956 further mathematics

STPM/S(E)956

PEPERIKSAAN

SIJIL TINGGI PERSEKOLAHAN MALAYSIA

(MALAYSIA HIGHER SCHOOL CERTIFICATE EXAMINATION)

FURTHER MATHEMATICS

Syllabus and Specimen Papers

This syllabus applies for the 2012/2013 session and thereafter until further notice.

MAJLIS PEPERIKSAAN MALAYSIA

(MALAYSIAN EXAMINATIONS COUNCIL)

NATIONAL EDUCATION PHILOSOPHY

“Education in Malaysia is an on-going effort towards further

developing the potential of individuals in a holistic and

integrated manner, so as to produce individuals who are

intellectually, spiritually, emotionally and physically

balanced and harmonious, based on a belief in and devotion

to God. Such effort is designed to produce Malaysian

citizens who are knowledgeable and competent, who possess

high moral standards, and who are responsible and capable

of achieving a high level of personal well-being as well as

being able to contribute to the betterment of the family, the

society and the nation at large.”

FOREWORD

This revised Further Mathematics syllabus is designed to replace the existing syllabus which has been

in use since the 2002 STPM examination. This new syllabus will be enforced in 2012 and the first

examination will also be held the same year. The revision of the syllabus takes into account the

changes made by the Malaysian Examinations Council (MEC) to the existing STPM examination.

Through the new system, the form six study will be divided into three terms, and candidates will sit

for an examination at the end of each term. The new syllabus fulfils the requirements of this new

system. The main objective of introducing the new examination system is to enhance the teaching

and learning orientation of form six so as to be in line with the orientation of teaching and learning in

colleges and universities.

The Further Mathematics syllabus is designed to cater for candidates who are competence and have

intense interest in mathematics and wish to further develop their understanding of mathematical

concepts and mathematical thinking and acquire skills in problem solving and the applications of

mathematics.

The syllabus contains topics, teaching periods, learning outcomes, examination format, grade

description and specimen papers.

The design of this syllabus was undertaken by a committee chaired by Professor Dr. Abu Osman bin

Md Tap from International Islamic University Malaysia. Other committee members consist of

university lecturers, representatives from the Curriculum Development Division, Ministry of

Education Malaysia, and experienced teachers who are teaching Mathematics. On behalf of MEC, I

would like to thank the committee for their commitment and invaluable contribution. It is hoped that

this syllabus will be a guide for teachers and candidates in the teaching and learning process.

Chief Executive

Malaysian Examinations Council

CONTENTS

Syllabus 956 Further Mathematics

Page

Aims 1

Objectives 1

Content

First Term: Discrete Mathematics 2 – 4

Second Term: Algebra and Geometry 5 – 7

Third Term: Calculus 8 – 11

Scheme of Assessment 12

Performance Descriptions 13

Mathematical Notation 14 – 18

Electronic Calculators 19

Reference Books 19

Specimen Paper 1 21 – 28



1

SYLLABUS

956 FURTHER MATHEMATICS

[May only be taken with 954 Mathematics (T)]

Aims

The Further Mathematics syllabus caters for candidates who have high competence and intense

interest in mathematics and wish to further develop the understanding of mathematical concepts and

mathematical thinking and acquire skills in problem solving and the applications of mathematics.

Objectives

The objectives of the syllabus are to enable candidates to:

(a) use mathematical concepts, terminology and notation;

(b) display and interpret mathematical information in tabular, diagrammatic and graphical forms;

(c) identify mathematical patterns and structures in a variety of situations;

(d) use appropriate mathematical models in different contexts;

(e) apply mathematical principles and techniques in solving problems;

(f) carry out calculations and approximations to an appropriate degree of accuracy;

(g) interpret the significance and reasonableness of results;

(h) present mathematical explanations, arguments and conclusions.

2

FIRST TERM: DISCRETE MATHEMATICS

Topic Teaching

Period Learning Outcome

1 Logic and Proofs 20 Candidates should be able to:

1.1 Logic 10 (a) use connectives and quantifiers to form

compound statements;

(b) construct a truth table for a compound

statement, and determine whether the

statement is a tautology or contradiction or

neither;

(c) use the converse, inverse and contrapositive of

a conditional statement;

(d) determine the validity of an argument;

(e) use the rules of inference;

1.2 Proofs 10 (f) suggest a counter-example to negate a

statement;

(g) use direct proof to prove a statement, including

a biconditional statement;

(h) prove a conditional statement by

contraposition;

(i) prove a statement by contradiction;

(j) apply the principle of mathematical induction.

2 Sets and Boolean Algebras 14 Candidates should be able to:

2.1 Sets 8 (a) perform operations on sets, including the

symmetric difference of sets;

(b) find the power set and the partitions of a set;

(c) find the cartesian product of two sets;

(d) use the algebraic laws of sets;

2.2 Boolean algebras 6 (e) identify a Boolean algebra;

(f) use the properties of Boolean algebras;

(g) prove that two Boolean expressions are

logically equivalent.

3

Topic Teaching


3 Number Theory 26 Candidates should be able to:

3.1 Divisibility 12 (a) use the divisibility properties of integers;

(b) find greatest common divisors and least

common multiples;

(c) use the properties of greatest common divisors

and least common multiples;

(d) apply Euclidean algorithm;

(e) use the properties of prime and composite

numbers;

(f) use the fundamental theorem of arithmetic;

3.2 Congruences 14 (g) use the properties of congruences;

(h) use congruences to determine the divisibility

of integers;

(i) perform addition, subtraction and

multiplication of integers modulo n;

(j) use the Chinese remainder theorem;

(k) use Fermat’s little theorem;

(l) solve linear congruence equations;

(m) solve simultaneous linear congruence

equations.

