1 SECURE AND OUTSTANDING FOR TERTIARY ADMISSION LOCALLY OR ABROAD Erected by : KHAIRIL ANUAR BIN MOHD RAZALI Enhanced, refined and continued by :
Sep 13, 2014
1
SECURE AND OUTSTANDING FOR TERTIARY
ADMISSION LOCALLY OR ABROAD
Erected by :
KHAIRIL ANUAR BIN MOHD RAZALI
Enhanced, refined and continued by :
2
SPM MATHEMATICS - PAPER 1 (1449 / 1)
A GENERAL GUIDE - Paper 1
1. Paper 1 SPM Mathematics covers selected topics from Form 1 to 3, all topics in Form 4 and 5, and requires BASIC,
INTERMEDIATE and HIGHER skills.
2. Topics in this paper covers the NUMBERS, SHAPES & SPACE and ALGEBRAIC themes.
3. Skills connected to the NUMBERS, SHAPES & SPACE and ALGEBRAIC themes sometimes complement each
other; without any one of the skill, others CAN’T BE acquired. Questions on SHAPES require a lot of ALGEBRAIC
and NUMBER skills whilst questions on ALGEBRA require skills on NUMBERS or vice versa.
4. From the above explaination, it is clear that skills on NUMBERS should be built first whilst skills on ALGEBRA are an
important tool to solve many problems.
5. Scope of questions covers topics that have been taught from Form 1 to 5 (please refer to topics analysis in Part C to get a
clear picture of topics posted)
NUMBERS
SHAPES AND SPACES
ALGEBRA
1. Whole Numbers
2. Fractions
3. Decimals
4. Percentages
5. Directed Numbers
6. Multiples & Factors
7. Squares, Square Roots,
Cubes, Cube Roots
8. Standard Form
9. Number Bases
1. Basic Measurements
2. Lines and Angles
3. Polygons
4. Perimeter and Area
5. Geometrical Constructions
6. Loci In Two Dimensions
7. Circles
8. Solid Geometry
9. Pythagoras‟ Theorem
10. Trigonometry
11. Bearings
12. Angles Of Elevation & Depression
13. Lines & Planes In 3-Dimentions
14. Plans And Elevations
15. Earth As A Sphere
16. Transformations
1. Indices
2. Algebraic Expressions
3. Algebraic Formulae
4. Linear Equations
5. Linear Inequalities
6. Quadratic Exp. & Equations
7. Coordinates
8. The Straight Line
9. Graphs Of Functions
10. Gradient & Area Under A Graph
11. Nisbah dan Kadar
12. Variations
13. Matrices
14. Sets
15. Mathematical Reasoning
16. Statistics
17. Probability
B EXAMINATION FORMAT – Paper 1
NO
ITEM
NOTES / DISCRIPTION
1 Type Of Instrument Objective Test
2 Type Of Item Multiple Choice
3 Number Of Question 40 questions (Answer all)
4 Total Marks 40
5 Test Duration 1 hour 15 minutes
6 Constructual Inclination Knowledge - 45 % / Skill - 55 %
7 Contextual Coverage 1. Lower secondaries field of studies that have
continuity at higher secondary.
2. All field of studies from form 4 to 5.
8 Level of Difficulty Easy : Moderate : Difficult = 5 : 3 : 2
9 Additional Tools 1. Scientific Calculators
2. Mathematical Tables Book
3
3. Geometrical Equipment
C ANALYSIS – Paper 1
TOPICS 2003 2004 2005 2006 2007 2008 2009
2010
2011
FORM 1 – 3 1. Polygons I and II
2. Algebraic Expressions
3. Linear Equations
4. Algebraic Formulae
5. Statistics I and II
6. Transformations I and II
7. Indices
8. Linear Inequalities
9. Trigonometry I
2
2
1
1
2
3
2
2
-
1
2
1
1
2
2
2
2
1
2
2
1
1
1
2
2
1
-
1
2
1
1
3
2
1
1
1
2
2
1
1
2
2
2
1
-
1
2
1
1
2
2
2
1
-
2
2
1
1
3
2
2
2
-
TOTAL 15 14 12 13 13 12 15
FORM 4 1. Standard Form
2. Quadratic Expr. & Equations
3. Sets
4. Mathematical Reasoning
5. The Straight Line
6. Statistics III
7. Probability I
8. Circles III
9. Trigonometry II
1100.. AAnngglleess ooff EElleevvaattiioonn && DDeepprreessss..
