956 Sukatan Pelajaran Matematik Lanjutan STPM (Baharu)

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.

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 berkeupayaan
mencapai kesejahteraan diri serta memberi sumbangan
terhadap keharmonian dan kemakmuran keluarga,
masyarakat dan negara.”

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 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



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

FIRST TERM: DISCRETE MATHEMATICS

Teaching
Topic Learning Outcome
Period

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

Teaching
Period

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 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 ( x1 + x2 + ⋅ ⋅ ⋅ + xk ) , where n, k ∈ +
n

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.

3

Teaching
Period

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.

4

SECOND TERM: ALGEBRA AND GEOMETRY

Teaching
Period

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.

5

Teaching
Period

9 Eigenvalues and
14 Candidates should be able to:
Eigenvectors

9.1 Eigenvalues and 6 (a) find the eigenvalues and eigenvectors of a
eigenvectors 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

Teaching
Period

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

7

THIRD TERM: CALCULUS

Teaching
Period

13 Hyperbolic and Inverse Candidates should be able to:
16
Hyperbolic Functions

13.1 Hyperbolic and 8 (a) use hyperbolic and inverse hyperbolic
inverse hyperbolic functions and their graphs;
functions
(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
sinh−1x, cosh−1x and tanh−1x;
(d) solve equations involving hyperbolic and
inverse hyperbolic expressions;

13.2 Derivatives and 8 (e) derive the derivatives of sinh x, cosh x, tanh x,
integrals sinh−1x, 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 20 Candidates should be able to:
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;

8

Teaching
Period

14.3 Applications of 12 (e) calculate arc lengths for curves with equations
integration 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 24 Candidates should be able to:
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 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.

9

Teaching
Period

16 Differential Equations 20 Candidates should be able to:

16.1 Linear differential 14 (a) find the general solution of a second order
equations 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 6 (f) use a Taylor series to find a polynomial
differential equations 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 solution for 2nd order
differential equations.

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

17.1 Vector-valued 6 (a) find the domain and sketch the graph of a
functions 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 2 (d) find the derivatives of vector-valued functions;
integrals (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.

10

Teaching
Period

18 Partial Derivatives 24 Candidates should be able to:

18.1 Functions of two 6 (a) find the domain and sketch the graph of a
variables 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

Scheme of Assessment

Term of Paper Code Mark
Type of Test Duration Administration
Study and Name (Weighting)
First 956/1 Written test 60
Term Further (33.33%)
Mathematics
Section A 45
Paper 1
Answer all 6 questions of variable
marks. Central
1½ hours
assessment
Section B 15
Answer 1 out of 2 questions.
All questions are based on topics 1
to 6.

Second 956/2 Written test 60
Mathematics
Section A 45
Paper 2
marks. Central
1½ hours
assessment
Section B 15
to 12.

Third 956/3 Written test 60
Mathematics
Section A 45
Paper 3
marks. Central
1½ hours
assessment
Section B 15
to 18.

12

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

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
a−b a minus b
a × b, a · b, ab a multiplied by b
a
a ÷ b, a divided by b
b
a:b ratio of a to b
n
a nth power of a
1
a , 2
a positive square root of a
1
a , n n
a positive nth root of a
|a| absolute value of a real number a
n

∑u
i =1
i u1 + u2 + · · · + un

n! n factorial for n ∈
⎛n⎞ n!
⎜ ⎟ binomial coefficient for n, r ∈ ,0 <r <n
⎝r⎠ r !( n − r )!
⎛ n ⎞ n!
⎜ r , r ,. . ., r ⎟ multinomial coefficient , where r1 + r2 + . . . + rk = n
⎝1 2 k⎠ r1 !r2 !...rk !

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

14

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
{x1, x2,. . ., xn} set with elements x1, x2, . . . , xn
{x | . . .} set of x such that . . .
set of natural numbers, {0, 1, 2, 3, . . .}
set of integers, {0, ±1, ±2, ±3, . . .}
+
set of positive integers, {1, 2, 3, . . .}
p +
set of rational numbers, { | p∈ and q∈ }
q
set of real numbers
[a, b] closed interval {x ∈ | a < x < b}
(a, b) open interval {x ∈ | a < x < b}
[a, b) interval {x ∈ | a < x < b}
(a, b] interval {x ∈ | a < x < b}
∅ empty set
∪ 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
AΔB 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

15

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

Relations
yRx 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
−1
a inverse of an element a
≅ is isomorphic to

Matrices
A a matrix A
I identity matrix
0 null matrix
−1
A inverse of a non-singular square matrix A
T
A transpose of a 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

