1 : 5 000 means1 cm represents 5 000 cm
So 1 cm represents 50 m
(i) The distance from St. Maria to St. Albert on the map = 5 cm
The distance from St. Maria to St. Albert on the ground = 5 × 50 m = 250 m
Model Exam 4 Paper 2
(ii) The distance from St. Albert to St. Raymond on the map = 8.6 cm
The distance from St. Albert to St. Raymond on the ground = 8.6 × 50 m
= 430 m
Model Exam 4 Paper 2
(iii) The area on the ground represented by 1 cm2 on the map = (50 m)2 = 2 500 m2
Model Exam 4 Paper 2
(iv) The approximate area of the recreation park on the map = (61 + 14) cm2
= 75 cm2
The approximate area of the recreation park on he ground = 75 × 2 500 m2 = 187 500 m2
Model Exam 4 Paper 2
4. (b) The diagram below, not drawn to scale, shows a prism of volume 450 cm3. The cross-section ABCD is a
square. The length of the prism is 18 cm.
(5 marks)Total 11 marks
Model Exam 4 Paper 2
Calculate
(i) the length of the edge AB, in cm
(ii) the total surface area of the prism, in cm2.ANSWER
4. (b) (i)
Model Exam 4 Paper 2
The volume of the prism:
Hence, the length of the edge AB is 5 cm.
(ii) The area of one rectangular surface A = lb
= 18 cm × 5 cm = 90 cm2
The total surface area of the prism, TSA = 2 × 25 cm2 + 4 × 90 cm2
= 50 cm2 × 360 cm2 = 410 cm2
Model Exam 4 Paper 2
5. Two variables x and y are related such that‘y varies inversely as the square root of x’.
(2 marks)
ANSWER
(a) Write an equation in x, y and k to describe the inverse variation, where k is the
constant of variation.
Model Exam 4 Paper 2
5. (a)
Model Exam 4 Paper 2
5. (b)
(6 marks) ANSWER
Using the information in the table of values above, calculate the value of
x 1 f 9
y r 12
3
(i) k, the constant of variation
(ii) r
(iii) r
Model Exam 4 Paper 2
5. (b) (i)
Hence, the value of k is 2.
Model Exam 4 Paper 2
(ii)
Hence, the value of r is 2.
Model Exam 4 Paper 2
(iii)
Hence, the value of f is 4.
Model Exam 4 Paper 2
5. (c) Determine the equation of the line which is parallel to the line y = 4x − 1 and passes
through the point (5, 2).
ANSWER
(4 marks)Total 12 marks
Model Exam 4 Paper 2
5. (c) y = 4x − 1 ⇒ m = 4
Using m = 4 and the point (5, 2), then
Hence, the equation of the line is y = 4x − 18.
Model Exam 4 Paper 2
(i) State
a) the scale factor for the enlargement.
b) the coordinates of the centre of enlargement.
L″M″N″ is the image of LMN under a
reflection in the line y = x.
(ii) Draw and label the triangle L″M″N″ on graph paper.
ANSWER
Model Exam 4 Paper 2
6. (a) L′M′N′ is the image of LMN under an enlargement.
(5 marks)
(i) a)
The image and the object are on the same side of the centre of enlargement.
the scale factor, k = 2.
Hence, the scale factor for the enlargement is 2.
b) The origin O(0, 0) is the centre of enlargement.
(ii) Triangle L″M″N″ was drawn on graphpaper as shown above.
Model Exam 4 Paper 2
6. (a)
6. (b)
ANSWER
Three parishes, P, Q, and R are such that the bearing of P from Q is O60°. R is 20 km due east of Q, and PQ = 8 km.
(i) Calculate, correct to one decimal place,
the distance PR.(ii) Given that ∠QPR = 125°, state the bearing of R from P.
Model Exam 4 Paper 2
(6 marks)Total 11 marks
6. (b) (i)
Model Exam 4 Paper 2
Considering ΔPQR and using the cosine rule:
Hence, the distance PR is 13.7 km.
6. (b) (ii)
Model Exam 4 Paper 2
Hence, the bearing of R from P is 115°.
7. The raw data for the number of minutes per day a class of 40 high school students spent watching TV is given below.
138 146 168 146 161 164 158 126173 145 150 140 138 142 135 132147 176 147 142 144 136 163 135150 125 148 119 153 156 149 152154 140 145 157 144 165 135 128
Model Exam 4 Paper 2
7. (a) Copy and complete the frequency table to represent this data.
ANSWER
Viewing Time in minutes Frequency118–126 3
127–135 5
136–144 9
145–153
154–162
163–171
172–180 (2 marks)
Model Exam 4 Paper 2
7. (a)Viewing Time in
minutes Frequency118–126 3
127–135 5
136–144 9
145–153 12
154–162 5
163–171 4
172–180 2
Model Exam 4 Paper 2
Model Exam 4 Paper 2
The frequency table way copied and completed as shown above.
Model Exam 4 Paper 2
7. (b) Using the raw scores, determine the range for the data. (2 marks)
ANSWER
Model Exam 4 Paper 2
7. (b) The range for the data = The largest observation
− The smallestobservation
= (176 − 119)minutes
= 57 minutes
Model Exam 4 Paper 2
7. (c) Using a scale of 1 cm to represent 5 minutes on the horizontal axis and a scale of 1cm to represent 1 student on the vertical axis, draw a frequency polygon to represent the data.
(6 marks)
ANSWER
Model Exam 4 Paper 2
7. (c) Class mid-point (minutes) Frequency
122 3
131 5
140 9
149 12
158 5
167 4
176 2
The frequency polygon was drawn using the table with class mid-points above and the given scales.
Model Exam 4 Paper 2
Model Exam 4 Paper 2
7. (d) What is the probability that a student chosen from this class watched TV for less than 145 minutes?
ANSWER
(2 marks)Total 12 marks
Model Exam 4 Paper 2
7. (b) The number of studentswho watched TV for = 3 + 5 + 9less than 145 minutes = 17
The total number of students = 40
P (student watched TV < 145 minutes)
Model Exam 4 Paper 2
8. Rectangle WXYZ below represents one whole plane figure which has been divided into seven smaller parts. These parts are labelled A, B, C, D, E, F and G.
Model Exam 4 Paper 2
8. (a) Copy and complete the following table, stating what fraction of the rectangle each part
represents.
ANSWER(5 marks)
Model Exam 4 Paper 2
8. (a)
The table was copied and completed as shown above.
Model Exam 4 Paper 2
8. (b) Write the parts in order of the size of their perimeters.
ANSWER
(2 marks)
Model Exam 4 Paper 2
8. (b)
Model Exam 4 Paper 2
Model Exam 4 Paper 2
The perimeter of F = (2 + 3 + 4+ 3.6) units = 12.6 units
The perimeter of G = 2 (3 + 2) units= 2 (5) units= 10 units
The parts in order of the size of their perimeters: G, C, B, F, D, E, A.
Model Exam 4 Paper 2
8. (c) The area of G is 6 square units. C, G and E are rearranged to form a trapezium.