11. Lines & Planes in 3-Dimension
4
-
3
-
1
-
2
1
2
1
1
4
-
3
-
2
-
3
1
2
2
1
3
-
3
-
2
1
2
1
3
1
1
4
-
3
-
2
-
2
1
2
2
1
3
-
3
-
2
-
2
1
3
1
1
4
-
3
-
2
-
2
1
3
2
1
4
-
2
-
2
-
2
1
2
2
1
TOTAL 15 18 17 17 16 18 16
FORM 5 1. Number Bases
2. Graphs of Functions II
33.. TTrraannssffoorrmmaattiioonnss IIIIII
4. Matrices
55.. VVaarriiaattiioonnss
66 GGrraaddiieenntt//AArreeaa UUnnddeerr GGrraapphhss
77.. PPrroobbaabbiilliittyy IIII
88.. BBeeaarriinnggss
99.. EEaarrtthh AAss AA SSpphheerree
10. PPllaannss AAnndd EElleevvaattiioonnss
2
1
-
3
2
-
-
1
1
-
2
1
-
1
2
-
-
1
1
-
2
1
-
2
3
-
-
1
2
-
2
1
-
2
3
-
-
1
1
-
2
1
-
2
3
-
-
1
2
-
2
1
-
2
3
-
-
1
1
-
2
1
-
2
2
-
-
1
1
-
TOTAL 10 8 11 10 11 10 9
☛ Bold printed topics are topics that are either not included in Paper 1 or topics that are seldom asked.
☛ Majority of questions are from Upper Secondary. 40 % Lower Secondary and 60 % Upper
Secondary
☛ Questions posted varies in terms of difficilty. There are simple and basic questions that touch only
the surface of a topic while others goes deeper and need higher skills.
☛ Familiarise yourself with the use of a scientific calculator. Fumbling with a calculator may invite
4
unwanted results.
D ANSWERING GUIDE – Paper 1 1. Paper 1 usually begins with simple and easy questions.
2. If any can‟t be answered, move to other questions and don‟t waste time an any one question.
3. For questions that involve Squares, Square Roots or other table readings, usually examples on how
to use the table are shown.
4. Table below shows instructions word in questions and what should be done.
QUESTION
INSTRUCTION
WHAT SHOULD BE
DONE
QUESTION
EXAMPLE
CALCULATE /
EVALUATE / FIND
You have to do calculation using formulae,
guide, theorem or law.
Calculate the probability of getting a
male student.
EXPRESS You have to give answer in the form requested. Express your answer in standard
form.
WRITE / STATE Write down answer without showing any working
method.
State the angle for this rotation.
ROUND You have to give answer to the nearest value. Round 0.0218 to two significant
figure.
SIMPLIFY You have to write certain expression in the
simplest form
Simplify m 2 – (k – m)
2
k
FACTORISE You have to write again expression in the form of
product of factors.
Factorise completely
4h2k – hk.
SOLVE Seeking the value of variable in a certain
equation.
Solve the equation f + 5 = 8
5. It is very important for candidates to study past years questions and try to answer them according to
the time and rules set. This will give us a clear picture of the form of question that will be given,
skills that must be grasp and topics that must be given priority.
6. Don‟t be too dependent on a certain method or skill to solve problems. Try to variate your technique
and skill.
7. THE MORE EXERCISE, THE BETTER METHOD OF SOLVING WE USE AND THE
FASTER WE SOLVE EXAMINATION QUESTIONS THAT HAVE THE SAME FORMAT
EACH YEAR .
E FORMS OF QUESTION – Paper 1
1. “COMMON SENSE” QUESTIONS (NEEDS NO CALCULATION)
EXAMPLE 1 :
A
B
P C
D
The diagram shows four lines drawn on a square grid. Which of the lines has a gradient of 2 ?
5
A. PA B. PB C. PC D. PD
2. QUESTIONS THAT CAN BE ANSWERED USING OTHER QUESTION BEFORE OR
AFTER IT AS A GUIDE
Example 2 below can be answered using Example 3 as a guide.