4 Counting 20 Candidates should be able to:

(a) use combinations and permutations to solve

counting problems;

(b) prove combinatorial identities;

(c) expand 1 2

n

kx x x , where n, k

and k 2;

(d) use the multinomial coefficients to solve

counting problems;

(e) apply the principle of inclusion and exclusion;

(f) apply the pigeonhole principle;

(g) apply the generalised pigeonhole principle.

>

4

Topic Teaching


5 Recurrence Relations 14 Candidates should be able to:

(a) find the general solution of a first order linear

homogeneous recurrence relation with constant

coefficients;

(b) find the general solution of a first order linear

non-homogeneous recurrence relation with

constant coefficients;

(c) find the general solution of a second order

linear homogeneous recurrence relation with


(d) find the general solution of a second order

linear non-homogeneous recurrence relation

with constant coefficients;

(e) use boundary conditions to find a particular

solution;

(f) solve problems that can be modelled by

recurrence relations.

6 Graphs 26 Candidates should be able to:

6.1 Graphs 10 (a) relate the sum of the degrees of vertices and

the number of edges of a graph;

(b) use the properties of simple graphs, regular

graphs, complete graphs, bipartite graphs and

planar graphs;

(c) represent a graph by its adjacency matrix and

incidence matrix;

(d) determine the subgraphs of a graph;

6.2 Circuits and cycles 10 (e) identify walks, trails, paths, circuits and cycles

of a graph;

(f) use properties associated with connected

graphs;

(g) determine whether a graph is eulerian, and find

eulerian trails and circuits;

(h) determine whether a graph is hamiltonian, and

find hamiltonian paths and cycles;

(i) solve problems that can be modelled by

graphs;

6.3 Isomorphism 6 (j) determine whether two graphs are isomorphic;

(k) use the properties of isomorphic graphs;

(l) apply adjacency matrices to isomorphism.

5

SECOND TERM: ALGEBRA AND GEOMETRY

Topic Teaching


7 Relations 20 Candidates should be able to:

7.1 Relations 12 (a) identify a binary relation on a set;

(b) determine the reflexivity, symmetry and

transitivity of a relation;

(c) determine whether a relation is an equivalence

relation;

(d) find the equivalence class of an element;

(e) find the partitions induced by an equivalence

relation;

(f) use the properties of equivalence relations;

7.2 Binary operations 8 (g) identify a binary operation on a set;

(h) use an operation table;

(i) determine the commutativity and associativity

of a binary operation, and determine whether a

binary operation is distributive over another

binary opration;

(j) find the identity element and the inverse of an

element.

8 Groups 24 Candidates should be able to:

8.1 Groups 6 (a) determine whether a set with a binary

operation is a group;

(b) identify an abelian group;

(c) determine the subgroups of a group;

8.2 Cyclic groups 6 (d) find the order of an element and of a group;

(e) determine the generators of a cyclic group;

(f) use the properties of a cyclic group;

8.3 Permutation groups 6 (g) determine the cycles and transpositions in a

permutation;

(h) determine whether a permutation is odd or

even;

(i) use the properties of a permutation group;

8.4 Isomorphism 6 (j) determine whether two groups are isomorphic;

(k) prove the isomorphism properties for identities

and inverses;

(l) use the properties of isomorphic groups.

6

Topic Teaching


9 Eigenvalues and

Eigenvectors 14 Candidates should be able to:

9.1 Eigenvalues and

eigenvectors

6 (a) find the eigenvalues and eigenvectors of a

matrix;

(b) use the properties of eigenvalues and

eigenvectors of a matrix;

(c) use the Cayley-Hamilton theorem;

9.2 Diagonalisation 8 (d) determine whether a matrix is diagonalisable,

and diagonalise a matrix where appropriate;

(e) find the powers of a matrix;

(f) use the properties of orthogonal matrices;

(g) determine whether a matrix is orthogonally

diagonalisable, and orthogonally diagonalise a

matrix where appropriate.

10 Vector Spaces 24 Candidates should be able to:

10.1 Vector spaces 8 (a) determine whether a set, with addition and

scalar multiplication defined on the set, is a

vector space;

(b) determine whether a subset of a vector space is

a subspace;

(c) determine whether a vector is a linear

combination of other vectors;

(d) find the spanning set for a vector space;

10.2 Bases and dimensions 8 (e) determine whether a set of vectors is linearly

dependent or independent;

(f) find a basis for and the dimension of a vector

space;

(g) use the properties of bases and dimensions;

(h) change the basis for a vector space;

10.3 Linear transformations 8 (i) determine whether a given transformation is

linear;

(j) use the properties of linear transformations;

(k) determine the null space and the range of a

linear transformation, and find a basis for and

the dimension of the null space and the range;

(l) determine whether a linear transformation is

one-to-one.

7

Topic Teaching


11 Plane Geometry 24 Candidates should be able to:

11.1 Triangles 8 (a) use the properties of triangles: medians,

altitudes, angle bisectors and perpendicular

bisectors of sides;

(b) use the properties of the orthocentre, incentre

and circumcentre;

(c) apply Apollonius’ theorem;

(d) apply the angle bisector theorem and its

converse;

11.2 Circles 10 (e) use the properties of angles in a circle and

tangency;

(f) apply the intersecting chords theorem;

(g) apply the tangent-secant and secant-secant

theorems;

(h) use the properties of cyclic quadrilaterals;

(i) apply Ptolemy’s theorem;

11.3 Collinear points and

concurrent lines

6 (j) apply Menelaus’ theorem and its converse;

(k) apply Ceva’s theorem and its converse.

12 Transformation Geometry 14 Candidates should be able to:

(a) use 2 2 and 3 3 matrices to represent linear

transformations;

(b) determine the standard matrices for

transformations;

(c) find the image and inverse image under a

transformation;

(d) find the invariant points and lines of

transformations;

(e) relate the area or volume scale-factor of a

transformation to the determinant of the

corresponding matrix;

(f) determine the compositions of transformations.