ANSWER
(i) What is the area of the trapezium in
square units?(ii) Sketch the trapezium clearly showing the
outline of each of the three parts.(3 marks)
Total 10 marks
Model Exam 4 Paper 2
8. (c) (i) The area of the trapezium = The area of (C + G + E)
= (4 + 6 + 9) square units
= 19 square units
or
Model Exam 4 Paper 2
The area of the trapezium,
Model Exam 4 Paper 2
(ii)
A sketch of the trapezium clearly showing the outline of each of the three parts is shown above.
Model Exam 4 Paper 2
SECTION II
Answer TWO questions in this section
Model Exam 4 Paper 2
9. (a) Given that and f (x) = x + 5
ANSWER
(i) calculate the value of g(−3)(ii) write an expression for g f (x) in
its simplest form
(iii) find the inverse function g−1(x).(7 marks)
Model Exam 4 Paper 2
9. (a) (i)
Model Exam 4 Paper 2
(ii)
Model Exam 4 Paper 2
(iii)
Model Exam 4 Paper 2
9. (b) The length of the rectangle below is (2x + 1) cm and its width is (x − 3) cm.
ANSWER
(i) Write an expression in the form ax2 + bx + c for the area of the rectangle.
(ii) Given that the area of the rectangle is 225 cm2, determine the value of x.
(iii) Hence, state the dimensions of the rectangle, in centimetres.
(8 marks)Total 15 marks
Model Exam 4 Paper 2
9. (b) (i)
Hence, an expression in the form ax2 + bx + c for the area of the rectangle is 2x2 − 5x − 3.
Model Exam 4 Paper 2
(ii) The area of the rectangle, A = (2x2 − 5x − 3) cm2 and the area of the rectangle = 225 cm2
We can form the equation of equal areas:
Model Exam 4 Paper 2
Hence, the value of x is 12.
Model Exam 4 Paper 2
(iii)
Hence the dimensions of the rectangle are 25 cm and 9 cm.
Model Exam 4 Paper 2
10. (a) Given that
ANSWER
(i) express in fractional or surd form
a) cos θ
b) tan θ
(ii) Hence, determine the exact value of
(6 marks)
Model Exam 4 Paper 2
10. (a) Given that
Model Exam 4 Paper 2
Considering a right-angled Δ and using Pythagoras’ theorem:
Model Exam 4 Paper 2
(ii)
Calculate the size of each of the followingangles, giving reasons for your answer
ANSWER
(i) angle TPQ
(ii) angle MTQ
(iii) angle TQS
(iv) angle SRQ
Model Exam 4 Paper 2
10. (b)
In the diagram above, not drawn to scale, LM is a tangent to the circle at the point, T. O is the centre of the circle and angle MTS = 25°.
(9 marks)Total 15 marks
10. (b)
Model Exam 4 Paper 2
Model Exam 4 Paper 2
Model Exam 4 Paper 2
(a) Sketch the diagram above. Show the approximate positions of P and Q such that
ANSWER
P is the mid-point of OL
Q in a point on OM such that (2 marks)1
.4
OQ OM����������������������������
Model Exam 4 Paper 2
OL and OM are position vectors such thatand .
OL l��������������
OM m��������������
11.
11. (a)
A sketch of the diagram is shown above.
.1
and 4
OP PL OQ OM ��������������������������������������������������������
Model Exam 4 Paper 2
11. (b) Write down, in terms of l and m the vectors
ANSWER
(i)
(ii)
(iii)
(iv) (8 marks)
ML��������������
PM��������������
LQ��������������
PQ��������������
Model Exam 4 Paper 2
11. (b)
Model Exam 4 Paper 2
Model Exam 4 Paper 2
Model Exam 4 Paper 2
11. (c) R is the mid-point of PM. Using a vector method, prove that PQ is not parallel to
LR.
ANSWER
(5 marks)Total 15 marks
Model Exam 4 Paper 2
11. (c)
Model Exam 4 Paper 2
Model Exam 4 Paper 2
Model Exam 4 Paper 2
is not a multiple of , they have different directions, so PQ is not parallel to LR.PQ��������������
LR��������������
Model Exam 4 Paper 2
CSEC MODEL EXAMINATION 5MATHEMATICS
Paper 190 minutes
Answer ALL the questions
NEXT
1. The decimal fraction 0.35 written as a common fraction is
ANSWER
(A)
(B)
(C)
(D)7
2
1
47
2013
20
Model Exam 5 Paper 1
1.100
0.35 0.35100
35
10035 5
1007
5
20
Multiply by which is 1, to write
the decimal as a common fraction.
Divide both the numerator and the
denominator by the common factor 5.
Common fraction reduced to its
lowest terms.
100
100
Model Exam 5 Paper 1
2. is the same as
ANSWER
(A)
(B)
(C)
(D)
23
4
6
8
8
99
162
4
3
Model Exam 5 Paper 1
2.2
2
2
2
3 1
4 34
41
3
4
3
A negative power is the reciprocal of the positive power.
Inverting the fraction which is the divisor and multiplying instead.
Model Exam 5 Paper 1
3. The number 3.142 857 143... written correct to 3 significant figures is
ANSWER
(A) 3.14
(B) 3.15
(C) 3.142
(D) 3.143
Model Exam 5 Paper 1
3. 3.14 2 857 143= 3.14 correct to 3 sf
The 4th significant figure which is the digit 2 is less than the digit 5, so we do not add 1 to the 3rd significant figure.
Model Exam 5 Paper 1
4. If 60% of a number is 312, what is the number?
ANSWER
(A) 208
(B) 520
(C) 832
(D) 1 280
Model Exam 5 Paper 1
4.
Model Exam 5 Paper 1
5. If $940 is shared in the ratio 3:5:12, then the difference between the largest and the smallest shares is
ANSWER
(A) $141
(B) $235
(C) $423
(D) $564
Model Exam 5 Paper 1
The total number of equal parts = 3 + 5 + 12 = 20
The difference between the largest andsmallest shares in terms of equal parts = 12 3 = 9
So 20 equal parts = $940
9 equal parts =9
$94020
= $47 9
= $423Hence, the difference between the largest and smallest shares is $423.
5.
Model Exam 5 Paper 1
6. Written in scientific notation 0.0057 104 is
ANSWER
(A) 5.7 103
(B) 5.7 101
(C) 5.7 103
(D) 5.7 107
Model Exam 5 Paper 1
The first number must be, between 1 and 10.A negative power is the reciprocal of a positive power.The product of numbers with the same base is simplifiedby adding the powers.
6.
Model Exam 5 Paper 1
7. What is the face value of the digit 5 in the number 45 368?
ANSWER
(A) 5
(B) 50
(C) 500
(D) 5 000
Model Exam 5 Paper 1
7. Place value 104 103 102 101 100
Face value 4 5 3 6 8
The face value is 5.
Model Exam 5 Paper 1
8. The next term in the sequence 1, 3, 7, 13, 21 . . . is
ANSWER
(A) 27
(B) 29
(C) 31
(D) 33
Model Exam 5 Paper 1
8. 1, 1 + 2, 3 + 4, 7 + 6, 13 + 8, 21 + 10,The next number in the sequence is 31.
