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SASBADISasbadi Sdn . Bhd. (139288-X)
Lot 12, Jalan Teknologi 3/4, Taman Sains Selangor 1,
Kota Damansara, 47810 Petaling Jaya,
Selangor Darul Ehsan , Malaysia.
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© Sasbadi Sdn. Bhd. (139288-X)
Cetakan 2007
ISBN 978-983-59-3350-9
The TOTAL PRO SPM Topical Assessment MATHEMATICS series has been published
specifically to fulfil the urgent needs of students who are preparing to sit for the SPM public
examination: These exam-oriented assessment materials are intensive and comprehensive,
covering all the topics prescribed in the KBSM syllabus for Forms 4 and 5. They also include
model papers which conform to the current SPM format. The book is systematically organised
according to topics and subtopics as in the textbooks, it also meets all the requirements of the
SPM and the KBSM syllabus.
The questions in this book are both original and challenging, incorporating a variety of
questioning techniques and levels of difficulty to meet the actual standard.of the SPM papers.
Moreover, the number of questions set on each topic is determined by the popularity or the
frequency of recurrence of the topic in the SPM examination as indicated by the trend observed
in the past three to five years.
Students who work consistently and systematically, using the TOTAL PRO SPM Topical
Assessment MATHEMATICS will be able to improve their understanding of the subjects and
evaluate their own performance. Consequently, this motivation will boost their confidence in
facing the real challenges of the public exam with greater success.
SUPERIOR
ADVANTACESS
OFTHE
TOTAL PRO.
SPM TOPI4AL,
ASSESSMENTMATHEMATICS;
THE PUBLISHER
Cover all the topics prescribed in the KBSM syllabus for Forms 4 and 5 and focus on popular examination topics in the SPM.
Enables students to predict the type and allocation of questions based on a careful
study of the past years' SPM papers.
Comprises challenging questions which incorporate a variety of questioning
techniques and levels of difficulty and conforms to the current SPM format.
Consist of two complete sets of SPM-quality Model Papers set in accordance with
the actual format and standard of the public exam.
Provide plausible answers to questions for students to check and evaluate their
own understanding and performance in the subject.
• ANALYSIS OF SPM
PAPERS-(2003 -2006)
(vi)
• MATHEMATICAL FORMULAE ( vii) - (viii)
FORM 4
1.1 Significant Figures 1.2 Standard Form
Quadratic Expressions and Equations
2.1 Quadratic Expressions
2.2 Factorisation of Quadratic Expressions. 2.3 Quadratic Equations
2.4 Roots of Quadratic Equations
Sets
3.1 Sets
3.2 Subset, Universal Set and Complement of a Set
3.3 Operations on Sets
Mathematical Reasoning
4.1 Statements
4.2 Quantifiers "All" and "Some" 4.3 Operations on Statements 4.4 Implications
4.5 Arguments
4.6 Deduction and Induction
The Straight Line
5.2 Gradient of a Straight Line in Cartesian Coordinates
5.3 Intercepts
5.4 Equation of a Straight Line 5.5 Parallel Lines
Statistics III
6.1 Class Intervals
6.2 Mode and Mean of Grouped Data 6.3 Histograms 6.4 Frequency Polygons 6.5 Cumulative Frequency 6.6 Measures of Dispersion 7.1 Sample Spaces 7.2 Events 7.3 Probability of an Event Circles Ill 8.1 Tangents to a Circle
8.2 Angles between Tangents and Chords 8.3 Common Tangents
9.1 The Values of Sin e, Cos 9 and Tan 0 9.2 Graphs of Sine, Cosine and Tangent
Lines and Planes in 3-Dimensions
"14,
11.1 Angles between Lines and Planes 11.2 Angles between Two Planes
12.1 Numbers in Bases Two, Eight and Five
Graphs of Functions II 13.1 Graphs of Functions
13.2 Solution of an Equation by the Graphical Method
13.3 Region Representing Inequalities in Two Variables
Transformations III
14.1 Combination of Two Transformations
Matrices 15.1 Concept of Matrices 15.2 Concept of Equal Matrices
15.3 Addition and Subtraction of Matrices 15.4 Multiplication of a Matrix by a Number 15.5 Multiplication of Two Matrices
15.6 Concept of Identity Matrices 15.7 Concept of Inverse Matrices
15.8 Solve Simultaneous Linear Equations. Using Matrices
Variations
16.1 Direct Variations 16.2 Inverse Variations 16.3 Joint Variations
Gradient and Area under IT a Graph
17.1 Quantity Represented by the Gradient of a Graph
17.2 Quantity Represented by the Area under a Graph
1] Probability II
18.1 Probability of an Event
18.2 Probability of the Complement of an Event 18.3 Probability of a Combined Event
19.1 Bearings
20.1 Longitudes 20.2 Latitudes
20.3 Location of a Place
20.4 Distance on the Surface of the Earth
21.1 Orthogonal Projections 21.2 Plans and Elevations
N
, !A ^ ILIVY'S11$ 0F`'S M^PA E- 15^
((l 00 )' I -j 'am' 0
Number of Questions Form Topic 2003 2004 2005 2006 Polygons I& II 2 1 2 1 Algebraic Expressions 2 2 2 2 Linear Equations 1 I 1 1 Algebraic Formulae 1 1 1 1 1-3 Statistics I & II 2 2 1 3 Transformations I & 11 3 2 2 2 Indices 2 2 2 1 Linear Inequalities 2 2 1 1 Trigonometry I - 1 - 1 Standard Form 4 4 3 4 Sets 3 3 3 3The Straight Line 1 2 2 2
Statistics III - - 1
-4 Probability I 2 1 2 1
Circles III 1 1 1 1
Trigonometry II 2 2 3 2
Angles of Elevation and Depression 1 2 1 2
Lines and Planes in 3-Dimensions 1 1 1 1
Number Bases 2 2 2 2 Graphs of Functions II 1 1 1 1 Matrices 3 1 2 2 5 Variations 2 2 3 3 Probability II - 2 - I Bearings 1 1 1 1 Earth as a Sphere 1 1 2 1
PAPER 2
Number of QuestionsSection Form Topic
2003 2004 2005 2006
Solid Geometry 1 1 1 1
1-3 Circles I&II 1 1 1 1
Linear Equations 1 1 1 1
Quadratic Expressions and Equations 1 1 1 1
Sets - I - 1
A 4 Mathematical Reasoning I 1 1 1
The Straight Line 1 1 1 1
Lines and Planes in 3-Dimensions 1 1 1 1
Graphs of Functions II 1 - 1
-Matrices 1 1 1 1
5
Gradient and Area under a Graph 1 1 1 1
Probability II 1 1 1 1
4 Statistics III 1 1 1 1
Graphs of Functions II 1 1 1 1
B 5 Transformations III I 1 1 1
Earth as a Sphere 1 1 1 1
The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used.
RELATIONS
I am x an = am+n 2 am+an=am-n 3 (am)n =a mn 4 A-'= 1 (d_b\ ad-bc -c a) 5 P(A) = n(A) n(S) 6 P(A') =1- P(A)7 Distance = (x2-x)2 + (y2 -y1)2
8 Midpoint, (x, y) = x1
+ x2
y1
+ y22 2
9 Average speed = distance travelled time taken 10 Mean- sum of data
number of data
11 Mean = sum of (class mark x frequency) sum of frequencies 12 Pythagoras' Theorem &=a2+b2
13 m
= 2-x1 X2-X1 14 m=- y-intercept x-intercept (xl> Y1) 0 a b Y y-intercept x-intercept x (x2, YZ)SHAPES AND SPACE
1 Area of trapezium = 2 x sum of parallel sides x height
2 Circumference of circle = TO = 2nr
3 Area of circle = nr2
4 Curved surface area of cylinder = 2inrh
5 Surface area of sphere = 4nr2
6 Volume of right prism = cross sectional area x length
7 Volume of cylinder = itr2h
8 Volume of cone = 1 nrzh 3
9 Volume of sphere = 3 nr3
10 Volume of right pyramid = 3 x base area x height
11 Sum of interior angles of a polygon = (n - 2) x 180° 12 arc length - angle subtended at centre
circumference of circle 360°
13 area of sector _ angle subtended at centre
area of circle 3600
14 Scale factor, k = PA
15 Area of image = kz x area of object
(viii)
1.1 Significant Figures
SECTION A Objective Questions
1.2 Standard Form
This section consists of 40 questions. Answer all the questions. For each question, choose only You may use a non -programmable scientific calculator.