8

THIRD TERM: CALCULUS

Topic Teaching


13 Hyperbolic and Inverse

Hyperbolic Functions 16

Candidates should be able to:

13.1 Hyperbolic and

inverse hyperbolic

functions

8 (a) use hyperbolic and inverse hyperbolic

functions and their graphs;

(b) use basic hyperbolic identities and the

formulae for sinh (x ± y), cosh (x ± y) and

tanh (x ± y), including sinh 2x, cosh 2x and

tanh 2x;

(c) derive and use the logarithmic forms for

sinh1x, cosh

1x and tanh

1x;

(d) solve equations involving hyperbolic and

inverse hyperbolic expressions;

13.2 Derivatives and

integrals

8 (e) derive the derivatives of sinh x, cosh x, tanh x,

sinh1x, cosh

1x and tanh

1x;

(f) differentiate functions involving hyperbolic

and inverse hyperbolic functions;

(g) integrate functions involving hyperbolic and

inverse hyperbolic functions;

(h) use hyperbolic substitutions in integration.

14 Techniques and

Applications of Integration

20 Candidates should be able to:

14.1 Reduction formulae 4 (a) obtain reduction formulae for integrals;

(b) use reduction formulae for the evaluation of

definite integrals;

14.2 Improper integrals 4 (c) evaluate integrals with infinite limits of

integration;

(d) evaluate integrals with discontinuous

integrands;

9

Topic Teaching


14.3 Applications of

integration

12 (e) calculate arc lengths for curves with equations

in cartesian coordinates (including the use of a

parameter);

(f) calculate areas of surfaces of revolution about

one of the coordinate axes for curves with

equations in cartesian coordinates (including

the use of a parameter);

(g) sketch curves defined by polar equations;

(h) calculate the areas of regions bounded by

curves with equations in polar coordinates;

(i) calculate arc lengths for curves with equations

in polar coordinates.

15 Infinite Sequences and

Series

24 Candidates should be able to:

15.1 Sequences 4 (a) determine the monotonicity and boundedness

of a sequence;

(b) determine the convergence or divergence of a

sequence;

15.2 Series 10 (c) use the properties of a p-series and harmonic

series;

(d) use the properties of an alternating series;

(e) use the nth-term test for divergence of a series;

(f) use the comparison, ratio, root and integral

tests to determine the convergence or

divergence of series;

15.3 Taylor series 10 (g) find the Taylor series for a function and the

interval of convergence;

(h) use a Taylor polynomial to approximate a

function;

(i) use the remainder term, in terms of the

(n + 1)th derivative at an intermediate point

and in terms of an integral of the (n + 1)th

derivative;

(j) use l’Hospital’s rule to find limits in

indeterminate forms.

10

Topic Teaching


16 Differential Equations 20 Candidates should be able to:

16.1 Linear differential

equations

14 (a) find the general solution of a second order

linear homogeneous differential equation with


(b) find the general solution of a second order

linear non- homogeneous differential equation

with constant coefficients;

(c) transform, by a given substitution, a

differential equation into a second order linear

differential equation with constant coefficients;

(d) use boundary conditions to find a particular

solution;

(e) solve problems that can be modelled by

differential equations;

16.2 Numerical solution of

differential equations

6 (f) use a Taylor series to find a polynomial

approximation for the solution of a first order

differential equation;

(g) use Euler’s method to find an approximate

solution for a first order differential equation,

and determine the effect of step length on the

error;

(h) find the series solutions for second order

differential equations.

17 Vector-valued Functions 16 Candidates should be able to:

17.1 Vector-valued

functions

6 (a) find the domain and sketch the graph of a

vector-valued function;

(b) determine the existence and values of the

limits of a vector-valued function;

(c) determine the continuity of a vector-valued

function;

17.2 Derivatives and

integrals

2

(d) find the derivatives of vector-valued functions;

(e) find the integrals of vector-valued functions;

17.3 Curvature 4 (f) find unit tangent, unit normal and binormal

vectors;

(g) calculate curvatures and radii of curvature;

17.4 Motion in space 4 (h) find the position, velocity and acceleration of a

particle moving along a curve;

(i) determine the tangential and normal

components of acceleration.

11

Topic Teaching


18 Partial Derivatives 24 Candidates should be able to:

18.1 Functions of two

variables

6 (a) find the domain and sketch the graph of a

function of two variables;

(b) determine the existence and values of the

limits of a function of two variables;

(c) determine the continuity of a function of two

variables;

18.2 Partial derivatives 8 (d) find the first and second order partial

derivatives of a function of two variables;

(e) use the chain rule to obtain the first derivative;

(f) find total differentials;

(g) determine linear approximations and errors;

18.3 Directional derivatives 4 (h) find the directional derivatives and gradient of

a function of two variables;

(i) determine the minimum and maximum values

of directional derivatives and the directions in

which they occur;

18.4 Extrema of functions 6 (j) use the second derivatives test to determine the

extremum values of a function of two

variables;

(k) use the method of Lagrange multipliers to

solve constrained optimisation problems.

12

Scheme of Assessment

Term of

Study

Paper Code

and Name Type of Test

Mark

(Weighting) Duration Administration

First

Term 956/1

Further

Mathematics

Paper 1

Written test

Section A Answer all 6 questions of variable

marks.

Section B Answer 1 out of 2 questions.

All questions are based on topics 1

to 6.

60

(33.33%)

45

15

1½ hours Central

assessment

Second

Term 956/2

Further

Mathematics

Paper 2

Written test


marks.



to 12.

60

(33.33%)

45

15

1½ hours Central

assessment

Third

Term 956/3

Further

Mathematics

Paper 3

Written test


marks.



to 18.