16

Geometry
AB length of the line segment with end points A and B
∠A angle at A
∠BAC angle between line segments AB and AC
ΔABC 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
ˆ
a unit vector in the direction of the 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
aib 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
f −1 inverse function of f
f g composite function of f and g which is defined by f g(x) = f[g(x)]
ex exponential function of x
loga x logarithm to base a of x
ln x natural logarithm of x, loge x
lg x logarithm to base 10 of x, log10 x
sin, cos, tan,
trigonometric functions
csc, sec, cot
sin−1, cos−1, tan−1,
inverse trigonometric functions
csc−1, sec−1, cot−1
sinh, cosh, tanh, hyperbolic functions
csch, sech, coth
sinh−1, cosh−1, tanh−1,
inverse hyperbolic functions
csch−1, sech−1, coth−1

17

Derivatives and integrals
lim f ( x) limit of f(x) as x tends to a
x→a

dy
first derivative of y with respect to x
dx
f '( x) first derivative of f(x) with respect to x
d2 y
second derivative of y with respect to x
dx 2
f ''( x) second derivative of f(x) with respect to x
dn y
nth derivative of y with respect to x
dx n
f (n ) ( x) nth derivative of f(x) with respect to x

∫ y dx indefinite integral of y with respect to x
b
∫ a
y dx definite integral of y with respect to x for values of x between a and b

Vector-valued functions
κ curvature
T unit tangent vector
N unit normal vector

Partial derivatives
∂y
partial derivative of y with respect to x
∂x
∂ ∂ ∂
∇ del operator, ∇ = i +j +k
∂x ∂y ∂z

18

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

19

SPECIMEN PAPER

956/1 STPM

FURTHER MATHEMATICS (MATEMATIK LANJUTAN)

PAPER 1 (KERTAS 1)
One and a half hours (Satu jam setengah)


(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

21

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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22

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

23

6 A graph is given as follows:
e1 v2
v1
e6 e2

e5 e3 v3
v4
e4
v5

(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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24

6 Satu graf diberikan seperti yang berikut:

e1 v2
v1
e6 e2

e5 e3 v3
v4
e4
v5

(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

25

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 x3 ≡ 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 x 3 + 15 xy = 12 152. [5 marks]

8 Let G be a simple graph with n vertices and m edges. Show that m < 1 n(n − 1).
2
[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

26

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 x3 ≡ 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 x 3 + 15 xy = 12152.
[5 markah]

8 Katakan G ialah satu graf ringkas dengan n bucu dan m tepi. Tunjukkan bahawa m < 1 n(n − 1).
2
[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]

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27

MATHEMATICAL FORMULAE
(RUMUS MATEMATIK)

Counting
(Pembilangan)

Multinomial theorem
(Teorem multinomial)
n!
( x1 + x2 + ⋅ ⋅ ⋅ + xk ) =∑ x1r x2 . . . xk , where r1 + r2 + ⋅ ⋅ ⋅ + rk = n
n r rk
1 2

r1 ! r2 !. . .rk !

n!
( x1 + x2 + ⋅ ⋅ ⋅ + xk ) =∑ x1r x2 . . . xk , dengan r1 + r2 + ⋅ ⋅ ⋅ + rk = n
n r rk
1 2

r1 ! r2 !. . .rk !

Principle of inclusion and exclusion
(Prinsip rangkuman dan eksklusi)

n ( A1 ∪ A2 ∪ ∪ Am ) = ∑ n( A ) − ∑ n( A ∩ A )
1< i < m
i
1< i < j < m
i j

+ ∑
1<i< j <k <m
(
n Ai ∩ A j ∩ Ak − ) + (−1) m +1 n ( A1 ∩ A2 ∩ ∩ Am )

956/1

28

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

29



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 + b2 is even, ∀ a, b ∈ S.
Show that R is an equivalence relation. [5 marks]

2 Let (G, ∗) be a group, a ∈ G and Ha = {x ∈ G| xa = ax}. Show that Ha is a subgroup of G.
[6 marks]

⎛ 2 −3 2 ⎞
⎜ ⎟
3 Find the eigenvalues and eigenvectors of the matrix ⎜ −3 3 3 ⎟ . [8 marks]
⎜ 2 3 2⎟
⎝ ⎠

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 ax 3 + bx 2 − bx + a . 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.
R

Q
A

X Z

P B Y C

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.
BY BP AZ AX
Show that = . Deduce similar expressions for and . [8 marks]
YC PC ZC XB
Hence, show that P, Q and R are collinear. [4 marks]