EXAMPLE 2 : EXAMPLE 3 :
Amir Amsyar bought a pair of pants Solve the following equation
at a price of RM 42 after discount. 60 – x (60) = 42
The original price is RM 60. Calculate 100
the percentage of discount given.
A. 20 % C. 30 % A. – 20 C. 45
B. 25 % D. 35 % B. 30 D. - 32
3. QUESTIONS THAT CAN BE ANSWERED BY TRYING OUT EACH CHOICE GIVEN (If possible,
try not using this method because it is time consuming)
EEXXAAMMPPLLEE 44 ::
A bag contains 624 balls which are either orange, purple
or white. If a ball is picked randomly from the bag, the
probability of picking a white ball is 3 . Find the number
8
of white balls in the bag.
A. 234 C. 324
B. 243 D. 423
4. QUESTIONS THAT CAN BE SOLVED USING ALGEBRAIC METHOD
EXAMPLE 5 :
The interior angles of a hexagon
are 2xo, 2x
o, 3x
o, 3x
o, 4x
o dan 4x
o.
The value of x is
A. 40o C. 80
o
B. 70o D. 90
o
EXAMPLE 6 :
In the following diagram, calculate the height of
the cylinder, h, given surface area of the cylinder
is 330 cm2 and its radius is 3.5 cm. r
A. 11.5 cm C. 15 cm B. 13.25 cm D. 26.5 cm h
EXAMPLE 7
Given M (k, 2) is the mid point for the
line that connects points P (-8, a) and
Q (2a, a). The value of k is
Try whether 234 is equal to 3 . If not, repeat with
624 8
other choices. (If possible make a RANDOM
choice because we might succeed at first try)
Form the equation
2x + 2x + 3x + 3x + 4x + 4x = 4(180)
and solve the equation.
Form the equation
2π(3.5)2 + 2π(3.5)h = 330 and sol
solve the equation. (Subtitute π =
22 / 7)
Form the simultaneous equation
a + a = 2 dan – 8 + 2a = k
2 2
and solve them.
A better and quicker method here is using
tthe algebraic method i.e by forming the
equation x = 3 and solving it.
624 8
6
A. 2 B. 3 C. – 2 D. – 3
5. QUESTIONS THAT HAD TO BE GUESSED
Before guessing, eliminate all the distractors first.
EXAMPLE 8 :
1. 6.27 x 10 –4
=
A. 0.0000627
B. 62700 Not possible because this is a big number!
C. 0.000627
D. 6270000 Not possible because this is a big number!
6. TRY THE FOLLOWING QUESTIONS INVOLVING NUMBERS, SHAPES & ALGEBRA :
1. Round 40450 to three
significant figure
A. 404 C. 40400
B. 405 D. 40500
2. 2.4 x 10 5 + 4.8 x 10
4 =
A. 7.2 x 10 9 C. 2.88 x 10
5
B. 2.88 x 10 9 D. 2.88 x 10
4
3. 1011101 2 – 10110 2 =
A. 10001 2 C. 10101 2
B. 10111 2 D. 11111 2
4. 3.47 x 10 3 =
A. 0.0034 C. 347
B. 34.7 D. 3470
5. The area of a square is 1.54 m2.
Its width is 250 cm. Find its
length in cm
A. 1.29 x 102 C. 6.16 x 10
1
B. 6.16 x 10-1
D. 6.16 x 103
6. If x + 2 = 3x then x =
5
A. – 1 / 3 C. – 5
B. – 2 / 5 D. 1
N
7. G H
S
G and H are two points on the
parallel of laltitude 72oN. Find
the shortest distance, in nautical
miles, between point G and H.
A. 1080 C. 4320
B. 2160 D. 8640
8. Factorise 6pq – 4q2
A. 2q(4p – 4q) C. 6p(p – 4q2)
B. 6q(p – 4q2) D. 2q(3p – 2q)
9. Factorise completely 2x2 - 8
A. 2(x2 – 4) B. 2(x – 2)
2
C. (x – 2)(x + 4)
D. 2(x – 2)(x + 2)
10. (4 – 3p)(2 + 5p) =
A. 8 + 26p – 15p2
B. 8 – 26p – 15p2
C. 8 + 14p – 15p2
D. 8 – 14p – 15p2
11. Express 2r _ r as a
k + 1 k
fraction in its lowest term
A. r(k – 1) C. r
k(k + 1) k
B. rk + r D. r .
k(k + 1) k + 1
12. Given w = 3a + 2b
a
then a =
A. 2b C. w – 2b
w – 3 3
B. 2b D. w
w + 3 6b
13.
