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1 SECURE AND OUTSTANDING FOR TERTIARY ADMISSION LOCALLY OR ABROAD Erected by : KHAIRIL ANUAR BIN MOHD RAZALI Enhanced, refined and continued by :
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TECHNIQUE APPLICATION MATHEMATICS SPM

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SECURE AND OUTSTANDING FOR TERTIARY ADMISSION LOCALLY OR ABROAD
Erected by :

KHAIRIL ANUAR BIN MOHD RAZALI
Enhanced, refined and continued by :

1

SPM MATHEMATICS - PAPER 1 (1449 / 1)
A
1. 2. 3.

GENERAL GUIDE - Paper 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. Topics in this paper covers the NUMBERS, SHAPES & SPACE and ALGEBRAIC themes. Skills connected to the NUMBERS, SHAPES & SPACE and AL
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Page 1: TECHNIQUE APPLICATION MATHEMATICS SPM

1

SECURE AND OUTSTANDING FOR TERTIARY

ADMISSION LOCALLY OR ABROAD

Erected by :

KHAIRIL ANUAR BIN MOHD RAZALI

Enhanced, refined and continued by :

Page 2: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 3: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 4: TECHNIQUE APPLICATION MATHEMATICS SPM

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 ?

Page 5: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 6: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 7: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 8: TECHNIQUE APPLICATION MATHEMATICS SPM

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.

Page 9: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 10: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 11: TECHNIQUE APPLICATION MATHEMATICS SPM

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

Page 12: TECHNIQUE APPLICATION MATHEMATICS SPM

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