Model Exam 5 Paper 1
9. What is the value of the digit 4 in the number 57.843?
ANSWER
(A)
(B)
(C) 4
(D) 400
4
100
4
10
Model Exam 5 Paper 1
9. Place value 101 100 101 102 103
Face value 5 7 8 4 3
The value ofthe digit 4 = Face value Place value
2
2
4 10
14
10
4
100
14
100
A negative power isthe reciprocal of thepositive power
Model Exam 5 Paper 1
10. What is the least number of carambolas that can be shared equally among 8, 12 or 16 children?
ANSWER
(A) 24
(B) 36
(C) 48
(D) 64
Model Exam 5 Paper 1
10. 2 8,12,16
2 4, 6, 8
2 2, 3, 4
2 1, 3, 2
3 1, 3, 1
1, 1, 1
The LCM of 8,12 and 16 = 2 2 2 2 3= 48
The least number of carambolas is 48.
Model Exam 5 Paper 1
11. The simple interest on $600 at 5% per annum for 3 years is given by
ANSWER
(A)
(B)
(C)
(D)
600 5$
3 100
600 3$
100 5
600 100$
3 5
600 5 3$
100
Model Exam 5 Paper 1
11. The simple interest,
600 5 3$
100
100
PRTI
FormulaP = $600, R = 5% and T = 3 years
Model Exam 5 Paper 1
12. A salesman is paid 4% of his sales as commission. His sales for last month were $8 700. How much commission was he paid?
ANSWER
(A) $34.80
(B) $174.00
(C) $348.00
(D) $435.00
Model Exam 5 Paper 1
The commission paid 4% of $8 700
4$8 700
100$348
12.
Model Exam 5 Paper 1
13. A man bought a goat for $700 and sold it for $525. What was his loss as a percentage of the cost price?
ANSWER
(A) 14.3%
(B) 20%
(C) 25%
(D) 33.3%
Model Exam 5 Paper 1
The loss = $(700 525) = $175
The cost price = $700The loss as a percentage of the cost price =
= 25%
$175100%
$700
13.
Model Exam 5 Paper 1
14. A sales tax of 8% is charged on an article marked at $140.00. How much does a customer pay for the article?
ANSWER
(A) $252.00
(B) $151.20
(C) $128.80
(D) $11.20
Model Exam 5 Paper 1
14. The sales tax = 8% of $140
= $11.20
The amount the customer pays = $(140 + 11.20)
= $151.20
8$140
100
$112
10
Model Exam 5 Paper 1
15. If the simple interest on $900 for 3 years is $108, what is the rate of interest per annum?
ANSWER
(A) 0.25%
(B) 2.8%
(C) 3.24%
(D) 4%
Model Exam 5 Paper 1
FormulaP = $900T = 3 years I = $108
15.
Model Exam 5 Paper 1
16. A lot of land is valued at $50 000. Land tax is charged at the rate of $0.60 per $100 value. What is the amount payable for land tax?
ANSWER
(A) $200
(B) $300
(C) $400
(D) $500
Model Exam 5 Paper 1
The number of $ 100 value =
The amount payable for land tax = $0.60 500
= $60 5
= $300
$50 000500
$10016.
Model Exam 5 Paper 1
17. A customer buys a computer on hire purchase. He makes a deposit of $350 and pays 30-monthly instalments of $115.50 each. The hire purchase price of the computer is
ANSWER
(A) $495.50
(B) $3 465
(C) $3 495
(D) $3 815
Model Exam 5 Paper 1
The deposit = $350
The amount of themonthly instalments = $115.50 30
= $3 465.00
The hire-purchase price = $(350 3 465.00)
= $3 815
17.
Model Exam 5 Paper 1
18. The exchange rate for one United States dollar (US $1.00) is two dollars and seventy cents in Eastern Caribbean currency (EC $2.70). What is the value of US $2 000 in EC currency?
ANSWER
(A) $1 993.70
(B) $2 002.70
(C) $2 700
(D) $5 400
Model Exam 5 Paper 1
US $1.00 = EC $2.70
US $2 000 = EC $2.70 2 000
= EC $270 20
= EC $5 400
18.
Model Exam 5 Paper 1
19.
ANSWER
(A) (P Q)
(B) P Q
(C) Q
(D) Q P
In the Venn diagram, the shaded region represents
Model Exam 5 Paper 1
19.
The unshaded region = Q
∴ the shaded region = Q
Model Exam 5 Paper 1
20. Which of the following sets is equivalent to {2, 3, 5, 7}?
ANSWER
(A) {p, q}
(B) {a, b, c}
(C) {k, l, m, n}
(D) {p, q, r, s, t}
Model Exam 5 Paper 1
20.
1 1 correspondence
The two sets are equivalent, since they have the same number of elements there exists a one-to-one correspondence.
Model Exam 5 Paper 1
21.
ANSWER
(A) 5
(B) 12
(C) 23
(D) 35
In the Venn diagram, if n(X) = 28, n(Y) = 17 and n(X Y) = 40, then n(X Y) =
Model Exam 5 Paper 1
21.
n(X Y) = n(X) + n(Y) n(X Y) Formula
40 = 28 + 17 n(X Y)
40 = 45 n(X Y)
n(X Y) = 45 40
= 5
Model Exam 5 Paper 1
22. If P = {2, 3, 5, 7, 11, 13}, Q = {3, 7, 13} and R = {2, 3, 7}, then P Q R =
ANSWER
(A) { }
(B) {3, 7 }
(C) {2, 5, 11}
(D) {5, 11, 13}
Model Exam 5 Paper 1
22. P Q = {2, 3, 5, 7, 11, 13} {3, 7, 13}
= {3, 7, 13} Common elements
(P Q) R = {3, 7, 13} {2, 3, 7} = {3, 7}
= P Q R Common elements
Model Exam 5 Paper 1
23. The circumference of a circular table-top is 110 cm.
Given that , then the radius of the circular
table-top, in centimeters, is
ANSWER
(A)
(B) 35
(C) 70
(D) 172
22π
7
117
2
Model Exam 5 Paper 1
23.
Model Exam 5 Paper 1
24. A man leaves piarco Airport at 21:45 h and reaches Miami International Airport at 03:15 h the next day. How many hours did the journey take?
ANSWER
(A) 5
(B)
(C)
(D) 25
15
21
182
Model Exam 5 Paper 1
The number of hoursto midnight = (24:00 21:45) h
= 2 h 15 min
The number of hoursfor the journey = 2 h 15 min + 3 h 15 min
= 5 h 30 min
= 1
5 h2
24.
Model Exam 5 Paper 1
25. A woman takes 30 minutes to drive to Cheddi Jagan International Airport which is 26 km away from her home. Her speed in km per hour is
ANSWER
(A)
(B)
(C)
(D)
26 60
1 30
30 1
26 60
30 60
26 1
26 30
60 1
Model Exam 5 Paper 1
25.
Substitute the value for d and for t.
Model Exam 5 Paper 1
26.
ANSWER
(A) 42 cm
(B) 44 cm
(C) 46 cm
(D) 48 cm
The perimeter of the shape is
Model Exam 5 Paper 1
26.
The perimeter, P = (14 + 2 + 5 + 7 + 4 + 7 + 5 + 2) cm
= 46 cm
Model Exam 5 Paper 1
27.
