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SPM MATHEMATICS 2011 PAPER 1

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SPM MATHEMATICS

2011

PAPER 1

A. GENERAL GUIDE - Paper 1 Covers some topics from Form 1 to 3, all topics in

Form 4 and 5, and requires BASIC, INTERMEDIATE and HIGHER skills.

Form of question - objective questions with 4 choices of answer.

Topics covers the NUMBERS, SHAPES & SPACE and ALGEBRAIC themes.

Skills on NUMBERS, SHAPES & SPACE and ALGEBRAIC themes 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.

Skills on NUMBERS should be built first whilst skills on ALGEBRA are an important tool to solve many problems.

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 asked).

Candidates must do an overall revision covering all these topics and don’t choose certain topics only

THEMES & TOPICSNUMBERS 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-D

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

Equations

7. Coordinates

8. The Straight Line

9. Graphs Of Functions

A Graph

11. Ratios and Rates

12. Variations

13. Matrices

14. Sets

15. Mathematical

Reasoning

16. Statistics

17. Probability

B. EXAMINATION FORMAT - Paper 1

NO. ITEM NOTES / DESCRIPTION

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 form’s field of studies

that have continuity at higher form.

2. All field of studies from form 4 to 5.

8 Level of Difficulty

Easy (E) Moderate (M)

Difficult (D)

E : M : D = 5 : 3 : 2

9 Additional Tools 1. Scientific Calculators

2. Mathematical Tables Book

3. Geometrical Equipment

C. ANALYSIS – Paper 1

TOPICS2003 2004 2005 2006 2007 2008 2009 2010 2011

FORM 1 – 31.Polygons I and II2.Algebraic Expressions3.Linear Equations4.Algebraic Formulae5.Statistics I and II6.Transformations I and II7.Indices8.Linear Inequalities9.Trigonometry I

22112322-

121122221

22111221-

121132111

22112221-

12112221-

22113222-

TOTAL 15 14 12 13 13 12 15FORM 41.Standard Form2.Quadratic Expr. & Equations3.Sets4.Mathematical Reasoning5.The Straight Line6.Statistics III7.Probability I8.Circles III9.Trigonometry II10.10. Angles of Elevation & Depress.Angles of Elevation & Depress.11. Lines & Planes in 3-Dimension

4-3-1-21211

4-3-2-31221

3-3-2121311

4-3-2-21221

3-3-2-21311

4-3-2-21321

4-2-2-21221

TOTAL 15 18 17 17 16 18 16FORM 51.Number Bases2.Graphs of Functions II3.3.Transformations IIITransformations III4.Matrices5. Variations5. Variations6 Gradient/Area Under Graphs6 Gradient/Area Under Graphs7. Probability II7. Probability II8.8.BearingsBearings9.9.Earth As A SphereEarth As A Sphere10.10.Plans And ElevationsPlans And Elevations

21-32--11-

21-12--11-

21-23--12-

21-23--11-

21-23--12-

21-23--11-

21-22--11-

TOTAL 10 8 11 10 11 10 9

D. ANSWERING GUIDE - Paper 1 Paper 1 usually begins with simple and easy questions.

If any question can’t be answered, move to other questions and don’t waste time an any one question.

For questions involving table readings, usually examples on how to use the table are shown.

Protractors and set squares are not permitted to be use in Geometrical Constructions. After constructions are made using compasses and ruler, candidates can use the protractor to check accuracy of the angles constructed. (NOTE : There are no more Geometrical Construction questions in Paper 1).

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.

Don’t be too dependent on a certain method or skill to solve problems. Try to variate your technique and skill.

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

“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 ?

A. PA B. PB C. PC D. PD

E. FORMS OF QUESTION – Paper 1

QUESTIONS THAT CAN BE ANSWERED USINGOTHER QUESTION BEFORE OR AFTER IT AS AGUIDE

Example 2 below can be answered using Example 3 as aguide.

EXAMPLE 2 : EXAMPLE 3 :

Amir Amsyar bought a pair of pants Solve the equationat a price of RM 42 after discount. 6 – 2x = 4The original price is RM 60. Calculate 3 The percentage of discount given.

A. 20 % C. 30 % A. – 2 C. 4B. 25 % D. 35% B. 3

D. - 3

E. FORMS OF QUESTION – Paper 1

QUESTIONS THAT CAN BE ANSWERED BY

TRYING OUT EACH CHOICE GIVEN (If possible,

try not using this method because it

is time consuming)

EXAMPLE 4 : 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

Try whether 234 is equal to 3 . If not 624 8 the same, repeat with other choices. (If possible make a RANDOM choice because we might succeed at first try)

A better and quicker method here is using tthe algebraic method i.e by forming the equation x = 3 and solving it. 624 8

E. FORMS OF QUESTION – Paper 1

QUESTIONS THAT CAN BE SOLVED USING ALGEBRAIC METHOD

EXAMPLE 5 :The interior angles of a hexagon are 2xo, 2xo, 3xo, 3xo, 4xo dan 4xo. The value of x is A. 40o C. 80o

B. 70o D. 90o

Form the equation2x + 2x + 3x + 3x + 4x + 4x = 4(180)and solve the equation.