Q is the image of triangle P
under a rotation. The
coordinates of the centre of
rotation are
Q
14. 60o
P 20o
R
R is due south of Q. The bearing
of P from R is
A. 080o C. 240
o
B. 100o D. 260
o
15.
☻denotes x students
Piktograph shows number of
students in class 3A. Find
value of x if total number of
students is 35 people
A. 4 C. 5
B. 6 D. 3
Chinese ☻☻☻☻
Indian ☻☻
Malay ☻☻☻☻☻
6
4
2
-2 0 2 4 6
Q P
7
A. (5, 1) C. (4, 2)
B. (2, 4) D. (0, 5)
SPM MATHEMATICS
PAPER 2 (1449/2)
A GENERAL GUIDE– Paper 2
1. Paper 2 SPM Mathematics contains two parts; Part A and Part B.
2. Test is in the form of written subjective and answers must be written in the question paper.
3. Questions are in the form of subjective and needs longer working method.
4. Scope of question covers certain particular topics from form 1 to form 5, different from Paper 1 that
has a wider coverage.
B EXAMINATION FORMAT – Paper 2
NO
ITEM
NOTES / DESCRIPTIONS
1
Type of Instrument
Subjective Test
2
Type of Item
Structure and Limited Response
3
Number of Question
Part A
11 questions (Answer all)
Part B
5 questions (Choose 4)
4
Total Marks
Part A : 52 marks
Part B : 48 marks (1 question 12 marks)
5
Test Duration
2 hours 30 minutes
6
Constructual Inclination
Knowledge - 25 %
Skill - 70 %
Value - 5 %
7
Contextual Coverage
☛ Lower secondary learning scope that has
continuity in upper secondary.
☛ All learning scope from Form 4 and 5.
8
Difficulty Level ☛ Easy (E)
☛ Moderate (M)
☛ Difficult (D)
E : M : D = 5 : 3 : 2
9
Additional Tools
☛ Scientific Calculator
☛ Mathematical Tables Book
8
☛ Geometrical Equipment
C GENERAL INSTRUCTION – Paper 2
1. Candidates must answer ALL 11 questions in Part A and 4 out of 5 questions in Part B (if more than
4 are answered, only 4 questions with the highest mark will be chosen).
2. Candidates can use a normal scientific calculator.
3. Candidates will be supplied with four digit tables book, graph papers, blank papers.
4. Final answer that involves decimals must be given correct to two decimal places.
5. Though not stated, candidates also have to bring along drawing tools like long rulers, geometry sets,
“flexi curve” and other tools thought to be useful.
D ANALYSIS – Paper 2
TOPICS
PART A
‟03 ‟04 ‟05 ‟06 ‟07 „08 ‟09 „10
PART B
‟03 ‟04 ‟05 ‟06 ‟07 ‟08 ‟09 „10
FORM 1 – 3 1. Simultaneous Linear Equations
2. Cicles (II)
3. Volume/Surface Area of Solids
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
TOTAL 3 3 3 3 3 3 3
FORM 4 1. Standard Form
2. Quadratic Expr & Equations
3. Sets
4. Mathematical Reasoning
5. The Straight Line
6. Statistics III
7. Probability I
8. Cicles III
9. Trigonometry II
10. Angl. of Elevation & Depress.
11. Lines & Planes in 3-Dimension
1 1 1 1 1 1 1
- 1 - 1 - 1 -
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
TOTAL 4 5 4 5 4 5 4 1 1 1 1 1 1 1
FORM 5 1. Number Bases
2. Graphs of Functions II
3. Transformations III
4. Matrices
5. Variations
6. Gradient/Area Under Graphs
7. Probability II
8. Bearings
9. Earth As A Sphere
10. Plans and Elevations
1 - 1 - 1 - 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
TOTAL 4 3 4 3 4 3 4 4 4 4 4 4 4 4
OVERALL TOTAL 11 11 11 11 11 11 11 5 5 5 5 5 5 5
☛ Topics from form 4 and 5 forms major questions.
☛ Candidates can make suitable choice of question in Part B and this can be done by looking at your skill and ability.