ANSWER
(A) (B)
(C) (D)
The figure above shows a sector of a circle centre O. The radius of the circle is 7 cm and the sector angle is 60°. The perimeter of the sector OPQ is
22Take π
7
228 cm
3
121 cm
3
122 cm
32
27 cm3
Model Exam 5 Paper 1
27. The length of the arc PQ,
θ2π
36022 60
2 7 cm7 360
12 22 cm
622
cm
7 c3
31
m
l r
FormulaSubstitute each valuefor , r and .
Model Exam 5 Paper 1
The perimeter of the sector OPQ
121 cm
3
2
17 2 7 cm
3
17 14 cm
3
l r
Substitute the valuefor l and for r.
Model Exam 5 Paper 1
28. Which of the following drawings represents a uniform solid?
ANSWER
(A) (B)
(C) (D)
Model Exam 5 Paper 1
28. The wedge is a uniform solid, since its ends or cross-section are identical.
Model Exam 5 Paper 1
29.
ANSWER
(A) 36 cm2
(B)
(C) 45 cm2
(D) 48 cm2
The area of the parallelogram PQRS is
2140 cm
2
Model Exam 5 Paper 1
29.
The area of parallelogramPQRS, A = bh
= 9 4 cm2
= 36 cm2
Formula
Substitute the valuefor b and for h.
Model Exam 5 Paper 1
30. If a square has the same area as a rectangle with sides 20 cm and 5 cm, then the length of a side of the square is
ANSWER
(A) 17.5 cm
(B) 15 cm
(C) 12.5 cm
(D) 10 cm
Model Exam 5 Paper 1
30.
The area of the rectangle,
A = lb
= 20 5 cm2
= 100 cm2
Substitute thevalue for l and for b.
Model Exam 5 Paper 1
Model Exam 5 Paper 1
31. Each of the letters of the word ‘STATISTICS’ is written on a slip of paper. One slip is randomly drawn. What is the probability of drawing a letter ‘S’?
ANSWER
(A)
(B)
(C)
(D)3
5
1
103
101
5
Model Exam 5 Paper 1
31. The number of letter ‘S’ = 3
The total number of letters = 10
P(letter is an S) = 3
10
Model Exam 5 Paper 1
32. A man throws a die twice. What is the probability that he will throw a ‘6’ followed by an odd number?
ANSWER
(A)
(B)
(C)
(D)
1
10
7
12
1
125
12
Model Exam 5 Paper 1
32. The two eventsare independent.
Model Exam 5 Paper 1
33.
ANSWER
(A) 55 and 59
(B) 54 and 60
(C) 54.5 and 59.5
(D) 54.5 and 59
The masses of 30 children were measured, to the nearest kg. The information is shown in the grouped frequency table above. The class boundaries of the class interval 55–59 are
Mass (kg) 40–44 45–49 50–54 55–59Frequency 5 8 13 4
Model Exam 5 Paper 1
33. 50 54 55 59 60 64
54.5 59.5
54 55The lower class boundary, LCB
254.5
59 60The upper class boundary, UCB
259.5
+=
=
+=
=
Hence, the class boundaries are 54.5 and 59.5.
Model Exam 5 Paper 1
34. The mean of a number of plums is 7. If x = 63, how many plums were used in the calculation of the mean?
ANSWER
(A) 3
(B) 6
(C) 9
(D) 12
Model Exam 5 Paper 1
34.
Formula
Substitute the value for and for x.Multiplying both sides by n.
Dividing both sides by 7.
x
The mean, 7
and 63
Now
63So 7
i.e. 7 63
9
63
7
x
x
xx
n
nn
n
Model Exam 5 Paper 1
Hence, 9 plums were used in the calculation of the mean.
Items 35–36 refer to the scores in the following box.
12 13 5 88 10 7 4
Model Exam 5 Paper 1
35. The mode of the scores presented in the box is
ANSWER
(A) 7
(B) 8
(C) 10
(D) 13
Model Exam 5 Paper 1
The score 8 occurs two times.
The mode = 8.
35.
Model Exam 5 Paper 1
36. The median of the scores presented in the box is
ANSWER
(A) 5
(B) 7
(C) 8
(D) 12
Model Exam 5 Paper 1
36. The scores in ascending order:
4, 5, 7, 8, 8, 10,12,13
Middle values
2
8 8The median scor ,
28
e Q
+=
Model Exam 5 Paper 1
37. The expression ‘y is equal to the square root of x’ can be written as
ANSWER
(A) y = 2x
(B)
(C) y = x2
(D) y2 = x
y x
Model Exam 5 Paper 1
37. The square root of
The equation is:
x x
y x
Model Exam 5 Paper 1
38. The expression 5(x 3) =
ANSWER
(A) 5x + 15
(B) 5x 15
(C) 5x + 15
(D) 5x 15
Model Exam 5 Paper 1
38. 5(x 3) = 5 x 5 (3)
= 5x + 15
Negative times positive is negative. Negative times negative is positive.
Model Exam 5 Paper 1
39. The expression (2a)4 =
ANSWER
(A) 16a4
(B) 2a4
(C) 16a
(D) 8a4
Model Exam 5 Paper 1
39. 4
4
4
(2 ) 2 2 2 2
2 2 2 2
1
6
6
1
a a a a a
a a a
a
a
a
Expanding using themeaning of the 4th power.Grouping like terms.Multiplying like terms.Simplifying.
Model Exam 5 Paper 1
40. If then 6 o (2) =
ANSWER
(A)
(B) 9
(C) 3
(D)
2
ο ,p
p qp q
9
2
1
2
Model Exam 5 Paper 1
40.2
2
Substitue
ο
66 ο ( 2)
6 ( 2)
36
6 23
6 and
9
2
6
8
2
pp
p q
qp q
Model Exam 5 Paper 1
41. If F = mv2; when m = 80 and v = 12, then F =
ANSWER
(A) 480
(B) 960
(C) 2 850
(D) 5 760
1
2
Model Exam 5 Paper 1
41. 2
2 Substitute the valuefor a
1
21
(80) (12)2
nd
40 (14 )
.
5 0
4
76
m v
F mv
Model Exam 5 Paper 1
42. If 9x 40 = 10 x, then x =
ANSWER
(A) 5
(B) 5
(C)
(D)
33
4
33
4
Model Exam 5 Paper 1
42.
Model Exam 5 Paper 1
43. Robert buys $x worth of gas each month. In December, he bought $8 less than twice the regular worth. The worth of gas he bought in December is
ANSWER
(A) $(8 2x)
(B) $(2x 8)
(C) $8x
(D) $16x
Model Exam 5 Paper 1
43. Twice the regular worth of gas = $x 2= $2x
$8 less than twice the regular worth of gas = $(2x 8)
Model Exam 5 Paper 1
44. If V = 4R2, then R is
ANSWER
(A)
(B)
(C)
(D)4π
V
1
2 π
V
4π
VR
21
2 π
V
Model Exam 5 Paper 1
44.
Model Exam 5 Paper 1
So
45. Using the distributive law, (p r) (q r) is
ANSWER
(A) p(q r)
(B) pq r2
(C) (p q)r
(D) pqr2
Model Exam 5 Paper 1
45. (p r) (q r)
= (p q)r
r is a common factor.Factorising using thedistributive law.