60

(33.33%)

45

15

1½ hours Central

assessment

13

Performance Descriptions

A grade A candidate is likely able to:

(a) use correctly mathematical concepts, terminology and notation;

(b) display and interpret mathematical information in tabular, diagrammatic and graphical

forms;

(c) identify mathematical patterns and structures in a variety of situations;

(d) use appropriate mathematical models in different contexts;

(e) apply correctly mathematical principles and techniques in solving problems;

(f) carry out calculations and approximations to an appropriate degree of accuracy;

(g) interpret the significance and reasonableness of results, making sensible predictions where

appropriate;

(h) present mathematical explanations, arguments and conclusions, usually in a logical and

systematic manner.

A grade C candidate is likely able to:

(a) use correctly some mathematical concepts, terminology and notation;

(b) display and interpret some mathematical information in tabular, diagrammatic and graphical

forms;

(c) identify mathematical patterns and structures in certain situations;

(d) use appropriate mathematical models in certain contexts;

(e) apply correctly some mathematical principles and techniques in solving problems;

(f) carry out some calculations and approximations to an appropriate degree of accuracy;

(g) interpret the significance and reasonableness of some results;

(h) present some mathematical explanations, arguments and conclusions.

14

Mathematical Notation

Miscellaneous symbols

= is equal to

≠ is not equal to

≡ is identical to or is congruent to

≈ is approximately equal to

< is less than

is less than or equal to

> is greater than

is greater than or equal to

∞ infinity

therefore

there exists

for all

Operations

a + b a plus b

a − b a minus b

a × b, ab a multiplied by b

a b, a

b a divided by b

a : b ratio of a to b

an nth power of a

12a , a positive square root of a

1na , n a positive nth root of a

|a| absolute value of a real number a

1

n

i

i

u u1 + u2 + ∙ ∙ ∙ + un

n! n factorial for n

n

r binomial coefficient

!

!( )!

n

r n r for n, r , 0 r n

1 2, ,. . ., k

n

r r r multinomial coefficient

1 2

!

! !... !k

n

r r r, where r1 + r2 + . . . + rk = n

Logic

p a statement p

p not p

p q p or q

p q p and q

p q p or q but not both p and q

<

>

< <

15

p q if p then q

p q p if and only if q

p q p is logically equivalent to q

p q p is not logically equivalent to q

Set notation

is an element of

is not an element of

empty set

{x | . . .} set of x such that . . .

set of natural numbers, {0, 1, 2, 3, . . .}

set of integers

set of positive integers

set of rational numbers

set of real numbers

[a, b] closed interval {x | x , a x b}

(a, b) open interval { x | x , a x b}

[a, b) interval { x | x , a x < b}

(a, b] interval { x | x , a < x b}

union

intersection

U universal set

A' complement of a set A

is a subset of

is a proper subset

is not a subset of

is not a proper subset

n(A) number of elements in a set A

P(A) power set of A

A B cartesian product of sets A and B, i.e. A B = {( a, b ) | a A, b B}

A B complement of set B in set A

A B symmetry difference of sets A and B, (A B) (B A)

Number theory

a b a divides b

a | b a does not divide b

gcd (a, b) greatest common divisor of integers a and b

lcm (a, b) least common multiple of integers a and b

< <

<

<

16

m ≡ n (mod d) m is congruent to n modulo d

n set of integers modulo n, {0, 1, 2, . . ., n 1}

x floor of x

x ceiling of x

Graphs

G a graph G

V(G) set of vertices of a graph G

E(G) set of edges of a graph G

deg (v) degree of vertex v

{v, w} edge joining v and w in a simple graph

n a complete graph on n vertices

,m n a complete bipartite graph with one set of m vertices and another set of

n vertices

Relations

y R x y is related to x by a relation R

y ~ x y is equivalent to x, in the context of some equivalence relation

[ a ] equivalence class of an element a

A / R a partition of set A induced by the equivalence relation R on A

Groups

(G, *) a set G together with a binary operation *

e identity element

a −1

inverse of an element a

is isomorphic to

Matrices

A a matrix A

0 null matrix

I identity matrix

AT transpose of a matrix A

A−1

inverse of a non-singular square matrix A

det A determinant of a square matrix A

Vector spaces

V a vector space V

2 set of real ordered pairs

3 set of real ordered triples

n set of real ordered n-tuples

T a linear transformation T

17

Geometry

AB length of the line segment with end points A and B

BAC angle between line segments AB and AC

ABC triangle whose vertices are A, B and C

// is parallel to

is perpendicular to

Vectors

a a vector a

| a | magnitude of a vector a

i, j, k unit vectors in the directions of the cartesian coordinates axes

AB

vector represented in magnitude and direction by the directed line

segment from point A to point B

| |AB

magnitude of AB

a b scalar product of vectors a and b

a × b vector product of vectors a and b

Functions

f a function f

f(x) value of a function f at x

f : A B f is a function under which each element of set A has an image in set B

f : x y f is a function which maps the element x to the element y

1f inverse function of f

f g composite function of f and g which is defined by f g( ) = f[g( )] x x

ex exponential function of x

loga x logarithm to base a of x

ln x natural logarithm of x, loge x

sin, cos, tan,

csc, sec, cot

sin1, cos

1, tan

1,

csc1, sec

1, cot

1

sinh, cosh, tanh,

csch, sech, coth

sinh1, cosh

1, tanh

1,

csch1, sech

1, coth

1

trigonometric functions

inverse trigonometric functions

inverse hyperbolic functions

hyperbolic functions

18

Derivatives and integrals

limf ( )x a

x limit of f(x) as x tends to a

d

d

y

x first derivative of y with respect to x

f '( )x first derivative of f(x) with respect to x

2

2

d

d

y

x second derivative of y with respect to x

f ''( )x second derivative of f(x) with respect to x

d

d

n

n

y

x nth derivative of y with respect to x

( )f ( )n x nth derivative of f(x) with respect to x

dy x indefinite integral of y with respect to x

db

ay x definite integral of y with respect to x for values of x between a and b

Vector-valued functions

curvature

T unit tangent vector

N unit normal vector

Partial derivatives

y

x partial derivative of y with respect to x

del operator, x y z

i j k

19

Electronic Calculators

During the written paper examination, candidates are advised to have standard scientific calculators

which must be silent. Programmable and graphic display calculators are prohibited.