Subtopic 1.1
1 Round off 2 794 to two significant figures.
A 2 700 B 2790 C 2 800 D 2 810
2 Round off 300 257 , to four significant figures.
A 3 003 B 300 200 C 300 300 D 300 357
3 Round off 4.17572 to three significant figures.
A 4.18 B 4.175 C 4.176 D 4.1757
4 Round off 0.060813 to three significant figures.
A 0.06 B 0.060 C 0.0608 D 0.06081
5 Round off 0.006197 to three significant figures.
A 0.006 B 0.0061 C 0.0062 D 0.00620
6 Round off 92.6142 to three significant figures.
A 92.6 C 92.614 B 92.61 D 93.0
7 Round off 0.00436 to two significant figures.
A 0.004 B 0.0043 C 0.0044 D 0.00436
8 Round off 73 208 to three significant figures. A 732 B 73 200 C 73 210 D 732 100 9 Convert 3 285 km to in and round off the answer to three significant figures.
A 32 850 in B 32900m C 329 000 in D 3290000m
10 Calculate the value of 5.801 - 9 x 0.417 and round off the answer to two significant figures. A 2 B 2.0 C 2.01 D 2.1 one answer.
12 Calculate the value of (5.19)2 and round off the answer to two significant figures. A 26.93 C 27 B 26.94 D 28 13 Calculate the value of
0.915 - 0.01 x 2.4 and round off the answer to three significant figures.
A 22 C 221 B 220 D 221.6 14 The volume of a cube is
50 cm3. Find the length, in cm, of the edge of the cube. Give the answer correct to two significant figures. A 3.6 C 7.1 B 3.7 D 7.2 15 The diagram shows a circle
with diameter 11.2 cm.
Calculate the area, in cm2, 11 Calculate the value of
of ans
the wer
71.82 + 16.2 - 0.03 and round significa off the answer to two
significant figures. Use it
A 61 A 35.2
B 610 B 98.5
C 6 100 C 98.6
D 6 200 D 99.0
circle and give the correct to three nt figures.
16 Mr Lee saves RM36 000 in a bank at an interest rate of 3.2% per annum. After one year, his total interest, correct to two significant figures, is A RM1 100 B RM1 150 C RM1 152 D RM1 200
Subtopic 1.2
17 Express 0.000000305 in the standard form. A 3.05 x 10-6 B 3.05x10-' C 3.05 x 10-8 D 30.5 x 10-8 18 Express 218 000 in the standard form. A 218 x 103 B 2.18 x 105 C 21.8 x 10-4 D 2.18 x 10-5 19 Express 0.00436 in the standard form. A 0.043 x 10-2 B 0.436 x 10-3 C 4.36 x 10-3 D 4.36 x 103 20 3.18x106= A 3180 B 31 800 C 318 000 D 3 180 000 21 7.3 x 10-4 = A 0.0000073 B 0.000073 C 0.00073 D 0.0073 22 Express 2.438 x 10-5 as a single number. A 0.002438 B 0.0002438 C 0.00002438 D 0.00000243823 Find the value of 0.00008 x 0.0045 and express the answer in the standard form.
31 Find the value of 3.6 x 10-4 - 7.8 x 10-5 and express the answer in the standard form. A 3.6 x 10-a
B 3.6 x 10-' C 3.6 x 10' D 3.6 x 108
24 Calculate the value of 0.00063 th d A 4.98 x 10-5 B 4.98 x 10-4 Y 2.82 x 10-5 D 2.82 x 10-4
32 Calculate the value of 65 600 and ex ress the express e
an 9 000
answer in standard form.
p 0.8 x 10-6
answer in the standard form.
A 7 x 10-8
A 8.2 x 108
B 7x10-' B 8.2 x 109
C 7 x 10-6 C 8.2 x 1010
D 7x108 D 8.2 x 1011
25 Calculate the value of 6.5x108x8x10-13and express the answer
standard form. in the A B C D 5.2 x 10'5 5.2 x 10-4 5.2 x 104 5.2 x 105 26 7.6 x 10-6 + 9.44 x 10-5 = A 1.704 x 10-s B 1.02 x 10-5 C 1.704 x 10-4 D 1.02 x 10-4 27 24 000 = 6 x 10-4 A 4x107 B 4x108 C 0.4 x 10' D 0.4 x 109 28 4.5 x 106 x 5 000 = A 2.25 x 1010 B 2.25 x 109 C 2.25x108 D 2.25 x 106 29 3.7x1012-8.9x10"= A 3.611 x 1012 B 3.611 x 1011 C 2.81x1012 D 2.81 x 1011 30 (5 x 10-3)2 = A 25 x 10-6 B 2.5 x 10-' C 2.5 x 1b-5 D 2.5 x 10-4
33 Calculate the value of 4 x 109+8.7x 1010 and express the answer in the standard form. A B C D 4.87 x 109 4.87 x 1010 9.1 x 109 9.1 x 1010 34 5.9 x 104 + 480 000 = A 1.07 x 105 B 1.07 x 106 C 5.39 x 104 D 5.39 x 105 35 12.28 x 105 = (4 x 10'3)2 A 3.07 x 1010 B 3.07 x 1011 C 7.675 x 109 D 7.675x1010 36 A motorcycle moved at a speed of 120 km h-'. Find the distance, in m, travelled by the motorcycle in 90 minutes. A 3x103
B 1.08x104 C 1.8 x 105 D 1.08 x 10'
37 The area of a rectangular piece of land is 8.4 km2. If its length is 3 500 m, find its width, in m. A 2.4 x 103 B 2.4 x 103 C 2.94 x 101 D 2.94x103
2
38 The wheel of a car has a radius of 28 cm. How many rotations does the wheel make if the car travels a distance of 88 km? Use it = 72) A 5 x 104 B 5x105 C 5 x 106 D 5 x 10-$
39 Given 1 g of metal Y contains 5.8 x 1020 atoms, Calculate the number of atoms in 2.5 kg of metal Y and express the answer in the standard form.
A 1.45 x 1021 B 1.45 x 1024 C 2.32 x 1017 D 2.32 x 1020
SECTION B
Subjective Questions
5 Calculate the value of 7.13 - 10 x 6.2 and round off the answer to three significant figures. This section consists of 20 questions. Answer all the questions. You may use a non-programmable scientific calculator.
Subtopic 1.1
1 Round off the following numbers correct to two significant figures.
(a) 0.07006 (b) 49 815 Answer:
2 Round off the following numbers correct to three significant figures.
(a) 50.761 (b) 83 249 Answer:
(a)
(b)
3 Calculate the value of each of the following and round off the answer to one significant figure. (a) 5 418 - 2 970
(b) 3.8 _ 800 Answer:
Answer:
6 (a) Calculate the value of 13.02 + 5.3 x 90. (b) Round off the answer in (a) to four
significant figures. Answer:
(a)
40 A rectangular floor has a length of 3 600 cm and a width of 2 000 cm. The floor needs to be covered with square tiles, measuring 20 cm x 20 cm. Calculate the number of tiles needed to cover the whole floor. A 1.8 x 103 C 3.6 x 103 B 1.8x104. D 3.6x104
(b)
7 (a) Calculate the value of 200 - 146.28 _ 16. (b) Round off the answer in (a) to four
significant figures. Answer:
(a)
8 The diagram shows a right-angled triangle.
4 (a) Calculate the value of 57 007 - 24 518 + 3 107. (b) Round off the answer in (a) to three
significant figures. Answer: (a)
(b)
X CM(b)
9 cm 7 cmCalculate the value of x correct to two significant figures.
9 A sphere has a radius of 8 cm. Find its total surface area, in cm2, and round off the answer to three significant figures.