Teachers‟ and friends‟ opinions have to be taken into account too.
9
☛ Bold printed topics are topics never being included in Paper 2 before.
E ANSWERING GUIDE AND MARKING SCHEME
In general, candidate will be awarded METHOD MARKS (for working method needed), ANSWER
MARKS (for the precise answer needed), INDEPENDENT MARKS (for answers that working
methods are not needed), TRANSFER MARKS (for transfering points into graph paper with
precision), GRAPH MARKS ( for smooth and flawless graph) QUALITY MARKS (for a good
drawings of Plans and Elevation ) and others.
Following are general guides that candidates have to practice in some questions in paper 2.
1. QUADRATIC EQUATION - Change to its standard form ax2 + bx + c = 0
- Factorise expression on the Left Hand Side.
- Use the fact that “If ab = 0, then a = 0 or b = 0”
Example 1 : Solve the equation (f – 1)(f + 3) = 5
2. SIMULTANEOUS LINEAR EQUATIONS
- Eliminate fractions first (if there is any) by multiplying the equation with the denominator.
- solve using the “subtitution” or “variable elimination” technique.
- recheck whether the answer satisfy the equation given.
Example 2 : Calculate the value of f and g that satisfy both the following equations
1 f + g = 1
2
3f – 2g = 22
3. MATRICES - candidates must be able to find Inverse of a Matix and know its characteristics.
- candidates must also be able to use that Inverse Matrix to solve simultaneous
equations or the matrix equation given.
- write final answer explicitly.
Example 3 : Given the matrix A = 5 3
-4 -2
( i) Find the inverse of matrix A
(ii) Hence, using matrices, calculate the values of x and y which
satisfy the following matrix equation
5 3 x = 0
-4 -2 y 2
10
4. SETS (ALTERNATES WITH REGION SHADING ON GRAPH’S QUESTION) - Usually, question is on shading region of intersection, union and complement of sets.
- Multiple hatchings are allowed.
Example 4 : On the diagrams in the answer space, shade
(a) the set P‟ ⋂ Q (b) the set (P ⋃ Q‟) ⋂ R
Q R Q R
5. GRADIENT AND AREA UNDER A GRAPH
- questions usually are based on Speed-Time or Distance-Time graphs.
- candidates must be able to
(a) write equation from the information given and hence solve that
equation.
(b) calculate speed from Distance-Time graph.
(c) calculate distance and acceleration from Speed-Time graph.
(d) caculate average speed from both graphs.
Example 5 : Example 6 :
Speed (m s-1)
14
12
Time (sec)
8 t
Diagram shows speed-time graph for a particle in a period of t s. Calculate
(i) rate of speed change for the particle in the first 8 seconds.
(ii) value of t, given total distance travelled by the particle in the period of t seconds is 248 m.
Distance from P (km) C N
280
154 M
D
O t 4 5 Time (hour)
Diagram shows distance-time graph for the route travelled by a bus and a
car. OMN represents the bus‟s route from town P to town R and CMD represents the car‟s route from town R to town P.
(i) Calculate average speed, in km h–1, travelled by the bus from P to R.
(ii) If the car travels at uniform speed, calculate value of t.
6. CIRCLES - candidates must be able use length of arc and area of a sector formulas with ease
where the use of angle at the centre is very important.
- answer must be given at least to 2 decimal place if decimals are involve.
C
Example 7 : In the diagram, ABD is a sector of a circle with centre A.
ADC is a straight line. By using π = 3.142, calculate
D
(a) perimeter of the shaded region
(b) area of the shaded region.
A B
P P
11
8 cm
7. SURFACE AREA AND VOLUME OF SOLIDS.
- Memorise formulaes on surface area and volume of solids.
- Skill on formulae application is also very important..
Example 8 : Example 9 : Example 10 :
(I) (II) Diagram (I) is a container in the shape of a
cuboid that is full with water. Base of the
cuboid in the shape of a rectangle with a length of 11 cm and breadth of 8 cm. Height of the
cuboid is 21 cm. Diagram (II) is an empty
container in the shape of a cylinder. Diameter of the base of the cylinder is 12 cm. All the water
in the cuboid container are poured into the cylinder container. Calculate the height of the
water level in the cylinder container.