Reference Books

Discrete Mathematics

1. Dolan, S., Neill, H. and Quadling, D., 2009. Further Mathematics for the IB Diploma:

Standard Level. United Kingdom: Cambridge University Press.

2. Gaulter, B. and Gaulter, M., 2001. Further Pure Mathematics. United Kingdom: Oxford

University Press.

3. Epp, S.S., 2011. Discrete Mathematics with Applications. 4th edition. Singapore:

Brooks/Cole, Cengage Learning.

4. Rosen, K.H., 2012. Discrete Mathematics and Its Applications. 7th edition. Kuala Lumpur:

McGraw-Hill.

Algebra and Geometry




University Press.

7. Fraleigh, J.B., 2003. A First Course in Abstract Algebra. 7th edition. Singapore: Pearson

Addison Wesley.

8. Nicholson, W.K., 2012. Indtroduction to Abstract Algebra. 4th edition. Singapore: John

Wiley.

9. Poole, D., 2010. Linear Algebra: A Modern Introduction. 3rd edition. Singapore:


10. Spence, L.E., Insel, A.J. and Frieberg, S.H., 2008. Elementary Linear Algebra: A Matrix

Approach. 2nd edition. Singapore: Pearson Prentice Hall.

Calculus




University Press.

13. Smith, R.T. and Minton, R.B., 2012. Calculus: Early Transcendental Functions.

4th edition. Kuala Lumpur: McGraw-Hill.

14. Stewart, J., 2012. Calculus: Early Transcendentals. 7th edition, Metric Version. Singapore:


21

SPECIMEN PAPER

956/1 STPM

FURTHER MATHEMATICS (MATEMATIK LANJUTAN)

PAPER 1 (KERTAS 1)

One and a half hours (Satu jam setengah)

MAJLIS PEPERIKSAAN MALAYSIA (MALAYSIAN EXAMINATIONS COUNCIL)

SIJIL TINGGI PERSEKOLAHAN MALAYSIA (MALAYSIA HIGHER SCHOOL CERTIFICATE)

Instruction to candidates:

DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.

Answer all questions in Section A and any one question in Section B. Answers may be written in

either English or Bahasa Malaysia.

All necessary working should be shown clearly.

Scientific calculators may be used. Programmable and graphic display calculators are

prohibited.

A list of mathematical formulae is provided on page of this question paper.

Arahan kepada calon:

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUAT

DEMIKIAN.

Jawab semua soalan dalam Bahagian A dan mana-mana satu soalan dalam Bahagian B.

Jawapan boleh ditulis dalam bahasa Inggeris atau Bahasa Malaysia.

Semua kerja yang perlu hendaklah ditunjukkan dengan jelas.

Kalkulator sainstifik boleh digunakan. Kalkulator boleh atur cara dan kalkulator paparan grafik

tidak dibenarkan.

Senarai rumus matematik dibekalkan pada halaman kertas soalan ini.

__________________________________________________________________________________

This question paper consists of printed pages and blank page.

(Kertas soalan ini terdiri daripada halaman bercetak dan halaman kosong.)

© Majlis Peperiksaan Malaysia

STPM 956/1

22

Section A [45 marks]

Answer all questions in this section.

1 Consider the following argument.

If Abu likes to drive to work or his father’s car is old, then he will buy a new car.

Abu does not buy a new car or he takes a train to work.

Abu did not take a train to work.

Therefore, Abu does not like to drive to work.

(a) Rewrite the argument using statement variables and connectives. [2 marks]

(b) Test the argument for validity. [5 marks]

2 Let set B with binary operations and be a Boolean algebra. Show that, for all x, y and z in B,

( ) ( ) ( ) ( ) .x y z x y z x y z x y z y

[5 marks]

3 Find the greatest common divisor of 2501 and 2173, and express it in the form 2501m + 2173n,

where m and n are integers to be determined. [6 marks]

Hence, find the smallest positive integer p such that 9977 + p = 2501x + 2173y, where x and y are

integers. [3 marks]

4 There are 20 balls of which 4 are yellow, 5 are red, 5 are white and 6 are black. The balls of the

same colour are identical.

(a) Find the number of ways in which all the balls can be arranged in a row so that all the white

balls are together to form a single block and there is at least one black ball beside the white block.

[3 marks]

(b) Find the number of ways in which 5 balls can be arranged in a row if the balls are selected

only from the red and yellow balls. [3 marks]

(c) Find the number of ways in which all the balls can be distributed to 4 persons so that each one

receives at least one ball of each colour. [4 marks]

(d) Determine the number of balls which must be chosen in order to obtain at least 4 balls of the

same colour. [2 marks]

5 Let an be the number of ways (where the order is significant) the natural number n can be written

as a sum of 1’s, 2’s or both.

(a) Explain why the recurrence relation for an, in terms of an 1 and an 2, is

an = an 1 + an 2, n > 2. [2 marks]

(b) Find an explicit formula for an. [6 marks]

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23

Bahagian A [45 markah]

Jawab semua soalan dalam bahagian ini.

1 Pertimbangkan hujah yang berikut.

Jika Abu suka memandu ke tempat kerja atau kereta ayahnya lama, maka dia akan

membeli kereta baharu.

Abu tidak membeli kereta baharu atau dia menaiki kereta api ke tempat kerja.

Abu tidak menaiki kereta api ke tempat kerja.

Oleh itu, Abu tidak suka memandu ke tempat kerja.