E. FORMS OF QUESTION – Paper 1

QUESTIONS THAT CAN BE SOLVED USINGALGEBRAIC METHOD

EXAMPLE 6 :In the following diagram, calculate theheight of the cylinder, h, given surfacearea of the cylinder is 330 cm2 and itsradius is 3.5 cm. r A. 11.5 cm C. 15 cmB. 13.25 cm D. 26.5 cm h

Form the equation2π(3.5)2 + 2π(3.5)h = 330and solve the equation. (Subtitute π = 22 / 7)

E. FORMS OF QUESTION – Paper 1

QUESTIONS THAT CAN BE SOLVED

USING ALGEBRAIC METHOD

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

A. 2 C. - 2

B. 3 D. – 3Form the simultaneous equation a + a = 2 dan – 8 + 2a = k 2 2 and solve them.

E. FORMS OF QUESTION – Paper 1

GUESSED

Before guessing, eliminate all the

distractors first.

EXAMPLE 8 :

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 !

TRY THE FOLLOWING QUESTIONS RELATING TO NUMBERS :

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

B. 2.88 x 10 5

C. 2.88 x 10 9

D. 2.88 x 10 4

3. 1011101 2

– 10110 2 =

A. 10001 2

B. 10101 2

C. 10111 2

D. 11111 2

4. 3.47 x 10 3 =

A. 0.0034

B. 347

C. 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

B. 6.16 x 101

C. 6.16 x 10-1

D. 6.16 x 103

TRY THE FOLLOWING QUESTIONS RELATING TO

ALGEBRA :

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 then a =

a

A. 2b C. w – 2b

w – 3 3

B. 2b D. w

w + 3 6b

6. If x + 2 = 3x then x =

5

A. – 1 / 3 C. – 5

B. – 2 / 5 D. 1

TRY THE FOLLOWING QUESTIONS RELATING TO

SHAPES :

7. N

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

13.

Q is the image of triangle P

under a rotation. The

coordinates of the centre of

rotation is

A. (5, 1) C. (4, 2)

B. (2, 4) D. (0, 5)

8

6 Q

4

2 P

0 2 4 6 8

SPM MATHEMATICS

2010

PAPER 2

A. GENERAL GUIDE – Paper 2

Paper 2 SPM Mathematics contains 2 parts; Part A & Part B.

Test is in the form of written subjective and answers must be written in the question paper.

Questions subjective and needs longer working method.

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 / DESCRIPTION

1 Type Of Instrument Subjective Test

2 Type Of Item Structured & Limited Response

3 Number Of Question Part A : 11 Question (Answer all)

Part B : 5 Questions (Choose 4)

4 Total Marks Part A : 52 marks

Part B : 48 marks (12 marks each)

5 Test Duration 2 hour 30 minutes

6 Constructual Inclination Knowledge - 25 % / Skill - 70 %

Value – 5 %

7 Contextual Coverage 1. Lower form’s field of studies

that have continuity at higher form.

2. All field of studies from form 4 to 5.

8 Level of Difficulty

Easy (E) Moderate (M)

Difficult (D)

E : M : D = 5 : 3 : 2

9 Additional Tools 1. Scientific Calculators

2. Mathematical Tables Book

3. Geometrical Equipment

C. GENERAL INSTRUCTION – Paper 2

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).

Candidates can use a normal scientific calculator.

Candidates will be supplied with four digit tables book, graph papers, blank papers.

Final answer that involves decimals must be given correct to two decimal places.

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 2TOPICS PART A

03 04 05 06 07 08 09 10

PART B03 04 05 06 07 08 09 10

FORM 1 – 31. Simultaneous Linear EqUATION

2. Cicles (II)

3. Volume/ Area of Solids

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

TOTAL 3 3 3 3 3

FORM 41. Standard Form

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

TOTAL 4 5 4 5 4 1 1 1 1 1

FORM 51. Number Bases

2. Graphs of Functions II

3. Transformations III

4. Matrices

5. Variations

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

TOTAL 4 3 4 3 4 4 4 4 4 4

OVERALL TOTAL 11 11 11 11 11 5 5 5 5 5

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

1. QUADRATIC - 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 - 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

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

3. MATRICES - candidates must be able to find Inverse of a Matrix and know its characteristics - candidates must also be able to use that Inverse Matrix to solve simultaneous equations @ matrices. - 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

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

4. SETS

- Usually, question is on shading region of intersection, union and complement of sets.

- Multiple hatchings are allowed.

(a) the set P’ Q (b) the set (P Q’) R ⋂ ⋃ ⋂

Q R Q R

P P

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

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) find speed from Distance-Time graph. (c) find distance & acceleration from Speed-Time graph. (d) caculate average speed from both graphs.

EXAMPLE 5 :

Speed (m s -1)

14

12

Time (s)

8 tDiagram 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.

EXAMPLE 6 :

Distance from P (km)

280 C N

154 M

Time (h)

D

O t 4 5Diagram 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) The car travels at uniform speed, calculate value of t.

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

6. CIRCLES

- candidates must be able use length of arc formulae and

area of a sector formulae 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.

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

(a) perimeter of the shaded region

(b) area of the shaded region. C

D

A B

8 cm

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

7. SURFACE AREA AND VOLUME OF SOLIDS. Formulas on volume of solids are supplied. Skill on formulae application is also very

important..

Example 8 : Example 9 : Example 10 :

VOLUME SURFACE AREA

SURFACE AREA

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

8. GRAPHS OF FUNCTIONS

- Graphs must be drawn within the lines of the 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 : EXAMPLE 12 :

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

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 corner 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.

A straight “lining” is very important.

Example 13 : Example 14 :

E. ANSWERING GUIDE & MARKING SCHEME – Paper 2

13. 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 extract informations from the ogive drawn.

EXAMPLE 15 : EXAMPLE 16 :