Model Exam 5 Paper 1
46. Which of the following equations represents a straight line?
ANSWER
(A) y = x2 1
(B) y = 5 2x
(C)
(D) y = 2 + 3x 5x2
2y
x
Model Exam 5 Paper 1
46. y = 5 2x
So y = 2x + 5
It is in the form y = mx + c, which is the equation of a straight line.
Model Exam 5 Paper 1
47. If g(x) = x2 x + 3, the g(4) =
ANSWER
(A) 17
(B) 9
(C) 15
(D) 23
Model Exam 5 Paper 1
2
2
( ) 3
( 4) ( 4) ( 4) 3
16 4 3
23
g x x x
g
47. Substitute –4 for x.The square of a numberis always positive.Negative timesnegative is positive.
Model Exam 5 Paper 1
48.
ANSWER
(A) one-to-one
(B) one-to-many
(C) many-to-one
(D) many-to-many
The relationship that best describes the arrow diagram is
Model Exam 5 Paper 1
48.
The relationship that best describes the arrowdiagram is one-to-many.
Model Exam 5 Paper 1
ANSWER
49. Which of the following graphs is that of a function?
Model Exam 5 Paper 1
49. Using the vertical line test for a function, graph (A) is the graph of a function.
Model Exam 5 Paper 1
50. The range of f: x x3 for the domain {3, 2, 1, 0, 1, 2}
ANSWER
(A) {3, 2, 1, 0, 1, 2}
(B) {9, 6, 3, 0, 3, 6}
(C) {0, 1, 8, 27}
(D) {27, 8, 1, 0, 1, 8}
Model Exam 5 Paper 1
50. 3
3
3
3
3
3
3
:
: 3 ( 3) 27
: 2 ( 2) 8
: 1 ( 1) 1
: 0 0 0
:1 1 1
: 2 2 8
f x x
f
f
f
f
f
f
The range of ƒ = {27, 8, 1, 0, 1, 8}
Model Exam 5 Paper 1
51. What is the gradient of the straight line 3y = 5 6x?
ANSWER
(A) 2
(B)
(C)
(D)5
3
1
2
3
5
Model Exam 5 Paper 1
51.
is in the
3 5 6
3 6 5
6 5
3 35
23
The gradient,
form
2.
y x
y x
y x
y x
m
y mx c
Model Exam 5 Paper 1
52. P(3, 8) is the image of P(9, 5) under a translation T. T is represented by the column matrix
ANSWER
(A)
(B)
(C)
(D) 16
13
6
13
6
13
1
16
3
Model Exam 5 Paper 1
52.
Model Exam 5 Paper 1
3 9
8 5
3 9
8 5
6
13
T P P
T P P
53. A ship was travelling on a bearing of 180º. In what direction was it travelling?
ANSWER
(A) North
(B) South
(C) East
(D) West
Model Exam 5 Paper 1
53.
The ship was travelling due south.
Model Exam 5 Paper 1
54.
ANSWER
(A) (1, 2) (B) (1, 2)
(C) (1, 2) (D) (3, 2)
ABC is rotated through 180º about the origin as centre. The coordinates of the image of A under the transformation is
Model Exam 5 Paper 1
54.
Hence, the coordinates of the image of A are (1, 2).
Model Exam 5 Paper 1
55.
ANSWER
(A) 45
(B) 120
(C) 135
(D) 150
In the figure above, O is the centre of the circle.The magnitude of angle PQR is
Model Exam 5 Paper 1
55.
Model Exam 5 Paper 1
Sum of in a
circle.
at centre 2.
135 at aircumferenc
ˆ (reflex) 360 90
270
e
270ˆ
.2
POR
PQ
s
R
56.
ANSWER
(A) 45 cos 48°
(B) 5 sin 48
(C) 9 sin 48
(D) 45 sin 48
In the parallelogram ABCD, AB = 9 cm, BC = 5 cm and angle ABC = 48º. The area of the parallelogram ABCD, in cm2, is
Model Exam 5 Paper 1
56.
The area of the parallelogram ABCD,
A = ab sin C Formula
= 9 5 sin 48° cm2
= 45 sin 48° cm2
Two sides and the included angle.
Model Exam 5 Paper 1
57.
ANSWER
(A) (1.5 + 15 cos 30°) m
(B) (1.5 + 15 sin 30°) m
(C) (1.5 + 15 tan 30°) m
(D) 15 tan 30° m
TAB represents a tower. The height of the tower, TB, is
Model Exam 5 Paper 1
57.
Model Exam 5 Paper 1
1.5 m
tan 3015 m
So 15 m tan 30
15 tan 30 m
And
15 tan 30 1
1.5 15 tan 3
.5 m
0 m
AB
TA
TA
TB TA AB
58.
ANSWER
(A) 311°
(B) 131°
(C) 98°
(D) 49°
AC and DE are straight lines intersecting at B.Angle CBE = 49º. The size of angle CBD is
Model Exam 5 Paper 1
58.
Sum of s on a st line.
Subtracting 49° from both sides.
Model Exam 5 Paper 1
ˆ 49 180
ˆ 180 49
131
CBD
CBD
Items 59–60 refer to the diagram below. OAB isa right-angled triangle.
Model Exam 5 Paper 1
59. OA B is the image of OAB under an enlargement by a scale factor of 3. The coordinates of the points A and B are
ANSWER
(A) (0, 2) and (3, 0)
(B) (0, 3) and (6, 0)
(C) (0, 3) and (4, 0)
(D) (0, 4) and (8, 0)
Model Exam 5 Paper 1
59.
Model Exam 5 Paper 1
60. Area of OAB : Area of OA B =
ANSWER
(A) 1:9
(B) 1:6
(C) 1:3
(D) 1:1.5
Model Exam 5 Paper 1
Area of OAB: Area of OAB = 1:9
60.
Model Exam 5 Paper 1
CSEC MODEL EXAMINATION 5
MATHEMATICS
Paper 2
2 hours 40 minutes
SECTION I
Answer ALL the questions in this section
All working must be clearly shown
NEXT
1. (a) Using a calculator, or otherwise, determine
the value of (15.4)2 − (0.217 ÷ 7) and write
the answer(2 marks)
ANSWER
(i) exactly
(ii) correct to two significant figures.
Model Exam 5 Paper 2
1. (a) (i) (15.4)2 − (0.217 ÷ 7)
= 237.16 − 0.031
= 237.129
Squaring anddividingSubtracting(exactly)
(ii) 23
7 .129
= 240
The third significant figure,7, is greater than 5, so weadd 1 to the 3.(correct to two significantfigures)
Model Exam 5 Paper 2
Calculate
ANSWER
(i) the values of k and l(ii) the book value of the Private Car after 2 years.
Model Exam 5 Paper 2
(6 marks)
1. (b) The table below gives information on the book values and the rates of depreciation for two motor vehicles.