(a) Tulis semula hujah itu dengan menggunakan pembolehubah dan penghubung penyataan.

[2 markah]

(b) Uji kesahan hujah tersebut. [5 markah]

2 Katakan set B dengan operasi dedua dan ialah algebra Boolean. Tunjukkan bahawa, bagi

semua x, y dan z dalam B,

( ) ( ) ( ) ( ) .x y z x y z x y z x y z y

[5 markah]

3 Cari pembahagi sepunya terbesar 2501 dan 2173, dan ungkapkannya dalam bentuk

2501m + 2173n, dengan m dan n integer yang perlu ditentukan. [6 markah]

Dengan yang demikian, cari integer positif terkecil p yang sebegitu rupa sehinggakan

9977 + p = 2501x + 2173y, dengan x dan y integer. [3 markah]

4 Terdapat 20 bola dengan 4 berwarna kuning, 5 berwarna merah, 5 berwarna putih dan 6 berwarna

hitam. Bola yang berwarna sama adalah secaman.

(a) Cari bilangan cara semua bola itu boleh disusun dalam satu baris supaya semua bola putih

bersama-sama membentuk satu blok tunggal dan terdapat sekurang-kurangnya satu bola hitam di sisi

blok putih. [3 markah]

(b) Cari bilangan cara 5 bola boleh disusun dalam satu baris jika bola itu dipilih hanya daripada

bola merah dan bola kuning. [3 markah]

(c) Cari bilangan cara semua bola itu boleh diagihkan kepada 4 orang supaya setiap orang

menerima sekurang-kurangnya satu bola bagi setiap warna. [4 markah]

(d) Tentukan bilangan bola yang mesti dipilih untuk memperoleh sekurang-kurangnya 4 bola

yang berwarna sama. [2 markah]

5 Katakan an ialah bilangan cara (tertib adalah bererti) nombor asli n boleh ditulis sebagai hasil

tambah 1, 2, atau kedua-duanya.

(a) Jelaskan mengapa hubungan jadi semula bagi an, dalam sebutan an 1 dan an 2, ialah

an = an 1 + an 2, n > 2. [2 markah]

(b) Cari satu rumus tak tersirat bagi an. [6 markah]

956/1

24

6 A graph is given as follows:

(a) Write down an incidence matrix for the graph. [2 marks]

(b) What can be said about the sum of the entries in any row and the sum of the entries in any

column of this incidence matrix? [2 marks]

956/1

v1

v2

v3 v4

v5

e1

e2

e3

e4

e5

e6

25

6 Satu graf diberikan seperti yang berikut:

(a) Tuliskan satu matriks insidens bagi graf itu. [2 markah]

(b) Apakah yang boleh dikatakan tentang hasil tambah kemasukan sebarang baris dan hasil

tambah kemasukan sebarang lajur matriks insidens ini? [2 markah]

956/1

v1

v2

v3 v4

v5

e1

e2

e3

e4

e5

e6

26

Section B [15 marks]

Answer any one question in this section.

7 Define the congruence a b (mod m). [1 mark]

Solve each of the congruences x3 2 (mod 3) and x

3 2 (mod 5). Deduce the set of positive

integers which satisfy both the congruences. [9 marks]

Hence, find the positive integers x and y which satisfy the equation 152. 12153 xyx [5 marks]

8 Let G be a simple graph with n vertices and m edges. Show that m 12

( 1).n n [4 marks]

(a) If m = 10 and G has all vertices of odd degrees, find the smallest possible value of n. [4 marks]

(b) If n = 11 and m = 46, show that G is connected. [7 marks]

956/1

<

27

Bahagian B [15 markah]

Jawab mana-mana satu soalan dalam bahagian ini.

7 Takrifkan kekongruenan a b (mod m). [1 markah]

Selesaikan setiap kekongruenan x3 2 (mod 3) dan x

3 2 (mod 5). Deduksikan set integer positif

yang memenuhi kedua-dua kekongruenan itu. [9 markah]

Dengan yang demikian, cari integer positif x dan y yang memenuhi persamaan 3 15 12152.x xy

[5 markah]

8 Katakan G ialah satu graf ringkas dengan n bucu dan m tepi. Tunjukkan bahawa m 12

( 1).n n

[4 markah]

(a) Jika m = 10 dan G mempunyai semua bucu dengan darjah ganjil, cari nilai n terkecil yang

mungkin. [4 markah]

(b) Jika n = 11 dan m = 46, tunjukkan bahawa G adalah berkait. [7 markah]

956/1

<

28

MATHEMATICAL FORMULAE (RUMUS MATEMATIK)

Counting (Pembilangan)

Multinomial theorem (Teorem multinomial)

1 2

1 2 1 2

1 2

!. . .

! !. . . !

kn rr r

k k

k

nx x x x x x

r r r, 1 2 kr r r n

Principle of inclusion and exclusion (Prinsip rangkuman dan eksklusi)

1 2

1 1

m i i j

i m i j m

n A A A n A n A A

11 2

1

( 1)mi j k m

i j k m

n A A A n A A A

956/1

< <

< < < <

29

SPECIMEN PAPER

956/2 STPM


PAPER 2 (KERTAS 2)










prohibited.




DEMIKIAN.





tidak dibenarkan.

Senarai rumus matematik dibekalkankan pada halaman kertas soalan ini.

__________________________________________________________________________________




STPM 956/2

30



1 Let S be the set of all prime numbers less then 20 and R the relation on S defined by

(a, b) R a2 + b

2 is even, a, b S.

Show that R is an equivalence relation. [5 marks]

2 Let (G, ) be a group, a G and H = {x G| xa = ax}. Show that H is a subgroup of G.

[6 marks]

3 Find the eigenvalues and eigenvectors of the matrix

2 3 2

3 3 3

2 3 2

. [8 marks]

4 Let P3 be the vector space of all polynomials with degree at most three and W be the set of all

polynomials of the form abxbxax 23 . Show that W is a subspace of P3, and find a basis for W.