(Use is = 3.142) Answer:
Subtopic 1.2
10 State the following numbers in the standard form. (a) 0.00076 (c) 359 (b) 0.0000204 (d) 8 003 000 Answer: (a)
(b)
(c)(d)
11 Write each of the following as a single number. (a) 1.8 x 10-3 (c) 6 x 105 (b) 5.04 x 10-4 (d) 9.815 x 106 Answer: (a) (b) (c) (d)
12 Find the value of 5.8 x 1012 - 5 x 1011 and express the answer in the standard form. Answer:
14 Find the value of 7.5 x 10-6 - 9.1 x 10-7 and express the answer in the standard form. Answer:
15 Find the value of 9.4 x 10-4 + 8 x 10-5 and express the answer in the standard form. Answer:
16 Find the value of 9 660 and express the 7x109
answer in the standard form. Answer:
17 Find the value of 6.3 x 104 x 2.5 x 10-9 and express the answer in the standard form. Answer:
18 Find the value of 6.8 x 7 x 106 and express the answer in the standard form.
Answer:
19 The thickness of a wooden plank is 5.4 x 10-5 m. Calculate the total thickness, in m, of 500 pieces of such planks and express the answer in the standard form.
Answer:
13 Find the value of 3.78 x 106 and express the 0.007
answer in the standard form. Answer:
20 The scale of a map is 1: 400 000. Find the actual length, in cm, of a river which measures 8.4 cm on the map. Express the answer in the standard form.
Form 4
2.1 Quadratic Expressions
2.2 Factorisation of Quadratic Expressions 2.4 Roots of Quadratic Equations
Equations
Quadratic Expressions and
2.3 Quadratic Equations
SECTION A
Objective Questionsyx
This section consists of 16 questions. Answer all the questions. For each question, choose only one answer. You may use a non-programmable scientific calculator.
Subtopic 2.1
1 Expand 3y(y + 2). A 3y+2 C 3y2+6 B 3y + 6y D 3y2 + 6y 2 (2p - 1) (p - 5) _ A 2p2 + 5 B 2p2+5p+5 C 2p2-5p+5 D 2p2 - 11p + 5 6 Factorise 3x2 - llx + 6. A (3x - 3) (x + 2) B (3x-2)(x-3) C (3x - 2) (x + 3) D (3x+2)(x+3) 7 Factorise 4r2 - 49 completely. A (2r + 7)2 B (2r - 7)2 C (2r + 7)( 2r - 1) D (2r + 7 )(2r - 7)Subtopic 2.4
11 The roots of the equation 8 - 3p2 = 2p are A -2 and 3 B -2 and 4 C -3 and2 D 3 and 2 3 Expand (5q - 2)2. A 5q2 +4 B 5q2 - 10 C 25g2 - 20q + 4 D 25q2 - 10q + 2
4 The diagram shows a trapezium. p cm ❑ 4p cm n (p + 5) cm
Express the area, in cm2, of the trapezium in terms of p. A 4p2 + p C 8p2 + 20 B 4p2+ 10p D 8p2 + 20p
Subtopic 2.2
5 Factorise 4 + 5t - 9t2. A (2 + 3t)(2 - 3t) B (4+9t)(1-t) C (4 + t)(1 - 9t) D (4 + t)(1 + 9t) 8 Factorise 3h2 - 18h + 15 completely . A (h-1)(h-5) B (h - 2)(h - 3) C 3(h - 1) (h - 5) D 3(h - 2)(h - 3)Subtopic 2.3
9 Which of the following is not a quadratic equation? A 2(m + l) = 8m2 B 2=(f-2)2 C 3a2 - 2ab = 5 D w2 = w 2 3 10 2n2 + 3(n - 1)2 = 0 when written in the general form is A 2n2-3n+3=0 B 2n2+3n+3=0 C 5n2-6n+1=0 D 5n2-6n+3=0
12 Solve the quadratic equation 3x(x - 5) = 0.
A x=-5orx=3 B x=-5orx=5 C x=Oorx=5 D x=3 orx=5
13 Solve the quadratic equation 4k2 = k.
A k= 4 ork=1 B k=Oork= C k=Oork=4 D k=fork=4
14 Solve the quadratic equation 2y2 - 5 =Y A y = -5 or y 1 2 B y -2 or y=1 C y=-1 ory= 2 D y=-2 ory=5
15 The diagram shows a triangle KLM.
(h + 5) cm
The area of the triangle KLM is equal to the area of a square with sides of 5 cm. Find the possible value of h. A 2 C 10 B 5 D 15 16 To prepare a bucket of
cement, Encik Kadir needs
Answer:
SECTION B Subjective Questions}
This section consists of 15 questions. Answer all the questions. You may use a non-programmable scientific calculator.
Subtopic 2.1
1 Determine whether each of the following is a quadratic expression. (a) 4x2 - y (c) p(p - 3) (b) 5-h-4h2 Answer: (a) (b) (c) (a)
(b)
(2x - 3) kg of sand and 2 kg of cement. Encik Kadir prepares x buckets of the mixture with a total mass of 66 kg of cement and sand. Find the value of x.A 3 C 6 B 5 D 7
(c)
(d)
5 Factorise the following completely. 2 Expand each of the following.
(a) 6y(y - 1) (c) (3k + 1)2 (a) (b) (b) (2 - m)(m + 3) (d) (1 - 2q)2 (c) (d) Answer: (a) (b) (c)
(d)
3 A salesperson sold y packets of book marks costing RM2 each and y cups costing RM(y + 3) each. Express the total sales of the salesperson in terms of y.
Answer:
Subtopic 2.2
4 Factorise completely each of the following quadratic expressions. (a) 8p2 + 6 (c) 64h2 - 1 n2 + n - 6 3m (m 5) - (m - 5) 2(5p2 - 1) - p 3q2 + 4q - 7 Answer: (a)
(b)
(c)(d)
6 Factorise the following completely. (a) a2 - 49 (b) 6b(3b + 1) - 2(3b + 1) (c) 16 + (k + 2) (k - 8)
(d) h2 - 2(3h - 4)
Answer: (a)(b)
(c)(d)
Subtopic 2.3
11 Solve each of the following quadratic equations. 2(a) 4p2 + 12p = 0 ( c) 3k + 4k = 5
7 Determine whether each of the following is
.k+2
a quadratic equation. - 5)2 = 9 (b) (
d
x 1(a) h2 - 2hk = 0
(c) (y - 2)2 = 9
y ( ) 36 = 4x(b) x = 3
Answer: (a) (b) (c) Answer: (a) ((b) (
)d)
8 Write each of the following quadratic equationsin the general form. (a) r2 = 7(r + 1) (b) (w + 1)(4w - 3) = 1 (c) x +x=2 Answer: (a)
(b)
(c)9 A taxi travels from the station to town P at an average speed of 20x km/h. The journey of 120 km takes (2x - 4) hours.
Form a quadratic equation from the information given tbove.
Answer:
Subtopic 2.4
10 Determine whether -3 and 2 are the roots of each of the following quadratic equations. (a) x2-2x-6=0 (c) 3(x2+1)=x (b) x(x + 1) = 6 Answer: (a) (b) (c)
12 Determine the roots of the following quadratic equations.
(a) (p - 1)(p+3)=5(p+3) (b) 3(2 - q) = 10 - q2 Answer:
13 Solve the quadratic equation 3p2 - 5 = 7p. 2
14 Solve the quadratic equation 2x 3- 1) = x + 4. Answer:
15 In the diagram, ABCD and AEFG are rectangles. (4 + x) cm
E
2 cmi F
x cm
B C (a) Express the following in terms of x.
(i) Length of AE, in cm.
(ii) Area of the shaded region, in cm2.
(b) Given that the area of the shaded region is
27 cm2, find the length of AG, in cm.
Answer:(a) (i)
3.1 Sets
3.2 Subset, Universal Set and Complement of a Set
SECTION A Objective Questions
3.3 Operations on Sets
This section consists of 35 questions. Answer all the questions. For each You may use a non-programmable scientific calculator.
Subtopic 3.1
1 Given P is a set of prime numbers between 10 and 30. The elements of set P are A {11, 13, 19)
B (11, 15, 19, 23, 29) C {11, 13, 17, 19, 23, 291 D {11, 13, 17, 19, 21, 23, 27,
29)
2 Given that Q = (factors of 18), find n(Q).
A 5 C 7 B 6 D 8
3 Given X = {colours of the Malaysian flag). Which of the following is false? A white E X B red E X C blue E X D green E X
Subtopic 3.2
4 Which of the following Venn diagrams represents the set S C T with the universal set E =SUT? A
sCD:T^]
B S T00
C S/ 5 Given setP={4,6,7),setQ= 13, 4, 5, 7) and set R = (4, 5, 7), which of the following is true? A PCQ C RcP B QCP D RCQ 6question, choose only one answer.