Diagram above shows a solid in the shape of a
cylinder with a hemisphere taken out from each
end of the cylinder. Base radius of the cylinder is the same as radius of the hemisphere, that is 5. 6
cm. Length of the cylinder is 13 cm. Calculate
the volume of the solid left..
V
Q P
M N Diagram above shows a solid erected from a
combination of a cuboid and a pyramid. Given
height of the vertice V from the base MNPQ is 13 cm, calculate the surface area of the solid..
8. GRAPHS OF FUNCTIONS - Graphs must be drawn on a graph paper.
- you must be able to calculate y values from the function given, obey scale
instruction, shift points in the table to graph and hence draw a smooth curve.
- skills on solving equation by graphical method are also needed.
Example 11 :
(a) Complete the following table for the function of y = x3 – 12x + 20.
(3 marks)
(b) Using the scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of y = x3 – 12x + 20 for – 3. 5 ≤ x ≤ 4. (3 marks)
(c) From your graph, find value of y when x = -1. 5 (1 mark)
(d) Draw a suitable straight line on your graph to find all the values of x in the range of
-3. 5 ≤ x ≤ 4 that satisfy the equation x3 – 12x – 5 = 0. State the values of x. (5 marks)
Example 12 :
(a) Complete the following table for the equation y = 24 .
x
x -4 - 3 -2 - 1 1 1. 5 2 3 4
y - 6 - 12 - 24 24 12 8 6
[2 marks]
(b) For this part of the question, use the graph paper provided. You may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of y = 24 for – 4 ≤ x ≤ 4.
x [5 marks]
(c) From your graph, find (i) the value of y when x = 2. 9 (ii) the value of x when y = -13 [2 marks]
(d) Draw a suitable straight line on your graph to find all the values of x which satisfy the equation 2x2 + 5x = 24 for - 4 ≤ x ≤ 4. State these values of x. [3 marks]
x -3.5 -3 -2 -1 0 1 2 3 3.5 4
y 19.1 29 20 9 4 20.9 36
12
9. PLANS AND ELEVATIONS - Drawings are done on the blank paper provided in the question paper.
- Drawings must be precise according to measurements given.
- All lines must be straight and drawn using a ruler.
- 90 o angle can be erected quickly using 90 edge of a ruler.
- Make sure there are no “extensions” and “gaps”.
- Construction lines must be differentiated with projection lines.
- Circles‟ curves must be drawn using compasses.
- “Line straightness” should be emphasize.
Example 13 : Example 14 :
10. STATISTICS - Candidates must be able to find mean, modes and medians.
- Candidates must be able to construct frequency table and hence draw histogram
or frequency polygons.
- Candidates must be able to construct cumulative frequency table and hence draw
an ogive.
- Candidate must also be able to find informations from the ogive drawn.
Example 15:
Example 16 :
(a) The table above shows number of appreciation certificate received by 40 students in a class. Find
(i) median
(ii) mean of the data. (3 marks)
(b) For this question, use the graph paper provided.
The table above shows the age (in years) distribution for 180 workers in an Electronic Factory.
(i) Construct a cumulative frequency table for the data.
(ii) Using the scale of 2 cm to 5 years on the x-axis and 2 cm to 20 workers on the y-axis, draw an ogive
for the data.
(iii) Workers in the first quartile are required to attend a course. State the oldest age of the worker
required to attend the course. (9 marks)
For this question, use the graph paper provided.
Data in the above table are height, in cm, for a group of 40 students.
(a) Construct a frequency table for this data using class intervals of the size of 5 cm, with 145-149 as
the first class interval. . (4 marks)
(b) Using a scale of 2 cm to 5 cm on the x-axis and 2 cm to 1 student on the y-axis, draw a frequency
polygon for the above data. (4 marks)
(c) From the frequency polygon,
(i) Find the modal class,
(ii) calculate the mean height for the group of students (4 marks)
152 173 167 172 168 174 166 178
176 164 154 167 162 155 151 163
160 176 168 175 174 177 171 159
171 174 179 169 153 173 156 172
160 154 164 158 167 178 169 154
Number of Appreciation Certificate 0 1 2 3 5 6 7 8
Number of Students 18 3 5 0 6 2 3 1
Age (years) 18-22 23-27 28-32 33-37 38-42 43-47 48-52 53-57
Number of workers 8 15 23 36 48 29 15 6