[8 marks]

5 The diagram below shows a circle inscribed in a triangle ABC, with the sides AB, BC and CA as

tangents to the circle at points X, Y and Z respectively.

The line segment ZX produced meets the line segment CB produced at a point P. The line segment

YX produced meets the line segment CA produced at a point Q. The line segment YZ produced meets

the line segment BA produced at a point R.

Show that .BY BP

YC PC Deduce similar expressions for

AZ

ZCand .

AX

XB [8 marks]

Hence, show that P, Q and R are collinear. [4 marks]

6 The matrices A and B are given by A =

3 12 2

312 2

and B =0 1

1 0. Describe the respective

plane transformation represented by A and B, and hence, describe the single transformation

represented by BA4B. [6 marks]

956/2

R

Z

A

Q

X

P B Y C

31



1 Katakan S ialah set semua nombor perdana yang kurang daripada 20 dan R ialah hubungan pada S

yang ditakrifkan oleh

(a, b) R a2 + b

2 adalah genap, a, b S.

Tunjukkan bahawa R ialah hubungan kesetaraan. [5 markah]

2 Katakan (G, ) ialah satu kumpulan, a G dan H = {x G| xa = ax}. Tunjukkan bahawa H ialah

satu subkumpulan G. [6 markah]

3 Cari nilai eigen dan vektor eigen matriks

2 3 2

3 3 3

2 3 2

. [8 markah]

4 Katakan P3 ialah ruang vektor semua polinomial berdarjah selebih-lebihnya tiga dan W ialah set

semua polinomial berbentuk abxbxax 23 . Tunjukkan bahawa W ialah subruang P3, dan cari

satu asas bagi W. [8 markah]

5 Gambar rajah di bawah menunjukkan satu bulatan yang terterap dalam segitiga ABC, dengan sisi

AB, BC, dan CA sebagai tangen kepada bulatan itu masing-masing di titik X, Y , dan Z.

Tembereng garis ZX yang dilanjurkan bertemu dengan tembereng garis CB yang dilanjurkan

di titik P. Tembereng garis YX yang dilajurkan bertemu dengan tembereng garis CA yang dilanjurkan

di titik Q. Tembereng garis YZ yang dilanjurkan bertemu dengan tembereng garis BA yang dilanjurkan

di titik R.

Tunjukkan bahawa .BY BP

YC PC Deduksikan ungkapan yang serupa bagi

AZ

ZCdan .

AX

XB

[8 markah]

Dengan yang demikian, tunjukkan bahawa P, Q dan R adalah segaris. [4 markah]

6 Matriks A dan B diberikan oleh A =

3 12 2

312 2

dan B =0 1

1 0. Perihalkan penjelmaan satah

yang masing-masing diwakili oleh A dan B, dan dengan yang demikian, perihalkan penjelmaan

tunggal yang diwakili oleh BA4B. [6 markah]

956/2

R

Z

A

Q

X

P B Y C

32



7 The set G = {e, a, a2, a

3, b, ab, a

2b, a

3b} and the operation are such that (G, ) is a group, where

e is the identity element, element a is of order 4, a2 = b

2 and ba = a

3b.

(a) Show, in any order, that ba2 = a

2b and ba

3 = ab. [4 marks]

(b) Construct a group table for (G, ). [6 marks]

(c) Find a subgroup of order 2 and a subgroup of order 4 for (G, ). [2 marks]

(d) Determine whether (H, +8) is isomorphic to (G, ), where H = {0, 1, 2, 3, 4, 5, 6, 7} and +8 is

an addition modulo 8. Justify your conclusion. [3 marks]

8 The linear transformation L : 3 3 is defined by

1 0 2

L 1 1 0 .

0 1 2

x x

y y

z z

(a) Determine a basis for the range of L, and a basis for the null space of L. [6 marks]

(b) Find the image of the line

1 2

2 2

2 0

r under L. [3 marks]

(c) Find, in the form r = a + b, the equation of a straight line whose image under L is the point

(5, 3, –2). [3 marks]

(d) Show that the image of the plane x – y + z = 2 under L is the plane x – y + z = 0. [3 marks]

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33



7 Set G = {e, a, a2, a

3, b, ab, a

2b, a

3b} dan operasi adalah sebegitu rupa sehinggakan (G, ) ialah

satu kumpulan, dengan e unsur identiti, unsur a berperingkat 4, a2 = b

2 dan ba = a

3b.

(a) Tunjukkan, mengikut mana-mana tertib, bahawa ba2 = a

2b dan ba

3 = ab. [4 markah]

(b) Bina satu jadual kumpulan bagi (G, ). [6 markah]

(c) Cari satu subkumpulan berperingkat 2 dan satu subkumpulan berperingkat 4 bagi (G, ).

[2 markah]

(d) Tentukan sama ada (H, +8) isomorfik dengan (G, ), dengan H = {0, 1, 2, 3, 4, 5, 6, 7} dan +8

penambahan modulo 8. Justifikasikan kesimpulan anda. [3 markah]

8 Penjelmaan linear L : 3 3 ditakrifkan oleh

1 0 2

L 1 1 0 .

0 1 2

x x

y y

z z

(a) Tentukan satu asas bagi julat L, dan satu asas bagi ruang nol L. [6 markah]

(b) Cari imej garis

1 2

2 2

2 0

r di bawah L. [3 markah]

(c) Cari, dalam bentuk r = a + b, persamaan garis lurus yang imejnya di bawah L ialah titik

(5, 3, –2). [3 markah]

(d) Tunjukkan bahawa imej satah x – y + z = 2 di bawah L ialah satah x – y + z = 0. [3 markah]

956/2

34


Plane Geometry (Geometri satah)

Radius of circle inscribed in a triangle (Jejari bulatan yang terterap dalam satu segitiga)

r = ( )( )( )s s a s b s c

s

Radius of circle circumscribing a triangle (Jejari bulatan yang menerap lilit satu segitiga)

R = 4 ( )( )( )

abc

s s a s b s c

956/2

35

SPECIMEN PAPER

956/3 STPM


PAPER 3 (KERTAS 3)










prohibited.