10 Given the universal set ^ = {x : 40 x , 50, x is an integer) and set A = {x : x is a number such that the sum of its digits is an even number), all the elements of set A' are
Givens= {x :5-- x-- 20,x
A (40, 42, 44, 46, 48)
is an integer} and P = (x : x is B 141, 43 , 45, 47, 49)
a multiple of 5), find n(P). C (41, 43, 45, 47, 49, 50) A 4 C 10 D 140, 41 , 43, 45, 47, 49, 501 B 8 D 12 7 Given= {x :36 -- x<48,x is an integer) and M = (x : x is a number such that the digit at tens < the digit at ones), the elements of set M are A (36, 37, 38, 39) B (37, 38, 39,40)
C (36, 37, 38, 39, 45, 46,
47)
D (36, 37, 38, 39, 45, 46, 47,48)
8 The Venn diagram shows the universal set E and set P.
t
List the elements of set P'. A {a, d, k}
B {a, d, k, g) C {b,c,g,h}
D (a, b, c, d, g, h)
9 The number of subsets in a set which contains 4 elements is A 6 C 12 B 8 D 16
11 The Venn diagram shows the elements of sets P, Q and R.
P Q R
Given the universal set ^ = P U Q U R, then set Q'= A {1,
3)
B{1,
3,5)
C12,
4, 6) D11,
2, 3, 5,6)
•Subtopic 3.3
12 The Venn diagram shows the universal set ;; set P and set Q.
The set which represents the shaded region is
A PnQ C PuQ B (P n Q)' D (P U Q)'
13 Given set P = {4, 5, 6, 81 and set Q = {2, 3, 5, 6, 9). The elements of set P n Q are A (5, 6)
B (4, 5, 6)
C (4, 5, 6, 7, 8) D (2, 3, 4, 5, 6, 8, 9)
14 The Venn diagram shows the universal sett = P U Q U R.
P Q
Which of the regions, A, B, C or D, represents the set PnQnR'?
15 Given that the universal set k = F U G U H, G C F and Fn H # 0, the Venn diagram that represents these relationships is A C F F MH
G0
D G HO
0
16 The Venn diagram shows the universal set g= X U Y U Z.
CJ D
Which of the regions, A, B, C or D, represents the set x n Y' n Z'?
17 The Venn diagram shows the universal set k, set K and set L.
k
The region (K n L)' is
equivalent to the region A KnL' C KUL'
B K'nL' D K'UL'
18 The Venn diagram shows the universal set ^ = X U Y.
Set X ' n Y' is equivalent to set
A X C Y B 0 D Y' 19 In which of the following
Venn diagrams does the shaded region represent the set P'nQUR?
A P Q R
20. The Venn diagram shows all the elements of the universal set :;, set L, set M and set N. E
Find the elements of set (MUN)f1L. A {6) B {5, 81 C {5, 6,8) D (4,-5, 8, 9) 21 Given ^ = {x : 36 -- x -- 45, x is an integer), set A = {x : x is a multiple of 31 and set B = {x : x > 41). The elements of set A U B are A {39, 421 B (36, 39, 42, 451
C 136, 39, 41, 42, 43)
D {36, 39 , 42, 43, 44, 45)22 It is given that the universal set ^ = {x : 1 -- x -- 15, x is an integer), set K = {x : x is a prime number), set L = {x : x is a multiple of 3) and set M = {2, 4, 7, 8). The elements of the set (L U M)' n K are A 1, 11, 13
B 5, 11, 13 C 5, 13, 15 D 1, 5, 11, 13
23 The Venn diagram shows the number of the elements in set P, set Q and set R.
P
It is given that the universal setk=P U Q U R and n(R') = n(P U R). Find the value of x. A 3 C 7 B 4 D 8 24 Given t = (x: 6 -- x 15, x is an integer), set P = (x : x is a multiple of 3) and set Q = (x : x is an odd number). The elements of set (P n Q)' are
A {6, 7, 9, 15)
B 16,7,9,12,151 C {6, 7, 8 , 9, 10, 11, 121 D (6, 7, 8, 10, 11, 12, 13, 141 25 The Venn diagram shows the sets X, Y and Z, where k=XUYUZ.
The elements of set (X U Z)' are
A {2, 4) B (5, 6) C {7, 8)
26 The Venn diagram shows the sets P, Q and R, where ^=PUQUR.
Given E = {x : 11 --x -- 20,x is an integer ), Q = {11, 13, 15, 17, 19, 201 and R = (prime numbers ), the elements of set P U (Q n R)' are
A {11,13,17,19)
B {12,14 , 16,18)
C 112, 14, 16, 18, 201
D {12, 14 , 15, 16, 18, 201 27 Given the universal set
^=(x:9-- x20,xisan integer), set P = {x : x is a perfect square) and set Q = {x : x is a factor of 361, list all the elements of set P' n Q. A { 12, 18) B (9, 16, 18} C {9, 12, 16, 181
D 19, 12, 16, 18, 20)
28 Given ={x:10-- x-- 30, x is an integer), set P = {x : x is a number such that the sum of its digits is 31 and set Q = {x : x is a number such that the digit at tens < the digit at ones), find n(P' n Q).A 10 C 14 B 12 D 15
29 The Venn diagram shows the number of elements in the universal set i;, set P, set Q and set R.
E
30 The Venn diagram shows the universal set t, set P, set Q and set R.
k
Which of the following sets represents the shaded region? A PUQ UR
B Pn(QUR) C (PUQ)nR' D P'UQUR
31 The Venn diagram shows all the elements of the set universal ^, set P and set Q.
t
The elements of set (P U Q)' are
A (2,7} C (2,7,8}
B {6, 81 D {4, 5, 91 32 The Venn diagram shows the
number students in a class who belong to at least one of the three societies. Given the universal set = E U M U S, set E = (members of the
English Society}, set M= (members. of the
Mathematics Society), set S = (members of the
Science Society).
33 Given that the universal set g = X U Y U Z, where X = (c, e, p, a, t}, Y= (p, i, n, t, a, s) and Z = {c, e, r, d, a , s}, find n(XUYnZ'). A 6 C 4 B 5 D 3
34 The Venn diagram shows the universal set ^ = (Form 3 students), set M = (students who like the Mathematics subject) and set S = (students who like the Science subject).
9
Given that n(M) = 120, n(S) = 85, n(M fl S) = 36 and the number of students who do not like both subjects is 15, find the total number of Form 3 students.
A 148 C 220 B 184 D 256
35 The table shows the data obtained from a survey of 100 students. The Venn diagram represents the information given in the table.
Favourite Number of Drinks Students Tea 56 Coffee 55 Tea and coffee only 15 Tea and milk
only
8 Tea only 24 Coffee only 25 Milk Find n[(P U Q) n R'}. A 9 B 10 C 11 D 12If 22 students are members of at least two societies, find the value of y.
A 1 C 11 B 8 D 19
Tea
Find the number of students who like tea or coffee and also milk.
A 9 C 15 B 14 D 23
Coffee
SECTION B Subjective Questions
This section consists of 15 questions. Answer all the questions. You may use a non-programmable scientific calculator.
Subtopic 3.1
1 Given M = {factors of 361, fill in the boxes with the symbol E or (4.
Answer:
(a) 1 M (c) 9 E M
(b) 8 M (d) 72 M
2 State the number of elements in each of the following sets.
(a) J = (the names of days in a week which begin with the letter S)
(b) K = {prime factors of 1051
(c) L={x :x is a multiple of4and20-_ x 50}
(c)
4 Find the complement of set for the following pairs of sets.
(a) ^ = {the months in a year)
B = (the months that have 31 days) (b) E = (x : 50 , x , 65, x is an integer)
G = (x : x is a number such that the sum of its digits is an even number)
Answer:
(a)
(b)
Subtopic 3.2
3 Draw a Venn diagram to show the relationship of each pair of sets in the following.
(a) M = {d, u, r, i, a, n) N- {a, i, u}
(b) P = {x : x is a negative integer and -10 < x < -2}
Q = {-4, -6, -81
(c) U = {prime numbers which are less than 10} V = {1, 2, 3, 4, 5, 6, 71 Answer: (a) Subtopic 3.3 5 Given k = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 121, P = {multiples of 2), Q = {1, 2, 3, 4, 5}, R = {factors of 6).