Motor Vehicle
Initial Book Value
Annual Rate of
Depreciation
Book Value after One Year
Taxi $75 000 15% $kPrivate car $90 000 l% $81 000
1. (b) (i) The book valueof the Taxi afterone year, $k = (100 − 15)% of
$75 000= 85% of $75 000= 0.85 × $75 000= $63 750
the value of k is 63 750
Model Exam 5 Paper 2
The amount of thedepreciation ofthe Private Carafter one year = $(90 000 – 81
000)= $9 000
The rate of depreciation of
the Private Car, l%
the value of l is 10.
Model Exam 5 Paper 2
(ii) The book value of thePrivate Car after one year = $81
000
The book valueof the PrivateCar aftertwo years = (100 − 10)% of
$81 000 = 90% of $81 000 = 0.9 × $81 000 = $72 900
Model Exam 5 Paper 2
1. (c) GUY $1.00 = US $0.005 andJAM $1.00 = US $0.011Calculate the value of
(2 marks)
ANSWER
(i) GUY $80 000 in US $
Model Exam 5 Paper 2
(ii) US $385 in JAM $. (2 marks)Total 12 marks
1. (c) (i) GUY $1.00 = US $0.005GUY $80 000 = US $0.005× $80 000
= US $400
Model Exam 5 Paper 2
(ii)
2. (a) Simplify
(3 marks)
ANSWER
Model Exam 5 Paper 2
2. (a)
Model Exam 5 Paper 2
2. (b) (i) Factorise
(1 mark)
ANSWER
a) p2 − 7p
Model Exam 5 Paper 2
b) p2 − 640 (1 mark)
(ii) Simplify (3 marks)
2. (b) (i) a)
Model Exam 5 Paper 2
b)
2. (b) (ii)
Model Exam 5 Paper 2
2. (c) Three DVDs and two Blue-ray discs cost $421, while two DVDs and one Blue-ray
disccost $242.
(2 marks)
ANSWER
(i) Given that one DVD costs $d and one
Blue-ray disc costs $b, write two equations
in b and d to represent the information.
Model Exam 5 Paper 2
(ii) Calculate the cost of one Blue-ray disc. (2 marks)Total 12 marks
2. (c) (i) The cost of one DVD = $dThe cost of one Blue-ray disc = $b
The two equations in b and d are:
3d + 2b= 421 (in dollars)
2d + b= 242 (in dollars)
Model Exam 5 Paper 2
Hence, the cost of one Blue-ray disc is $116.
2. (c) (ii)
Model Exam 5 Paper 2
3. (a)
Model Exam 5 Paper 2
Giving the reason for each step of your
answer, calculate the size of
ANSWER
(2 marks)(i) LNK
Model Exam 5 Paper 2
3. (a) In the quadrilateral KLMN, not drawn to scale, KL = LN = LM, NKL = 35° and KLM = 140°.
(ii) NLM(iii) KNM
(2 marks)
(2 marks)
(i) LNK = NKL = 35°Base s ofisoscelesΔ KLN
Model Exam 5 Paper 2
3. (a)
3. (a) (ii) KLN = 180° − (35° + 35°) = 180° − 70°= 110°
NLM = 140° − 110°= 30°
Sum of the s of a Δ
Model Exam 5 Paper 2
3. (a) (iii)
= 75°
Model Exam 5 Paper 2
(i) Copy and complete the Venn diagram torepresent the information.
ANSWER(5 marks)Total 11 marks
(ii) Write an expression in x for the number of students in the survey.
(iii) Calculate the value of x.
Model Exam 5 Paper 2
3. (b) In a survey of a class of 38 students, it was found that
15 like Karate12 like Judox like Karate and Judo2x like neither Karate nor Judo.
K is the set of students in the survey who like Karate, and J is the set of students who like Judo.
3. (b) (i)
The Venn diagram represent the information given.
Model Exam 5 Paper 2
(ii) The number of students in the survey
= 15 − x + x + 12 – x + 2x= 2x + x – x – x + 15 + 12 = x + 27
(iii) The number ofstudents in the survey = 38So x + 27 = 38i.e. x = 38 − 27
= 11
Hence, the value of x is 11.
Model Exam 5 Paper 2
4. (a) Using a ruler, a pencil and a pair of compasses, construct the triangle ABC
in which AB = 10 cm, BAC = 60°, and AC = 7 cm.
ANSWER
(4 marks)
Model Exam 5 Paper 2
4. (a)
Construct AB = 10 cm, then BAC = 60°.Construct AC = 7 cm, then complete the ΔABC.
Model Exam 5 Paper 2
4. (b) Measure and state the length of BC.
ANSWER
(1 mark)
Model Exam 5 Paper 2
4. (b) By measurement, the length of BC = 8.9 cm
Model Exam 5 Paper 2
4. (c) Find the perimeter of Δ ABC.
ANSWER
(1 mark)
Model Exam 5 Paper 2
4. (c) The perimeter of ΔABC = (10 + 7 + 8.9) cm
= 25.9 cm
Model Exam 5 Paper 2
4. (d) Construct on your diagram the line CD which is perpendicular to AB, and
meets AB at D.
ANSWER
(2 marks)
Model Exam 5 Paper 2
4. (d)
The line CD which is perpendicular to AB was constructed in the preceding diagram.
Model Exam 5 Paper 2
4. (e) Determine the length of CD.
ANSWER
(2 marks)
Model Exam 5 Paper 2
4. (e) Considering the right-angled ΔCAD:
Hence, the length of CD is 6.1 cm.
Model Exam 5 Paper 2
4. (f) Calculate the area of ΔABC, giving your
answer correct to one decimal place.
ANSWER
(2 marks)Total 12 marks
Model Exam 5 Paper 2
4. (f)
Model Exam 5 Paper 2
Or
Model Exam 5 Paper 2
Or
Model Exam 5 Paper 2
Use the graph to determine the(a) values of a and b which define the domain of
the graph
ANSWER
(2 marks)
Model Exam 5 Paper 2
5. The diagram below shows the graph of the function f (x) = 2 − x − x2 for a ≤ x ≤ b. The tangent to the graph at (1, 0) is also drawn.
5. (a) From the graph, the domain is −3 ≤ x ≤ 2.
The value of a is −3; this value is obtained from the left-hand side of the graph.
The value of b is 2; this value is obtained from the right-hand side
of the graph.
Model Exam 5 Paper 2
(b) values of x for which x2 + x − 2 = 0
ANSWER
(2 marks)
Model Exam 5 Paper 2
5. The diagram below shows the graph of the function f (x) = 2 − x − x2 for a ≤ x ≤ b. The tangent to the graph at (1, 0) is also drawn.
5. (b) Since f (x) = 2 − x − x2,if f (x) = 0, then
0 = 2 − x − x2
So x2 + x – 2 = 0
The solution to the quadratic equation are the
intercepts on the x-axis of the graph.
Hence, x = −2 or x = 1.
Model Exam 5 Paper 2
(c) coordinates of the maximum point on thegraph
ANSWER(2 marks)
Model Exam 5 Paper 2
5. The diagram below shows the graph of the function f (x) = 2 − x − x2 for a ≤ x ≤ b. The tangent to the graph at (1, 0) is also drawn.