⎛ 2
3
−1⎞
2 ⎛ 0 −1⎞
6 The matrices A and B are given by A = ⎜ ⎟ and B = ⎜ ⎟ . Describe the respective
⎜ 1 3⎟
⎝ −1 0 ⎠
⎝ 2 2 ⎠
plane transformation represented by A and B, and hence, describe the single transformation
represented by BA4B. [6 marks]

956/2

30



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 + b2 adalah genap, ∀ a, b ∈ S.
Tunjukkan bahawa R ialah hubungan kesetaraan. [5 markah]

2 Katakan (G, ∗) ialah satu kumpulan, a ∈ G dan Ha = {x ∈ G| xa = ax}. Tunjukkan bahawa Ha
ialah satu subkumpulan G. [6 markah]

⎛ 2 −3 2 ⎞
⎜ ⎟
3 Cari nilai eigen dan vektor eigen matriks ⎜ −3 3 3 ⎟ . [8 markah]
⎜ 2 3 2⎟
⎝ ⎠

4 Katakan P3 ialah ruang vektor semua polinomial berdarjah selebih-lebihnya tiga dan W ialah set
semua polinomial berbentuk ax 3 + bx 2 − bx + a . 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.
R

Q
A

X Z

P B Y C
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.
BY BP AZ AX
Tunjukkan bahawa = . Deduksikan ungkapan yang serupa bagi dan .
YC PC ZC XB
[8 markah]
Dengan yang demikian, tunjukkan bahawa P, Q dan R adalah segaris. [4 markah]

⎛ 2
3
−1⎞
2 ⎛ 0 −1⎞
6 Matriks A dan B diberikan oleh A = ⎜ ⎟ dan B = ⎜ ⎟ . Perihalkan penjelmaan satah
⎜ 1 3⎟
⎝ −1 0 ⎠
⎝ 2 2 ⎠
yang masing-masing diwakili oleh A dan B, dan dengan yang demikian, perihalkan penjelmaan
tunggal yang diwakili oleh BA4B. [6 markah]

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31



7 The set G = {e, a, a2, a3, b, ab, a2b, a3b} and the operation are such that (G, ) is a group, where
e is the identity element, element a is of order 4, a2 = b2 and ba = a3b.
(a) Show, in any order, that ba2 = a2b and ba3 = 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]

3 3
8 The linear transformation L : → is defined by
⎛ x⎞ ⎛ 1 0 −2 ⎞⎛ x ⎞
⎜ ⎟ ⎜ ⎟⎜ ⎟
L⎜ y ⎟ = ⎜ 1 1 0 ⎟⎜ y ⎟ .
⎜z⎟ ⎜ 0 1 2 ⎟⎜ z ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
(a) Determine a basis for the range of L, and a basis for the null space of L. [6 marks]
⎛ 1⎞ ⎛ 2⎞
⎜ ⎟ ⎜ ⎟
(b) Find the image of the line r = ⎜ 2 ⎟ + λ ⎜ −2 ⎟ under L. [3 marks]
⎜ −2 ⎟ ⎜ 0⎟
⎝ ⎠ ⎝ ⎠
(c) Find, in the form r = a + μb, 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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32



7 Set G = {e, a, a2, a3, b, ab, a2b, a3b} dan operasi adalah sebegitu rupa sehinggakan (G, ) ialah
satu kumpulan, dengan e unsur identiti, unsur a berperingkat 4, a2 = b2 dan ba = a3b.
(a) Tunjukkan, mengikut mana-mana tertib, bahawa ba2 = a2b dan ba3 = 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]

3 3
8 Penjelmaan linear L : → ditakrifkan oleh
⎛ x ⎞ ⎛ 1 0 −2 ⎞⎛ x ⎞
⎜ ⎟ ⎜ ⎟⎜ ⎟
L ⎜ y ⎟ = ⎜ 1 1 0 ⎟⎜ y ⎟ .
⎜z⎟ ⎜ 0 1 2 ⎟⎜ z ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
(a) Tentukan satu asas bagi julat L, dan satu asas bagi ruang nol L. [6 markah]
⎛ 1⎞ ⎛ 2⎞
⎜ ⎟ ⎜ ⎟
(b) Cari imej garis r = ⎜ 2 ⎟ + λ ⎜ −2 ⎟ di bawah L. [3 markah]
⎜ −2 ⎟ ⎜ 0⎟
⎝ ⎠ ⎝ ⎠
(c) Cari, dalam bentuk r = a + μb, 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

33

(RUMUS MATEMATIK)