List all the elements of the set (a) Q n R, (b) PnQnR, (c) (P n Q)' . Answer: (a)
(b)
(b)
(c)6 The Venn diagrams in the answer space shows the universal set i; = P U Q U R. On the diagram, shade the region which represents
(a) set P U R, (b) set (P fl R) U Q.
Answer:
(a) ^
P-^ Q
(b)
7 The Venn diagrams in.the answer space show the universal sett A U B U C. On the diagram, shade the region which represents
(a) set B U. C.
(b) set (A fl B)' U C.
Answer: (a)(b)
Subtopics 3.1 - 3.3
8 It is given that set K = ( 1, 3, 4), set L = (1, 2, 3, 4, 5) set M = (0, 1, 2, 3 , 4, 6, 8, 9) and the universal set = K U L U M.
(a) List all the elements of K fl M. (b) Find n (K' fl L).
Answer:
(a)
(b)
9 The Venn diagram shows the universal set , set P and set Q.
(a) Complete the statements in the answer space, using the symbol set fl, U or C to show the relationship between set P and set Q.
(b) State the set for Q fl P'.
(c) State the set which represents the shaded region. Answer: (a) (i) Q 0 P
(ii)
P a Q = Q(b)
(c)10 The Venn diagram shows all the elements of the universal set t, set X, set Y and set Z.
t
(a) List all the elements of (X fl Y) U (Y fl Z). (b) Find
(i) n[(X fl Y) U Z], (ii) n(X' fl Z').
Answer:
(a)
11 Given
t = {x : 60 -- x -- 80, x is an integer), P = {x : x is a prime number),
Q = {x : x is a number such that the product of its digits is an odd number),
R = {x : x is a multiple of 5).
(a) List all the elements of set P and set Q. (b) Find n(P U Q). (c) Find n(P' fl R). Answer: (a)
(b)
(c)12 Given that the universal set = P U Q U R, set P = ( s, o, n, a, r),
set Q = { s, o, n, i, c), set R = {t, o, n, i, c).
(a) List all the elements of set P fl R. (b) Find n (Q U R').
Answer:
(a)
(b)
13 Given that the universal set t = A U B U C, set C = {x : 10 -- x -- 25, x is an integer), set A = {x : x is a factor of 24),
set A U B = {x : x is a multiple of 3), ACBandBflC=0.
(a) List all the elements of set C.
(b) Find n (A U C)'.
Answer:
(a)
(b)
14 The Venn diagram shows the number of elements in sets J, K and L. Given the universal setk=1UKUL.
Find
(a) the value of y when n (K) = n(K' U L), (b) n(T n L),
(c) n(J U K fl L'). Answer:
(a)
(b)
15 The Venn diagram shows the relationship between the universal set k, set B and set C. Given universal set t = {Form 3 students), set B = (members of the Red Crescent Society), set C = {members of the Chess Society). Given n(^) = 200, n(B) = 80, n(C) = 110 and n(B fl C) = 30.
C
Find
(a) n(C),
(b) n(B U C),
(c) the number of students who are members of the Chess Society but are not members of the Red Crescent Society.
Answer:
(a)
(b)
4.1 Statements 4.4 Implications 4.2 Quantifiers "All" and "Some" 4.5 Arguments
4.3 Operations on Statements 4.6 Deduction and Induction
SECTION A
Objective Questions \.
This section consists of 15 questions . Answer all the questions . For each question, choose only one answer. You may use a non-programmable scientific calculator.
Subtopic 4.1
1 Which of the following is a statement? A 3+5 B C x+2=5 sin 60° - cos 30° D 62<12
2 Which of the following is a true statement? A B C -6 > -5 9 is a prime number. 16 is a perfect square. D 0.25 > 0.33
3 Which of the following is a false statement? A pis a consonant. B 6 is a factor of 258. C 2 is a prime number. D A trapezium is a regular polygon.
Subtopic 4.2
4 Which of the following is a false statement?
A Some odd numbers are prime numbers.
B All multiples of 6 are multiples of 3.
C Some perfect squares are negative numbers. D Some pyramids have a
square base.
5 Which of the following is a true statement?
A All even numbers are multiples of 4.
B Some birds have wings. C All fish eat meat. D Some polygons have 5
sides.
Subtopic 4.3
6 Which of the following is a true statement?
A 52=10 or 72=8=6. B 8 + 9 = 17 and 8 x 4 =32. C 5 is a factor of 20 and 7 is
a factor of 10.
D Hexagons have 8 sides and octagons have 9 sides. 7 Which of the following is a
true statement?
A (42)4 = 46 or 6 - (-5) = 1. B 4 = 0.75 and sin 60° = 0.5. C { } C {2, 3} or -4(-4) = 8. D A trapezium has four sides of equal length and two of its sides are parallel.
Subtopic 4.4
8 "If m = -7, then m2 = 49" - The antecedent in the above
implication is
A m=-7 C m2=-49 B m=7 D m2=49
9 Which of the following are implications for the sentence "h > k if and only if h + 5 > k+5."?
I If h > k, then h + 5 >
k + 5.
II Ifh <k,then5h>5k. III If h + 5 > k + 5, then h > k. 'A I only B III onlyC I and III only D II and III only
Subtopic 4.5
10Premise 1: If x > 10, then x2 > 100. Premise 2: 12 > 10 Form a conclusion based on the two given premises. A 122> 10 B 122> 100 C 144> 10 D 144 < 100 11 Premise 1 : If 4x < 20, then x < 5. Premise 2: ... Conclusion: 4x > 20
Premise 2 in the above argument is
A x<5 C x<20 B x>5 D x>20
12
Premise 1: All squares have four sides of equal length. Premise 2: EFGH is a
square.
Which of the following is the conclusion for the above argument?
A All squares have four sides of equal length.
B All quadrilaterals are EFGH.
C EFGH has four sides of equal length.
D EFGH has four right
angles.
13 Premise 1: All multiples
of 12 are multiples of 6. Premise 2: ...
Conclusion: 42 is a
multiple of 6.
Which of the following is Premise 2 in the argument given? A 42 is a multiple of 12. B 42 is a multiple of 18. C 72 is a multiple of 6. D 7 is a multiple of 12.
Subtopic 4.6
14 All acute angles are less
than 90°.
ZKLM is an acute angle. The conclusion by deduction for the argument above is A ZKLM = 900
B LKLM > 900 C LKLM < 900 D LKLM a 90°
15 Given a number sequence, 5, 14, 29, 50, ..., has the following pattern. 5=3(12)+2 14=3(22)+2 29 = 3(32) + 2 50 = 3(42) + 2
The general conclusion by induction for the number sequence is A n2, where n = 0, 1, 2, 3, ... B 3n2, where n = 1, 2, 3, 4,... C n2 + 2, where n = 0, 1, 2, 3,... D 3n2 + 2, where n = 1, 2, 3, 4,...
SICMN B Subjective Questions
This section consists of 25 questions. Answer all the questions . You may use a non-programmable scientific
calculator.
Subtopic 4.1
1 Determine whether each of the following is a statement.
(a) Help me!
(b) Kuantan is in Pahang. (c) (-3)3 = 27 Answer: (a) (b) (c)
2 Complete the following mathematical sentences by using the symbol > or < in the empty box to form
(a) a true statement, (b) a false statement.
Answer :
(a) -15
(b) 7
J -8
2 3
3 Construct true mathematical statements by using the following numbers and symbols.
(a) -12, -11 and > (b) 0.25, and < (c) {3, 6, 9}, 3 and E Answer: (a) (b) (c)
4 Determine whether each of the following statements is true or false.
(a) 47 is a perfect square.
(b) The highest common factor of 18 and 27 is 9.
Answer: 8 Combine the following pair of statements to (a) (c) form a true statement.
Statement 1: -8 x (-3) = 11 (b) Statement 2: 35 is a multiple of 7.
Subtopic 4.2
5 Determine whether each of the following statements is true or false.
(a) All even numbers are multiples of 4. (b) All factors of 12 are factors of 60. Answer:
6 Construct a true statement using the quantifier "all" or "some" based on the given object and property in each of the following.
(a) Object: regular polygons Property: 8 sides of equal length (b) Object: workers in a
factory-Property: wear spectacles (c) Object: triangles
Property: right angles
(d) Object: trapeziums
Property: two parallel sides
Answer:
Subtopic 4.4
9 State the antecedent and consequent in each of the following implications.
(a) If k = -4, then k3 = -64.
(b) If it rains today, then the football match will be cancelled.