5. (c) The coordinates of the maximum point on the
graph are
Model Exam 5 Paper 2
(d) gradient of f (x) = 2 − x − x2 at x = 1
ANSWER
(3 marks)
Model Exam 5 Paper 2
5. The diagram below shows the graph of the function f (x) = 2 − x − x2 for a ≤ x ≤ b. The tangent to the graph at (1, 0) is also drawn.
5. (d) Using the points (0, 3) and (3, −6) on the tangent to the graph, then the gradient of
the tangent, m
Hence, the gradient of f (x) = 2 − x − x2 at x = 1 is –3.
Model Exam 5 Paper 2
(e) values of x for which 2 − x − x2 > −2, where x is a whole number. ANSWER
(2 marks)
Model Exam 5 Paper 2
5. The diagram below shows the graph of the function f (x) = 2 − x − x2 for a ≤ x ≤ b. The tangent to the graph at (1, 0) is also drawn.
5. (e) Draw the line f (x) = −2, then the values of x for which 2 − x
− x2 > −2 is given by the inequality −2.6 ≤ x ≤ 1.6.
Since x is a whole number, then the possible values of x are 0, 1.
Model Exam 5 Paper 2
(a) Copy the diagram and indicate on the diagram, the distances x km, (x + 2) km and 10 km. ANSWER
Model Exam 5 Paper 2
6. A man walks x km due north, from point K to point L. He then walks (x + 2) km due west from point L to point M. The distance along a straight line from K to M is 10 km. The diagram below, not drawn to scale, shows the relative positions of K, L and M. The direction of north and west are also shown.
(2 marks)
6. (a)
The diagram was copied and the distancesindicated as shown above.
Model Exam 5 Paper 2
(b) Apply Pythagoras’ theorem to the diagram to obtain an equation in x. Show that the equation can be
simplified to x2 + 2x – 48 = 0.ANSWER
(3 marks)
Model Exam 5 Paper 2
6. A man walks x km due north, from point K to point L. He then walks (x + 2) km due west from point L to point M. The distance along a straight line from K to M is 10 km. The diagram below, not drawn to scale, shows the relative positions of K, L and M. The direction of north and west are also shown.
6. (b) Using Pythagoras’ theorem:
Model Exam 5 Paper 2
(c) Solve the equation and state the distance KL. ANSWER(2 marks)
Model Exam 5 Paper 2
6. A man walks x km due north, from point K to point L. He then walks (x + 2) km due west from point L to point M. The distance along a straight line from K to M is 10 km. The diagram below, not drawn to scale, shows the relative positions of K, L and M. The direction of north and west are also shown.
6. (c)
Hence, the distance KL is 6 km.
Model Exam 5 Paper 2
(d) Determine the bearing of K from M. ANSWER(4 marks)Total 11 marks
Model Exam 5 Paper 2
6. A man walks x km due north, from point K to point L. He then walks (x + 2) km due west from point L to point M. The distance along a straight line from K to M is 10 km. The diagram below, not drawn to scale, shows the relative positions of K, L and M. The direction of north and west are also shown.
6. (d)
Model Exam 5 Paper 2
Hence, the bearing of K from M is 126.9°.
Model Exam 5 Paper 2
(a) Copy and complete the mid-interval valuescolumn.
ANSWER
Model Exam 5 Paper 2
7. In an experiment, the mass of 100 children were recorded in kilograms as shown in the table below.
Mass in kilograms Frequency
Mid-Interval Values (kg)
5–9 5 7
10–14 23 12
15–19 29
20–24 14
25–29 13
30–34 11
35–39 5
(1 mark)
Mid-IntervalValues (kg)
7
12
17
22
27
32
37
7. (a)The mid-intervalvalues column wascopied and completedas shown.
Model Exam 5 Paper 2
(b) (i) Calculate an estimate of the mean mass
of the 100 children.
ANSWER(3 marks)
Model Exam 5 Paper 2
7. In an experiment, the mass of 100 children were recorded in kilograms as shown in the table below.
Mass in kilograms Frequency
Mid-Interval Values (kg)
5–9 5 7
10–14 23 12
15–19 29
20–24 14
25–29 13
30–34 11
35–39 5
7. (b) (i)
Frequency f
Mid-Interval Values (kg)
x fx 5 7 35
23 12 276
29 17 493
14 22 308
13 27 351
11 32 352
5 37 185 n = f = 100 fx = 2 000
Model Exam 5 Paper 2
An estimate of themean mass of the100 children,
Model Exam 5 Paper 2
(b) (ii) Draw a frequency polygon to represent
the mass of the children.
ANSWER(5 marks)
Model Exam 5 Paper 2
7. In an experiment, the mass of 100 children were recorded in kilograms as shown in the table below.
Mass in kilograms Frequency
Mid-Interval Values (kg)
5–9 5 7
10–14 23 12
15–19 29
20–24 14
25–29 13
30–34 11
35–39 5
The frequency polygon above represents the mass of the children.
7. (b) (ii)
Model Exam 5 Paper 2
(c) Calculate the probability that a student chosen at random from the experimental group had a mass of 25 kg or more.
ANSWER(2 marks)Total 11 marks
Model Exam 5 Paper 2
7. In an experiment, the mass of 100 children were recorded in kilograms as shown in the table below.
Mass in kilograms Frequency
Mid-Interval Values (kg)
5–9 5 7
10–14 23 12
15–19 29
20–24 14
25–29 13
30–34 11
35–39 5
7. (c) The number of children witha mass of 25 kg or more =
13 + 11 + 5= 29
The total number of children = 100
P (student’s mass $ 25 kg)
Model Exam 5 Paper 2
8. The first three diagrams in a sequence are shown below.Diagram 1 has four disks forming a square pattern.
Diagram 2 is a square pattern formed by using five additional disks.
Diagram 3 is a square pattern formed by using seven additional disks.
(2 marks)
Model Exam 5 Paper 2
(a) Draw Diagram 4 in the sequence.ANSWER
8. (a)
Diagram 4 in the sequence can be seen above.
Model Exam 5 Paper 2
8. (b) Complete the table by inserting the appropriate values at
the rows marked (i), (ii) and (iii).
ANSWER
(6 marks)
Diagram Number
Number of Disks Forming
the Square
Number of Additional
Disks Forming the
Square
Pattern for Calculating the
Number of Additional Disks
1 2 × 2 ~ ~2 3 × 3 5 2 × 3 − 13 4 × 4 7 2 × 4 − 14 5 × 5 — —— 8 × 8 15 —n (n + 1) (n + 1) — —
Model Exam 5 Paper 2
8. (b)
The completed table can be seen above.
Diagram Number
Number of Disks Forming
the Square
Number of Additional
Disks Forming the
Square
Pattern for Calculating the
Number of Additional Disks
1 2 × 2 ~ ~2 3 × 3 5 2 × 3 − 13 4 × 4 7 2 × 4 − 14 5 × 5 9 2 × 5 − 17 8 × 8 15 2 × 8 − 1n (n + 1) (n + 1) (n + 1) + n 2(n + 1) − 1
Model Exam 5 Paper 2
8. (c) Hence, determine a formula consisting of a
single term in n for calculating the total number of disks, N, in a diagram.
ANSWER
(2 marks)Total 10 marks
Model Exam 5 Paper 2
8. (c) The total number of disks in a diagram,
Hence, the formula is N = (n + 1)2.