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

s ( s − a )( s − b)( s − c )
r=
s

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

abc
R=
4 s ( s − a )( s − b)( s − c )

956/2

34

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

35



1 Show that
ln 2

∫
255 1
sinh 2 x cosh 2 x dx = − ln 2. [6 marks]
0 1024 8

0
Deduce the exact value of
∫ ln 1
2
sinh 2 x cosh 2 x dx. [2 marks]

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

∞
1
3 Using a comparison test, show that the series ∑ 3k 2 + k is convergent. [4 marks]
k =1

dy
4 It is given that = x( x 2 + y + 2) with the initial condition y = 1 when x = 0. Using the Euler
dx
formula y1 ≈ y0 + hf ( x0 , y0 ), 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]

( x − 1) 2 ln x
6 Show that lim exists, and find its value. [6 marks]
( x , y ) → (1, 0 ) ( x − 1) 2 + y 2

956/3

36



1 Tunjukkan bahawa
ln 2

∫
255 1
sinh 2 x kosh 2 x dx = − ln 2. [6 markah]
0 1024 8

0
Deduksikan nilai tepat
∫ ln 1
2
sinh 2 x kosh 2 x dx. [2 markah]

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

∞
1
3 Dengan menggunakan ujian bandingan, tunjukkan bahawa siri ∑ 3k 2 + k adalah menumpu.
k =1
[4 markah]

dy
4 Diberikan bahawa = x( x 2 + y + 2) dengan syarat awal y = 1 apabila x = 0. Dengan
dx
menggunakan rumus Euler y1 ≈ y0 + hf ( x0 , y0 ), 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]

( x − 1) 2 ln x
6 Tunjukkan bahawa had wujud, dan cari nilainya. [6 markah]
( x , y ) →(1,0) ( x − 1) 2 + y 2

956/3

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7 Find the general solution of the differential equation
d2 y dy
2
+4 + 12 y = 8 cos 2t . [8 marks]
dt dt
(a) Find the approximate values of y when t = nπ and t = ( n + 1 ) π , where n is a large positive
2
integer. [3 marks]
(b) Show that, whatever the initial conditions, the limiting solution as t → ∞ may be expressed in
the form y = k sin(2t + α ), where k is a positive integer and α 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]

956/3

38



7 Cari selesaian am persamaan pembezaan
d2 y dy
+4 + 12 y = 8 kos 2t . [8 markah]
dt 2
dt
(a) Cari nilai hampiran y apabila t = nπ dan t = ( n + 1 ) π , dengan n integer positif yang besar.
2
[3 markah]
(b) Tunjukkan bahawa, walau apa pun syarat awal, selesaian pengehad semasa t → ∞ boleh
diungkapkan dalam bentuk y = k sin(2t + α ), dengan k integer positif dan α 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

39

(RUMUS MATEMATIK)

Inverse hyperbolic functions
(Fungsi hiperbolik songsang)

sinh−1x = ln ( x + x2 + 1 )

cosh−1x = ln ( x + x 2 − 1 ), x > 1

1 ⎛1+ x ⎞
tanh−1x = ln ⎜ ⎟ , |x| < 1
2 ⎝ 1− x ⎠

sinh−1x = ln ( x + x2 + 1 )

kosh−1x = ln ( x + x 2 − 1 ), x > 1

1 ⎛1+ x ⎞
tanh−1x = ln ⎜ ⎟ , |x| < 1
2 ⎝ 1− x ⎠

Integrals
(Pengamiran)
1 1 ⎛x⎞
∫a 2
+x
dx = tan −1 ⎜ ⎟ + c
a2
⎝a⎠
1 ⎛x⎞
∫ a −x2
dx = sin −1 ⎜ ⎟ + c
2⎝a⎠
1 ⎛x⎞
∫ x2 − a2
dx = cosh −1 ⎜ ⎟ + c
⎝a⎠
1 ⎛x⎞
∫ x +a2
dx = sinh −1 ⎜ ⎟ + c
2⎝a⎠

1 1 ⎛x⎞
∫a 2
+x
dx = tan −1 ⎜ ⎟ + c
a2
⎝a⎠
1 ⎛x⎞
∫ a2 − x2
dx = sin −1 ⎜ ⎟ + c
⎝a⎠
1 ⎛x⎞
∫ 2
x −a
dx = kosh −1 ⎜ ⎟ + c
2 ⎝a⎠
1 ⎛x⎞
∫ x2 + a2
dx = sinh −1 ⎜ ⎟ + c
⎝a⎠

956/3

40

956 Sukatan Pelajaran Matematik Lanjutan STPM (Baharu)

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