Answer:
(a)
(b)
10 State the antecedent and consequent in each of the following implications.
(a) If Ismail 's father comes to school late, then Ismail will go home late.
(b) If > 8, then m > 82. Answer: (a) Answer: (a)
(b)
(c)(b)
(d)
Subtopic 4.3
7 Determine whether each of the following statements is true or false.
(a) A cat has four legs and a chicken has four legs.
(b) 32 + 52 > 42 and -0.43 < -0.34. (c) 30 is a multiple of 4 and 6. (d) 53= 125 or 25=9=3 Answer:
11 Write two implications from each of the following sentences.
(a) x > y if and only if 3x > 3y.
(b) p is a negative number if and only if p3 is a negative number. Answer: (a) Implication 1: Implication 2: (b) Implication 1: Implication 2:
Subtopic 4.5
12 Form a conclusion based on the following premises.
Premise 1: All students of the Form 5K class passed the SPM examination. Premise 2: Azlina is a student of the Form 5K
class.
Conclusion: ... Answer:
13 Complete the premise in the following argument: Premise 1 : All regular pentagons have five sides
of equal length.
Premise 2: ...
Conclusion: PQRST have five sides of equal length.
Answer:
14 Complete the conclusion in the following argument:
Premise 1: If 5x = 20, then x = 4. Premise 2: x # 4
Conclusion: ... Answer:
15 Complete the premise in the following argument: Premise 1: If a number is a factor of 18, then
the number is a factor of 54. Premise 2: ... Conclusion: 6 is a factor of 54.
Answer:
16 Complete the premise in the following argument:
Premise 1: ... Premise 2: p < 6
Conclusion : p + 4 < 10
Answer:
Subtopic 4.6
17 Form a conclusion by induction for the number sequence , 5, 17, 37, 65, ..., which follows the pattern: 5=4(12)+1 17 = 4(22) + 1 37 = 4(32) + 1 65 = 4(42) + 1 Answer:
18 Form a conclusion by induction for the number sequence, 0, 3, 8, 15, ..., which follows the pattern:
0=12-1 3=22-1 8=32-1 15=42- 1
Answer:
19 Form a conclusion by deduction for the following statements.
All parallelograms have opposite sides that are parallel. (general statement)
ABCD is a parallelogram. (specific case) Answer:
Subtopics 4.1- 4.6
20 (a)(b)
Complete the premise in the following argument:
Premise 1:If the length of each side of a square is x cm, then the area of the square is x2 cm.
Premise 2: ... Conclusion: The area of the square PQRS
is 25 cm2.
Write down two implications based on the following sentence. "pq>0ifandonly ifp>0andq>0." Answer: (a) Premise 2: (b) Implication 1: Implication 2:
21 (a) Complete the premise in the following argument:
Premise 1: ... 0 ... Premise 2: Jamal is not Juliana 's brother. Conclusion: Juliana is not a doctor. (b) Given (Xm)n = x mn, where m, n and x are
positive numbers. Find the value of
(i) (24)2, (ii ) (7 4 )8
Answer:
(a) Premise 1:
22 (a) Complete the conclusion in the following argument:
Premise 1: All triangles have the sum of interior angles of 180°.
Premise 2: ABC is a triangle.
... Conclusion: ...
(b) Form a conclusion by induction for the number sequence, 3, 10, 21, 36, ..., which follows the pattern:
3=2(12)+1 10=2(22)+2 21=2(32)+3 36 = 2(42) +4 Answer: (a) Conclusion:
(b)
23 (a) State whether the following statement is true or false.
5>3or21=6.
(b) Write down two implications based on the following sentence.
x'= 64 if only ifx=4.
(c) Complete the premise in the following argument:
Premise 1: All pentagons have five sides. Premise 2: ... Conclusion: ABCDE has five sides. Answer: (a) (b) Implication 1: 4 Implication 2: (c) Premise 2:
24 (a) State whether each of the following
statements is true or false. (i) 4x2= 8and52=10.
(ii) The elements of set P = ( 10, 15 , 20) are divisible by 5 or the elements of set Q = (1, 2, 3) are factors of 4.
(b) Write down Premise 2 to complete the following argument:
Premise 1 : If y is less than zero , then y is
a negative number.
Premise 2 : ... i ... Conclusion : -2 is a negative number. (c) Write down two implications based on the
following sentence.
5p > 20 if only if p > 4.
Answer:(a) (i) (ii)
(b) Premise 2:
(c) Implication 1:
Implication 2:
25 (a) Complete each statement in the answer space with the quantifier "all" or "some" so that it will become a true statement.
(b) State the converse of the following statement and hence, determine whether its converse is true or false.
Ifx>7,then x>4.
(c) Complete the premise in the following argument:
Premise 1 : If set P is a subset of set Q, then P fl Q = P.
Premise 2 : ... Conclusion : Set P is not a subset of set Q.
Answer:
(a) (i) ... of the multiples of 5 are even numbers.
(ii) ... hexagons have six sides.
(b)
(c) Premise 2:
5.1 Gradient of a Straight Line
5.2 Gradient of a Straight Line in Cartesian Coordinates
3 Given the gradient of a straight line which passes through points (1, 4) and (2, k) is -3. The value of k is A -7 C 0
B -i D 1
SECTION A
Objective Questions
This section consists of 25 questions. Answer all the questions. For each question, choose only one answer. You may use a non-programmable scientific calculator.
Subtopic 5.1
1 In the following, which straight line PQ has a gradient of 4 ? A Q B P D 5 cm Subtopic 5.2
2 The diagram shows a straight line AB on a Cartesian plane.
y B(6,3) ^ x A(-2, -5) f The gradient of AB is A -1 C 1 B 2 D 2
4 The diagram shows a straight line PQ on a Cartesian plane.
The gradient of PQ is
A -2
5.3 Intercepts
5.4 Equation of a Straight Line 5.5 Parallel Lines
C 1 2
B -2 D 2
5 In the diagram , PQR is a straight line on a Cartesian plane. y 0 The value of h is Al C 5 B 3 D 7 ^x
6 The coordinates of point Q are (-1, 2) and the gradient of the straight line QR is 3. The coordinates of point R could be
A (2, -11) C (2, 9) B (2, -9) D (2, 11) 7 In the diagram, 0 is the origin.
The straight line which has the largest gradient is
A PQ C TU
B RS D VW
Subtopic 5.3
8 In the diagram, PQ is a straight line on a Cartesian plane.
y 01 10 Qx The gradient of PQ is A -5 C 3 3 5 B -3 D 5 .5 3
9 In the diagram, FG is a straight line.
y
Cc 0 *x
What is the gradient of FG? A -3 C 1
3 B -3 D 3
10 In the diagram, RSTis a straight line on a Cartesian plane.
y \R(0, P) S(1, 0) ^x T(2, -3) The value of p is A 2 C 4 B 3 D 6
11 The gradient of a straight line VW is 3. If they-intercept of the straight line VW is 12, the x-intercept is
A -36 C 4 B -4 D 36
12 In the diagram, PQ is a straight line with the gradient
1 3
13 The diagram shows two straight lines, MN and NP, on a Cartesian plane.
y
M O
P(3, 3)
+x
The gradient of NP i s -3 and the distance of MN is 13 units. Find the x -intercept of MN. A -12 C _ 13
5 B -5 D -12
5 14 In the diagram , OPQR is a
trapezium . Given 2PQ = OR.
y R(0, 8)
.x OI P(6, 0) The x-intercept of the straight line QR is
A 8 C 12 B 10 D 16
Subtopic 5.4
15 Which of the following points does not lie on the straight line y = 3x - 5?
A (-2, -1) C (0, -5) B (-1, -8) D (1, -2) 16 The equation of a straight line
which has a gradient of and passes through point (0, -3) is
A y= 2x-2
B y= 2x+3
C 2y=x-6
D 2y=x-3
18 The gradient of the straight line 3x-5y= 15 is
A -3 C 3 5
B - 3 D 5
19 Find the y-intercept of the straight line 2x - 5y = 20. A -5 C 2
5 B -4 D 10
20 In the diagram , the straight line HK intersects the straight line KL at K.
y
OI L(4, 0)x
The equation of the straight line KL is 3 A y=-4x3 3 B y=-4x+3 C y=-12x-2 D y=-21x+2
Subtopic 5.5
21 Which of the following pairs of straight lines are parallel? A y=5+1 y=0.4x+2 B y=-2x+1 2y=4x+2 C 3y+9x=-3 3x - y = 9 D x + Y =5 2 3 2y + 3x = 2
22 Given the straight line y = 17 The equation of a straight line mx - 5 is parallel to the Find the x-intercept of the which passes through points straight line 4x + 6y = 8. The
straight line PQ.
