956 sp furthermath

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STPM/S(E)956

MAJLIS PEPERIKSAAN MALAYSIA

(MALAYSIAN EXAMINATIONS COUNCIL)

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.


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FALSAFAH PENDIDIKAN KEBANGSAAN

Pendidikan di Malaysia adalah satu usaha berterusan

ke arah memperkembangkan lagi potensi individu secara

menyeluruh dan bersepadu untuk mewujudkan insan yang

seimbang dan harmonis dari segi intelek, rohani, emosi,

dan jasmani. Usaha ini adalah bagi melahirkan rakyat

Malaysia yang berilmu pengetahuan, berakhlak mulia,

bertanggungjawab, berketerampilan, dan berkeupayaanmencapai kesejahteraan diri serta memberi sumbangan

terhadap keharmonian dan kemakmuran keluarga,

masyarakat dan negara.


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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, sixth-form 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 in sixth form 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 sample questions.

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

Mad Tap of International Islamic University of Malaysia. Other committee members consist of

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

Education Malaysia, and experienced teachers teaching Mathematics. On behalf of the Malaysian

Examinations Council, 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.

OMAR BIN ABU BAKAR

Chief ExecutiveMalaysian Examinations Council


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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




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SYLLABUS

956 FURTHER MATHEMATICS

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 this syllabus are to enable the 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.

1


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FIRST TERM: DISCRETE MATHEMATICS

TopicTeaching

PeriodLearning 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.

2


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TopicTeaching


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 Chinese remainder theorem;

(k) use Fermats little theorem;

(l) solve linear congruence equations;

(m) solve simultaneous linear congruenceequations.

4 Counting 20 Candidates should be able to:

(a) use combinations and permutations to solve

counting problems;

(b) prove combinatorial identities;

(c) expand ( )1 2n

kx x x+ + + , where n, k +

and k 2;

(d) use the multinomial coefficients to solvecounting problems;

(e) apply the principle of inclusion and exclusion;

(f) apply the pigeonhole principle;

(g) apply the generalised pigeonhole principle.

>

3


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TopicTeaching


5 Recurrence Relations 14 Candidates should be able to:

(a) find the general solution of a first order linearhomogeneous 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 relationwith 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.

4


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SECOND TERM: ALGEBRA AND GEOMETRY

TopicTeaching


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 isomorphics groups.

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TopicTeaching


9 Eigenvalues and

Eigenvectors14 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.

6


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TopicTeaching


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 Ptolemys theorem;

11.3 Collinear points and

concurrent lines

6 (j) apply Menelaus theorem and its converse;

(k) apply Cevas theorem and its converse.

12 Transformation Geometry 14 Candidates should be able to:

(a) use 2 2 and 3 3 matrices to represent lineartransformations;

(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.

7


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THIRD TERM: CALCULUS

TopicTeaching


13 Hyperbolic and InverseHyperbolic 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 andinverse hyperbolic expressions;

13.2 Derivatives and

integrals

8 (e) derive the derivatives of sinhx, coshx, tanhx,

sinh1x, cosh

1x and tanh

1x;

(f) differentiate functions involvinghyperbolic

and inverse hyperbolic functions;

(g) integrate functions involvinghyperbolic and

inverse hyperbolic functions;

(h) use hyperbolic substitutions in integration.

14 Techniques and

Applications of Integration


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;

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TopicTeaching


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


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 ap-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 lHospitals rule to find limits in

indeterminate forms.

9


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TopicTeaching


16 Differential Equations 20 Candidates should be able to:

16.1 Linear differentialequations

14 (a) find the general solution of a second orderlinear 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 Eulers 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 solution for 2nd orderdifferential 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.

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TopicTeaching


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.

11


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Scheme of Assessment

Term of

Study

Paper Code

and NameType of Test

Mark

(Weighting)Duration Administration

First

Term956/1

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

1 hoursCentral

assessment

SecondTerm

956/2Further

Mathematics

Paper 2

Written test

Section A


marks.

Section B



to 12.

60(33.33%)

45

15

1 hoursCentral

assessment

ThirdTerm 956/3Further

Mathematics

Paper 3

Written test

Section A


marks.

Section B



to 18.

60(33.33%)

45

15

1 hoursCentral

assessment

12


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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.