Model Exam 5 Paper 2
SECTION II
Answer TWO questions in this section
Model Exam 5 Paper 2
9. (a) Solve the pair of simultaneous equations y = x + 3
y = x2
ANSWER
(5 marks)
Model Exam 5 Paper 2
9. (a)
where a = 1, b = −1 and c = −3.
Model Exam 5 Paper 2
Using the quadratic formula:
Model Exam 5 Paper 2
Model Exam 5 Paper 2
Hence, x = 2.30 or x = −1.30
Model Exam 5 Paper 2
(i) Write an expression, in terms of l and x, for the length of the strip of wire.
ANSWER(2 marks)
Model Exam 5 Paper 2
9. (b) A strip of wire of length 58 cm is cut into two pieces. One piece is bent to form a square of
side x cm. The other piece is bent to form arectangle of length l cm and width 5 cm.
The diagrams below, not drawn to scale, show the square and the rectangle.
9. (b) (i) The length of thestrip of wire = [4x + 2l +
2(5)] cm= (4x + 2l + 10) cm
An expression, in terms of l and x, for the
length of the strip of wire is 4x + 2l + 10.
Model Exam 5 Paper 2
(ii) Show that l = 24 − 2x
ANSWER(2 marks)
The sum of the areas of the square and the rectangle is represented by A.
Model Exam 5 Paper 2
9. (b) A strip of wire of length 58 cm is cut into two pieces. One piece is bent to form a square of
side x cm. The other piece is bent to form arectangle of length l cm and width 5 cm.
The diagrams below, not drawn to scale, show the square and the rectangle.
9. (b) (ii)
Model Exam 5 Paper 2
(iii) Show that A = x2 − 10x + 120.
ANSWER
(2 marks)
Model Exam 5 Paper 2
9. (b) A strip of wire of length 58 cm is cut into two pieces. One piece is bent to form a square of
side x cm. The other piece is bent to form arectangle of length l cm and width 5 cm.
The diagrams below, not drawn to scale, show the square and the rectangle.
9. (b) (iii) The sum of the areasof the square andthe rectangle
Model Exam 5 Paper 2
(iv) Calculate the values of x for which A = 96. ANSWER(4 marks)
Total 15 marks
Model Exam 5 Paper 2
9. (b) A strip of wire of length 58 cm is cut into two pieces. One piece is bent to form a square of
side x cm. The other piece is bent to form arectangle of length l cm and width 5 cm.
The diagrams below, not drawn to scale, show the square and the rectangle.
9. (b) (iv)
Hence, the values of x are 4 and 6.
Model Exam 5 Paper 2
10. (a) The diagram below, not drawn to scale, shows a vertical cellphone tower, TF, and a vertical antenna, WT, mounted on the
top of the tower.
A point P is on the same horizontal ground as F, such that PF = 30 m, and the angles of elevation of T and W from P are 42° and 55° respectively.
Model Exam 5 Paper 2
10. (a) (i) Copy and label the diagram clearly showing
a) the distance 30 m
b) the angles of 42° and 55°
c) any right angles.
ANSWER
(7 marks) (ii) Calculate the length of the antenna WT.
Model Exam 5 Paper 2
10. (a) (i)
Model Exam 5 Paper 2
The diagram was copied and labelledabove, clearly showinga) the distance 30 mb) the angles of 42° and 55°c) the right angle TFP
Model Exam 5 Paper 2
10. (a) (ii)
Model Exam 5 Paper 2
Considering the right-angled ΔPWF:
Model Exam 5 Paper 2
Considering the right-angled ΔPTF:
The length of the antenna WT = WF − TF= (42.8 − 27) m= 15.8 m
Model Exam 5 Paper 2
10. (b) The diagram below, not drawn to scale,shows a circle with centre O. The lines
BDand DCE are tangents to the circle. AngleBCD = 65°
Model Exam 5 Paper 2
Calculate, giving reasons for each step ofyour answer.
ANSWER
(8 marks)Total 15 marks
(i) OCE
(ii) BAC
(iii) BOC
(iv) BDC
Model Exam 5 Paper 2
10. (b)
Model Exam 5 Paper 2
10. (b) (i)
Model Exam 5 Paper 2
10. (b) (ii)
Model Exam 5 Paper 2
10. (b) (iii)
Model Exam 5 Paper 2
10. (b) (iv)
Model Exam 5 Paper 2
11. (a) The value of the
determinant of M = is 22.
ANSWER
(3 marks)(i) Calculate the value of x.
Model Exam 5 Paper 2
11. (a) (i) The value of the determinant of M,
Hence, the value of x is 3.
Model Exam 5 Paper 2
(ii) For this value of x, find M–1.
ANSWER
(2 marks)
Model Exam 5 Paper 2
11. (a) The value of the
determinant of M = is 22.
11. (a) (ii)
Model Exam 5 Paper 2
(iii) Show that M–1 M = I.
ANSWER
(2 marks)
Model Exam 5 Paper 2
11. (a) The value of the
determinant of M = is 22.
11. (a) (iii)
Hence M –1 M = I.
Model Exam 5 Paper 2
11. (b) The graph shows the line segment AC andits image A′C′ after a transformation by the
matrix
Model Exam 5 Paper 2
Model Exam 5 Paper 2
(i) Write in the form of a single 2 × 2 matrix,the coordinates ofa) A and C ANSWER
(2 marks)
Model Exam 5 Paper 2
11. (b) The graph shows the line segment AC andits image A′C′ after a transformation by the
matrix
11. (b) (i) a)
Model Exam 5 Paper 2
(i) b) A′ and C′.
ANSWER
(2 marks)
Model Exam 5 Paper 2
11. (b) The graph shows the line segment AC andits image A′C′ after a transformation by the
matrix
11. (b) (i) b)
Model Exam 5 Paper 2
(ii) Using matrixes only, write an equation torepresent the transformation of AC intoA′C′.
ANSWER(2 marks)
Model Exam 5 Paper 2
11. (b) The graph shows the line segment AC andits image A′C′ after a transformation by the
matrix
11. (b) (ii) The matrix equation that represents thetransformation of AC onto A′C′ is:
Model Exam 5 Paper 2
(iii) Determine the values of p, q, r and s.
ANSWER(2 marks)Total 15 marks
Model Exam 5 Paper 2
11. (b) The graph shows the line segment AC andits image A′C′ after a transformation by the
matrix
11. (b) (iii)
Model Exam 5 Paper 2
Equating corresponding elements in the first row:
Model Exam 5 Paper 2
Substituting −1 for p in :
Model Exam 5 Paper 2
Equating corresponding elements in the second row:
Model Exam 5 Paper 2
Substituting 0 for r in :
Hence, p = −1, q = 0, r = 0 and s = 1.
Model Exam 5 Paper 2
Model Exam 5 Paper 2
MATHEMATICS: A COMPLETE COURSE WITH CXC QUESTIONSText © Raymond ToolsieFirst Published in 1996Reprinted in 1997, 1998, 1999, 2000, 2001, 2002, 2003Second Edition November 2004Third Edition 2009
ISBN: 976-8014-13-0
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