DEMIKIAN.





tidak dibenarkan.

Senarai rumus matematik dibekalkan pada halaman kertas soalan ini.

__________________________________________________________________________________




STPM 956/3

36



1 Show that

ln 2

2 2

0

255 1

1024 8sinh cosh d ln 2.x x x [6 marks]

Deduce the exact value of 12

02 2

ln

sinh cosh d .x x x [2 marks]

2 The polar equations of a circle and a cardioid are r = sin and r = 1 – cos respectively.

(a) Sketch, on the same axes, the two curves. [4 marks]

(b) Calculate the area of the intersecting region bounded by both curves. [4 marks]

(c) Calculate the perimeter of the region in (b). [4 marks]

3 Using a comparison test, show that the series2

1

1

3k k k is convergent. [4 marks]

4 It is given that 2d

( 2)d

yx x y

x with the initial condition y = 1 when x = 0. Using the Euler

formula 1 0 0 0f ( , ),y y h x y obtain an estimate of y at x = 0.1 in five steps correct to four decimal

places. [5 marks]

5 The path of an object is defined by r(t) = ti + 2tj + t2k.

(a) Find the tangential and normal components of the acceleration of the object. [7 marks]

(b) Determine the curvature at t = 1. [3 marks]

6 Show that 22

2

)0,1(),( )1(

ln)1(lim

yx

xx

yxexists, and find its value. [6 marks]

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37



1 Tunjukkan bahawa

ln 2

2 2

0

255 1

1024 8sinh kosh d ln 2.x x x [6 markah]

Deduksikan nilai tepat 12

02 2

ln

sinh kosh d .x x x [2 markah]

2 Persamaan kutub satu bulatan dan satu kardioid masing-masing ialah r = sin dan r = 1 – kos .

(a) Lakar, pada paksi yang sama, dua lengkung itu. [4 markah]

(b) Hitung luas rantau bersilang yang dibatasi oleh kedua-dua lengkung itu. [4 markah]

(c) Hitung perimeter rantau dalam (b). [4 markah]

3 Dengan menggunakan ujian bandingan, tunjukkan bahawa siri2

1

1

3k k k adalah menumpu.

[4 markah]

4 Diberikan bahawa 2d

( 2)d

yx x y

x dengan syarat awal y = 1 apabila x = 0. Dengan

menggunakan rumus Euler y1 y0 +0 0f ( , ),h x y dapatkan satu anggaran y di x = 0.1 dalam lima

langkah betul hingga empat tempat perpuluhan. [5 markah]

5 Lintasan satu objek ditakrifkan oleh r(t) = ti + 2tj + t2k.

(a) Cari komponen tangen dan komponen normal pecutan objek itu. [7 markah]

(b) Tentukan kelengkungan di t = 1. [3 markah]

6 Tunjukkan bahawa

2

2 2( , ) (1,0)

( 1) lnhad

( 1)x y

x x

x y wujud, dan cari nilainya. [6 markah]

956/3

38



7 Find the general solution of the differential equation

2

2

d d4 12 8cos 2 .

d d

y yy t

t t [8 marks]

(a) Find the approximate values of y when t = n and 12

,t n where n is a large positive

integer. [3 marks]

(b) Show that, whatever the initial conditions, the limiting solution as t may be expressed in

the form ,sin(2 )y k t where k is a positive integer and an acute angle which are to be

determined. [4 marks]

8 A right pyramid has a rectangular base of length 2x and width 2y. The slant edges are each of

length 5 units.

(a) Find an expression for the total surface area, S, of the pyramid in terms of x and y. [3 marks]

(b) Find the total differential of S. Interpret your answer. [6 marks]

(c) Determine the error in estimating the change in S using the total differential if x changes from

4.00 to 4.04 and y changes from 3.00 to 2.94. [6 marks]

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39



7 Cari selesaian am persamaan pembezaan

2

2

d d4 12 8kos 2 .

d d

y yy t

t t [8 markah]

(a) Cari nilai hampiran y apabila t = n dan 12

,t n dengan n integer positif yang besar.

[3 markah]

(b) Tunjukkan bahawa, walau apa pun syarat awal, selesaian pengehad semasa t boleh

diungkapkan dalam bentuk ,sin(2 )y k t dengan k integer positif dan sudut tirus yang perlu

ditentukan. [4 markah]

8 Satu piramid tegak mempunyai tapak segiempat dengan panjang 2x dan lebar 2y. Setiap tepi

sendeng mempunyai panjang 5 unit.

(a) Cari satu ungkapan bagi jumlah luas permukaan, S, piramid itu dalam sebutan x dan y.

[3 markah]

(b) Cari pembeza seluruh S. Tafsirkan jawapan anda. [6 markah]

(c) Tentukan ralat dalam menganggar perubahan dalam S dengan menggunakan pembeza seluruh

jika x berubah dari 4.00 ke 4.04 dan y berubah dari 3.00 ke 2.94. [6 markah]

956/3

40


Inverse hyperbolic functions(Fungsi hiperbolik songsang)

sinh1x = ln ( x +

21x )

cosh1x = ln ( x +

21x ), x 1

tanh1x =

1 1ln

2 1

x

x, |x| 1

Integrals (Kamiran)

1

2 2

1 1d tan

xx c

a aa x

1

2 2

1d sin

xx c

aa x

1

2 2

1d cosh

xx c

ax a

1

2 2

1d sinh

xx c

ax a

956/3

>

956 further mathematics

Documents