13


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Mathematical Notation

Miscellaneous symbols

= is equal to

is not equal to

is identical to or is congruent to

is approximately equal to

is proportional 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

ab a minus b

a b, a b, ab a multiplied by b

a b, ab

a divided by b

a : b ratio ofa to b

an nth power ofa12a , a positive square root ofa

1na , n a positive nth root ofa

|a| absolute value of a real numbera

u1 + u2 + + un1

n

i

i

u=

n! n factorial for n

binomial coefficientnr

!!( )!

nr n r

forn, r , 0 r n


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pq ifp then q

pq p if and only ifq

p q p is logically equivalent to qp q p is not logically equivalent to q/

Set notation

is an element of

is not an element of

{x1,x2,. . .,xn} set with elementsx1,x2, . . . ,xn

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

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

set of integers, {0, 1, 2, 3, . . .}

set of positive integers, {1, 2, 3, . . .}+

set of rational numbers, {p

q|p and q + }

set of real numbers

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


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

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

x floor ofx

x ceiling ofx

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

a complete graph on n verticesn

a complete bipartite graph with one set ofm vertices and another set ofn vertices

,m n

Relations

yR x y is related tox by a relationR

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

[ a ] equivalence class of an element a

A /R a partition of setA induced by the equivalence relationR onA

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

I identity matrix

0 null matrix

A1

inverse of a non-singular square matrix A

AT

transpose of a matrix A

det A determinant of a square matrix A

Vector spaces

V a vector space V

set of real ordered pairs2

set of real ordered triples3

set of real ordered n-tuplesn

T a linear transformation T

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Geometry

AB length of the line segment with end pointsA andB

angle atAA

BAC angle between line segmentsAB andAC

triangle whose vertices areA,B and CABC// is parallel to

is perpendicular to

Vectors

a a vectora

| a | magnitude of a vectora

unit vector in the direction of the vectoraa

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

vector represented in magnitude and direction by the directed linesegment from pointA to pointB

AB

| magnitude ofAB|AB

scalar product of vectors a and ba bi

a b vector product of vectors a and b

Functions

f a function f

f(x) value of a function f atx

f is a function under which each element of setA has an image in setBf :A Bf :x y f is a function which maps the elementx to the elementy

inverse function of f1f

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

ex exponential function ofx

logax logarithm to base a ofx

lnx natural logarithm ofx, logex

lgx logarithm to base 10 ofx, log10x

sin, cos, tan,

csc, sec, cot trigonometric functions

sin1, cos1, tan1,

csc1

, sec1

, cot1

inverse trigonometric functions

sinh, cosh, tanh,

csch, sech, cothhyperbolic functions

sinh1

, cosh1

, tanh1

,

csch1, sech

1, coth

1

inverse hyperbolic functions

17


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Derivatives and integrals

lim f ( )x a

x

limit of f(x) asx tends to a

d

d

y

xfirst derivative ofy with respect tox

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

2

2

d

d

y

xsecond derivative ofy with respect tox

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

d

d

n

n

y

x nth derivative ofy with respect tox

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

indefinite integral ofy with respect toxy x d

definite integral ofy with respect tox for values ofx between a and bdb

ay x

Vector-valued functions

curvature

T unit tangent vector

N unit normal vector

Partial derivatives

y

x

partial derivative ofy with respect tox

del operator,x y z

= + +

i j k

18


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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



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

University Press.

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

Wiley.

8. Poole, D., 2010. Linear Algebra: A Modern Introduction. 3rd edition. Singapore:Brooks/Cole, Cengage Learning.

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

Approach. 2nd edition. New Jersey: Pearson Prentice Hall.

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

Addison Wesley.

Calculus



12. Gaulter, B. and Gaulter, M., 2001. Further Pure Mathematics. United Kingdom: OxfordUniversity Press.

13. Smith, R.T. and Minton, R.B., 2012. Calculus: Early Transcendental Function. 4th edition.

Kuala Lumpur: McGraw-Hill.

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

Brooks/Cole, Cengage Learning.

15. Tan, S. T., 2011. Calculus: Early Transcendentals. California: Brooks/Cole, Cengage

Learning.

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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.

Answerall questions in Section A and any one question in Section B. Answers may be written ineither 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 MalaysiaSTPM 956/1

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Section A [45 marks]

Answerallquestionsinthissection.

1 Consider the following argument.

If Abu likes to drive to work or his fathers 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 setB with binary operations and be a Boolean algebra. Show that, for allx,y andz inB,

( ) ( ) ( ) ( ) .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 integerp such that 9977 +p = 2501x + 2173y, wherex 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 selectedonly 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 numbern can be written

as a sum of 1s, 2s or both.

(a) Explain why the recurrence relation foran, in terms ofan1 and an2, is

an = an1 + an2, n > 2. [2 marks]

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

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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, bagisemuax,y danz dalamB,

( ) ( ) ( ) ( ) .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, denganx 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 sisiblok 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 an1 dan an2, ialah

an= an1 + an2, n > 2. [2 markah]

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

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6 A graph is given as follows:

e6

e5

e4

e3

e2

e1

v5

v4

v2

v1

v3

(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]

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6 Satu graf diberikan seperti yang berikut:

v2e1v1

(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]

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v3v4

e2

e3

e4

e5

e6

v5

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Section B [15 marks]

Answer anyonequestioninthissection.

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

Solve each of the congruences x3 2 (mod 3) and x3 2 (mod 5). Deduce the set of positiveintegers which satisfy both the congruences. [9 marks]

Hence, find the positive integersx and y which satisfy the equation [5 marks]152.12153 =+ xyx

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

( 1)n n .

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[4 marks]

956 sp furthermath

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