(-3, -3) and (4, 11) isvalue of m is
A -12 C 4 A y=-2x-3 A -3 C 2
3 B y=-2x+3 2 3
B -4 D _ 1 C y=2x-3 B -2 D 3
12 D y=2x+3 3 2
23 In the diagram, OKLM is a parallelogram. Given the gradient of the straight line OK is 3. y O 24 In the diagram, PQRS is a parallelogram and RST is a straight line.
The value of t is The coordinates of point Tare
A 5 C 7 A (-12, 0) C (0, -6)
B 6 D 8 B (-6, 0) D (0, -12)
25 Which of the following straight lines is parallel to the straight line 3y = x + 6 and passes through point (-6, -4)? A y= 3+2
B y= Z+3 C 3y=x-6 D 3y=x-3
SECTION B Subjective Questionsy
This section consists of 20 questions. Answer all the questions. You may use a non-programmable scientific calculator.
Subtopic 5.1
Im,
Based on the diagram, find the gradient of the straight line MN.
Answer:
Subtopic 5.2
2 Find the gradient of the straight line that passes through points (-4, 2) and (-8, 6).
Answer:
Subtopic 5.3
4
Based on the diagram, state (a) the x-intercept,
(b) the y-intercept, of the straight line RS. Answer:
(a)
(b)
5 A straight line PQ intersects the y-axis at point R. If the x-intercept of the straight line PQ is 4 and its gradient is -2, find the coordinates of point R. Answer:
3 Given points (1, -7), (4, k) and (6, 4) lie on a straight line. Find the value of k.
Subtopic 5.4
6 On the diagram in the answer space, draw the straight line y = -ix + 1.
Answer:
7 Find the points of intersection of the following pairs of straight lines by solving the simultaneous linear equations.
(a) y=x+5 (b) 3x+2y=12 y= 1x+4 2x-y=1 Answer:
8 Find the equation of a straight line which passes through each of the following pairs of points. (a) (0, -7) and (3, 2)
(b) (-2, 4) and (-8, 1) Answer:
Subtopic 5.5
9 Determine whether each of the following pairs of straight lines are parallel.
(a) y=x-3 (b) 2x-5y=1 3y=x-6 5y=2x+3
Answer:
10 Find the equation of the straight line which is parallel to y = 4 - 6x and passes through point (-1, 1). Answer:
Subtopics 5.1 - 5.5
11 In the diagram, the
straight line MN is parallel to the x-axis and the length of OK is 2 units.
Find
(a) the y-intercept of the straight line MN, (b) the gradient of the straight line KN. Answer:
(a)
(b)
12 The diagram shows a rectangle PQRS drawn on a Cartesian plane.
y
(a) Calculate the gradient of the straight line PR. (b) Find the y-intercept of the straight line QS.
P(-1, 0) 0
Answer:
(a)
(b)
In the diagram, the gradient of the straight line KLM is - 2. Find (a) the value of p,
x (b) the x-intercept of
M the straight line MN.
13
(a) (b) I Answer:
14 Answer: (a) (a) 16 Answer: (a) 17 Answer:
y In the diagram, 0 is the
origin. The gradient of the straight line ST is 2 Find
(a) the gradient of the
18 y
O ^ x
straight line ROS,
(a)
(b) the y-intercept of
H(10 -6)
the straight line ST. , (b)
(b)
In the diagram, 0 is the origin. OPQR is a parallelogram. Find (a) the equation of the
straight line PQ, (b) the coordinates of .x point Q.
(b)
In the diagram, the straight line PQ is parallel to the straight line OR. Find
(a) the gradient of the straight line OR, (b) the y-intercept of
the straight line QR.
(b)
In the diagram, OPQR is a parallelogram and 0 is the origin. Find (a) the equation of the
straight line QR. (b) the y-intercept of
the straight line PQ.
Answer:
(a)
In the diagram , EF, FG and GH are straight lines. OE is parallel to FG and EF is parallel to GH. Given the equation of EF is 2x+y=6.
State the equation of the straight line FG. Find the equation of the straight line GH and hence, state its y-intercept.
(b)
19 In the diagram, 0 is the origin. Q lies on the x-axis and P lies on the y-axis. The straight line PT is parallel to the x-axis and the straight line PQ is parallel to the straight line RS. The equation of the straight PQ is x + 2y = 10.
y
(a) State the equation of the straight line PT. (b) Find the equation of the straight line RS
and hence, state its x-intercept. Answer: (a) 20 E(0, 6) G M(2, 10) y 0 F(3, 0) Answer: (a)
(b)
(a)
S(4, -10) N .x(b)
The diagram shows a straight line EF and a straight line MN drawn on a Cartesian plane. EF is parallel to MN. Find
(a) the equation of the straight line MN, (b) the x-intercept of
the straight line MN.
6.1 Class Intervals
6.2 Mode and Mean of Grouped Data 6.3 Histograms
6.4 Frequency Polygons 6.5 Cumulative Frequency 6.6 Measures of Dispersion
SECTION A
Objective Questions
This section consists of 16 questions. Answer all the questions. For each question, choose only one answer. You may use a non-programmable scientific calculator.
Subtopic 6.1
Age Group (years 11-20 21-30 31 -40 41-50
Questions 4 and 5 are based on the following frequency table.
Score Frequency
1-10 5
11 - 20 15
21 - 30 10
31-40 8
Based on the frequency table, the lower limit of the class interval 31 - 40 is
A 30.5 C 39.5 B 31 D 40
Length of Rope (m)
0.4-0.8
4 Calculate the size of the class interval in the table.
A 7 C 9 B 8 D 10
5 State the lower boundary of the class interval 11 - 20.
7 The frequency table shows the heights of 30 students in a class. Height (cm) Frequency 101 - 110 2 111 - 120 5 121-130 6 131 - 140 9 141-150 4 151-160 3 161 - 170 1 A 10 C 11
The midpoint of the modal
B 10.5 D 11.5
class is
0.9 - 1.3
1.9-2.3
In the table, the missing class interval is
A 1.3 - 1.8 B 1.3 - 1.9 C 1.4 - 1.8 D 1.4 - 1.9
3 The largest value in a data is 38 and the smallest value is 12. If the number of class intervals required is 6, then the suitable size of the class interval is
A 4 C 6 B 5 D 7
Subtopic 6.2
6 The frequency table shows the lengths of 30 ribbons. Length (cm) Frequency 40 - 44 3 45 - 49 8 50 - 54 10 55-59 9
The modal class is A 45 - 49 B 49.5 - 54.5 C 50 - 54 D 50.5 - 54.5 A 135 C 136 B 135 .5 D 136.5 8 The frequency table shows the
marks obtained by 20 students in a game. Marks Frequency 11 - 15 3 16-20 5 21 - 25 6 26-30 4 31-35 2
Calculate the mean mark. A 20.55 C 21.65 B 21.55 D 22.25
9 The frequency table shows the masses of 20 baskets of mangoes collected by a farmer. Mass of Mangoes ( kg) Frequency 9-11 3 12-14 6 15-17 5 18-20 4 21-23 2
Calculate the mean mass, in kg, of a baseket of mangoes. A 14.4 C 16.4 B 15.4 D 30.8
Subtopic 6.3
Questions 10 and 11 are based on the following histogram. The histogram shows the distribution of the ages of a group of participants in a competition.
0
0
kr L0 0 0
IqAge (years) ,
10 The mean age, in years, of the group is
A 32.0 B 36.8 C 37.3 D 38.6
11 The percentage of the number of participants who are more than 40 years old is
A 17 B 20 C 32 D 34
Subtopic 6.4
Questions 12 and 13 are based on the following frequency polygon.
The frequency polygon shows the thickness of 100 books in a library.
Thickness of book (mm)
12 The modal class is A 10.5 - 15.5 B 11 - 15 C 12 - 15 D 12.5 - 15.5
them are written in Malay and the rest are in English. The number of English books with a thickness of between 5 mm and 20 mm is
A 32 C 48 B 40 D 60
Subtopic 6.5
14 Which Of the following is not a step to draw an ogive? A Add one class interval
with a cumulative frequency of 0 before the first class interval. B Find the upper boundary
of each class interval. C Find the cumulative
frequency of each class interval.
D Plot the graph of frequency against the upper boundary.
15 The table shows the distribution of the masses of sugar sold in a market on a certain day.
Mass of
Sugar (kg) 1 2 3 4 5
Frequency 6 12 10 18 2
Based on the table above, a cumulative frequency table is constructed as follows: Mass of Sugar (kg) Upper Boundary Cumulative Frequency 0 0.5 0 1 1.5 6 2 2.5 18 3 3.5 x 4 4.5 46 5 5.5 48 The value of x is
Subtopic 6.6
16 The ogive shows the total donation that is collected from 100 donors.
Cumulative frequency
The first quartile is A 25
B 30 C 35 D 40
10 13 Of the total number of books
B 22 with a thickness of between
C 28 5 mm and 20 mm each, 5 of D 30
SECTION B Subjective Questions'
This section consists of 16 questions. Answer all the questions. You may use a non-programmable scientific calculator.
Subtopic 6.1
1 The data shows the masses, in kg, of 40 boxes that are shipped by a transport company.
Subtopic 6.2
3 The data shows the circumferences, in cm, of 30 rubber tree trunks that are sent to a factory to be processed to produce furniture wood.
68 42 67 55 56 44 53 56 11 52 45 28 48 26 35 45 140 165 152 153 157 160 59 43 25 46 52 24 44 37 148 143 133 146 127 157 34 61 20 33 42 18 32 57 135 140 122 142 151 161 22 60 36 58 23 48 35 47 128- 147 137 142 163 131 139 149 141 145 162 150 Construct a frequency table for the data by using
the class intervals, 11 - 20, 21 - 30, 31 40 and so on. State the size of the class interval. Answer:
2 The data shows the time, in minutes, required by 25 students to solve the mathematical problems in a set of questions. 8 24 15 17 28 7 12 8 11 14 16 12 16 7 21 11 10 13 14 8 26 27 18 30 20
(a) Construct a grouped frequency table for the data by using the class intervals, 120 - 129, 130 - 139, 140 - 149 and so on.
(b) (i) State the size of the class interval. (ii) State the modal class.
(iii) Calculate the mean circumference, in cm, of the trunks.
Answer: (a)
4 The frequency table shows the distribution of the masses of _40 watermelons harvested by a farmer.
Construct a frequency table for the data by using the class intervals, 6 - 10, 11 - 15, 16 - 20 and so on. Answer: Mass (kg) Frequency 1.5- 1.9 6 2.0-2.4 10 2.5 - 2.9 12 3.0-3.4 8 3.5 -3.9 4
(a) State the modal class.
(b) Calculate the mean mass, in kg. Answer: (a)
Subtopic 6.3
5 The frequency table shows the distribution of the periods of complete oscillation of 55 pendulums. Period of Oscillation (minutes ) Frequency 2.5 -2.9 6 3.0-3.4 5 3.5 - 3.9 18 4.0-4.4 11 4.5 - 4.9 10 5.0 -5.4 5
Based on the frequency table, draw a histogram. 6 The frequency table shows the heights of 100
students in a school. Height ( cm) Frequency 120- 124 8 125 - 129 10 130 - 134 22 135 - 139 13 140 - 144 12 145 - 149 20 150 - 154 15
Based on the frequency table, draw a histogram.
Subtopic 6.4
7 The frequency table shows the distances travelled by a group of students to school.
Distance ( km) Frequency 31 - 40 7 41-50 8 51 - 60 12 61 - 70 16 71-80 9 81 - 90 5 91-100 3
8 The frequency table shows the masses of the baskets of prawns transported by a fisherman's boat.
Mass (kg) Frequency 12 - 15 5 16 - 19 7 20-23 8 24-27 12 28-31 4 32-35 3 36 - 39 6
Construct a frequency polygon based on the frequency table.
Subtopic 6.5
9 Complete the cumulative frequency table below. Distance (km) Frequency Cumulative Frequency Upper Boundary 3-5 4 6-8, 5 9-11 6 12 - 14 10 15 - 17 8 18-20 2
Subtopic 6.6
10 The frequency table shows the distribution of the masses of 45 watermelons in a stall.
Mass (kg) Frequency 1.6 - 2.0 5 2.1 - 2.5 8 2.6 -3.0 9 3.1 - 3.5 12 3.6 -4.0 7 4.1 -4.5 4
(a) Draw an ogive based on the data given. (b) From the ogive, find
(a) Draw a histogram. (i) the median,
(b) Construct a frequency polygon based on the (ii) the first quartile,
Subtopics 6.1- 6.6
11 The frequency table shows the lengths of 42 pieces of ribbon used to tie presents.
Length (cm) Frequency 61-63 2 64-66 3 67 - 69 8 70- 72 10 73-75 7 76- 78 6 79 - 81 5 82-84 1
(a) Draw an ogive based on the data given. (b) From the ogive, find
(i) the median, (ii) the first quartile, (iii) the third quartile.
Answer: (b) (i) (ii) (iii)
12 The table shows the scores obtained by a group of shooters in a shooting competition.
Score Frequency x Score
10 30 12 60 14 x 16 48 18 36 20 20
If the total frequency is 20, find the value of x. Answer:
13 The data shows the number of foreign workers employed by 40 factories in an industrial area.
2 4 34 22 25 33 15 43 58 2 42 2 1 28 65 48 28 11 29 37 65 28 5 0 13 32 17 48 17 16 35 38 56 1 3 3 23 5 28 44 54 68 22 34 57
(a) Based in the data and using a class interval of 10, complete the table in the answer space. (b) Based on the table in (a),
(i) state the modal class,
(ii) calculate the estimated mean number of foreign workers in each factory. (c) For this part of the question, use graph paper.
Using a scale of 2 cm to 10 foreign workers on the horizontal axis and 2 cm to 5 factories on the vertical axis, draw an ogive for the data.
From the ogive, find (i) the median, (ii) the third quartile. Answer:
(a)
Class Frequency Mid- Cumulative Interval point Frequency
1 - 10
(b) (i)
Upper Boundary
14 The data shows the volume, in me, of water that is collected in each bottle by 50 participants in a telematch. 142 160 152 145 146 156 151 150 142 153 142 148 154 141 152 149 152 138 147 149 151 151 137 155 151 140 153 141 138 139 154 144 147 154 145 157 158 137 140 151 146 158 163 164 157 146 152 153 157 152
(a) Construct a grouped frequency table for the data by using the class intervals, 130 - 134, 135 - 139 and so on.
(b) For this part of the question, use graph paper.
Using a scale of 2 cm to 5 me on the horizontal axis and 2 cm to 5 participants on the vertical axis, draw an ogive for the data.
(c) From the ogive, find (i) the median,
(ii) the interquartile range. Answer:
15 The data shows the marks scored by 36 students in a monthly mathematics test.
51 65 58 62 68 48 60 68 59 61 48 56 63 55 44 59 54 40 60 57 49 45 58 46 67 62 64 69 61 50 55 66 56 65 58 64
(a) Using the data and a class interval of 5 marks, complete the table in the answer space.
(b) Based on the table in (a), (i) state the modal class,
(ii) calculate the mean score of the group and give the answer correct to 2 decimal places.
(c) For this part of the question, use graph paper.
Using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 1 student on the vertical axis, draw a histogram for the data.
Answer:
(a) Marks Midpoint Frequency
40 - 44 42 45-49 50-54 55-59 60 - 64 65-69
16 The data shows the monthly savings, in RM, of 40 students. 46 53 44 60 42 38 41 55 45 37 54 32 46 56 40 60 52 40 34 45 52 35 50 36 47 38 40 48 45 42 53 44 50 44 58 51 36 48 56 32
(a) Based on the data and using a class interval of 5, complete the table in the answer space. (b) Based on the table in (a), calculate the estimated mean of the monthly savings of each student.
(c) For this part of the question, use graph paper. Using a scale of 2 cm to RM5 on the horizontal axis and 2 cm to 1 student on the vertical axis, draw a frequency polygon to represent the data.
(d) Based on the frequency polygon in (c), state one piece of information about the montly savings.
Answer:
(a)
(b)
(d)
Class Interval Midpoint Frequency
31 - 35 36-40
