956 Sukatan Pelajaran Matematik Lanjutan STPM (Baharu)

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956 Sukatan Pelajaran Matematik Lanjutan STPM (Baharu)

  1. 1. STPM/S(E)956 MAJLIS PEPERIKSAAN MALAYSIA (MALAYSIAN EXAMINATIONS COUNCIL) PEPERIKSAANSIJIL 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.
  2. 2. FALSAFAH PENDIDIKAN KEBANGSAAN“Pendidikan di Malaysia adalah satu usaha berterusanke arah memperkembangkan lagi potensi individu secaramenyeluruh dan bersepadu untuk mewujudkan insan yangseimbang dan harmonis dari segi intelek, rohani, emosi,dan jasmani. Usaha ini adalah bagi melahirkan rakyatMalaysia yang berilmu pengetahuan, berakhlak mulia,bertanggungjawab, berketerampilan, dan berkeupayaanmencapai kesejahteraan diri serta memberi sumbanganterhadap keharmonian dan kemakmuran keluarga,masyarakat dan negara.”
  3. 3. FOREWORDThis revised Further Mathematics syllabus is designed to replace the existing syllabus which has beenin use since the 2002 STPM examination. This new syllabus will be enforced in 2012 and the firstexamination will also be held the same year. The revision of the syllabus takes into account thechanges 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 foran 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 andlearning orientation in sixth form so as to be in line with the orientation of teaching and learning incolleges and universities.The Further Mathematics syllabus is designed to cater for candidates who are competence and haveintense interest in mathematics and wish to further develop their understanding of mathematicalconcepts and mathematical thinking and acquire skills in problem solving and the applications ofmathematics. 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 binMad Tap of International Islamic University of Malaysia. Other committee members consist ofuniversity lecturers, representatives from the Curriculum Development Division, Ministry ofEducation Malaysia, and experienced teachers teaching Mathematics. On behalf of the MalaysianExaminations Council, I would like to thank the committee for their commitment and invaluablecontribution. It is hoped that this syllabus will be a guide for teachers and candidates in the teachingand learning process.OMAR BIN ABU BAKARChief ExecutiveMalaysian Examinations Council
  4. 4. CONTENTS Syllabus 956 Further Mathematics PageAims 1Objectives 1Content First Term: Discrete Mathematics 2–4 Second Term: Algebra and Geometry 5–7 Third Term: Calculus 8 – 11Scheme of Assessment 12Performance Descriptions 13Mathematical Notation 14 – 18Electronic Calculators 19Reference Books 19Specimen Paper 1 21 – 28Specimen Paper 2 29 – 34Specimen Paper 3 35 – 40
  5. 5. SYLLABUS 956 FURTHER MATHEMATICSAimsThe Further Mathematics syllabus caters for candidates who have high competence and intenseinterest in mathematics and wish to further develop the understanding of mathematical concepts andmathematical thinking and acquire skills in problem solving and the applications of mathematics.ObjectivesThe 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
  6. 6. FIRST TERM: DISCRETE MATHEMATICS Teaching Topic Learning Outcome Period1 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
  7. 7. Teaching Topic Learning Outcome Period3 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
  8. 8. Teaching Topic Learning Outcome Period5 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 constant coefficients; (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
  9. 9. SECOND TERM: ALGEBRA AND GEOMETRY Teaching Topic Learning Outcome Period7 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
  10. 10. Teaching Topic Learning Outcome Period9 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
  11. 11. Teaching Topic Learning Outcome Period11 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
  12. 12. THIRD TERM: CALCULUS Teaching Topic Learning Outcome Period13 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
  13. 13. Teaching Topic Learning Outcome 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
  14. 14. Teaching Topic Learning Outcome Period16 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 constant coefficients; (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
  15. 15. Teaching Topic Learning Outcome Period18 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
  16. 16. 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 Term Further (33.33%) Mathematics Section A 45 Paper 2 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 7 to 12. Third 956/3 Written test 60 Term Further (33.33%) Mathematics Section A 45 Paper 3 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 13 to 18. 12
  17. 17. Performance DescriptionsA 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
  18. 18. 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
  19. 19. p→q if p then qp↔q p if and only if qp ≡ q p is logically equivalent to qp ≡ q / p is not logically equivalent to qSet 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∩ intersectionU universal setA complement of a set A⊆ is a subset of⊂ is a proper subset⊆⁄ is not a subset of⊄ is not a proper subsetn(A) number of elements in a set AP(A) power set of AA×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 AAΔB symmetry difference of sets A and B, (A − B) ∪ (B − A)Number theorya⏐b a divides ba / b | a does not divide bgcd (a, b) greatest common divisor of integers a and blcm (a, b) least common multiple of integers a and b 15
  20. 20. 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 xGraphsG a graph GV(G) set of vertices of a graph GE(G) set of edges of a graph Gdeg (v) degree of vertex v{v, w} edge joining v and w in a simple graphκn a complete graph on n verticesκ m,n a complete bipartite graph with one set of m vertices and another set of n verticesRelationsyRx y is related to x by a relation Ry~x y is equivalent to x, in the context of some equivalence relation[a] equivalence class of an element aA/R a partition of set A induced by the equivalence relation R on AGroups(G, *) a set G together with a binary operation *e identity element −1a inverse of an element a≅ is isomorphic toMatricesA a matrix AI identity matrix0 null matrix −1A inverse of a non-singular square matrix A TA transpose of a matrix Adet A determinant of a square matrix AVector spacesV a vector space V 2 set of real ordered pairs 3 set of real ordered triples n set of real ordered n-tuplesT a linear transformation T 16
  21. 21. GeometryAB length of the line segment with end points A and B∠A angle at A∠BAC angle between line segments AB and ACΔABC triangle whose vertices are A, B and C// is parallel to⊥ is perpendicular toVectorsa a vector a|a| magnitude of a vector aˆa unit vector in the direction of the vector ai, j, k unit vectors in the directions of the cartesian coordinates axesAB vector represented in magnitude and direction by the directed line segment from point A to point B| AB | magnitude of ABaib scalar product of vectors a and ba×b vector product of vectors a and bFunctionsf a function ff(x) value of a function f at xf : A→ B f is a function under which each element of set A has an image in set Bf :x y f is a function which maps the element x to the element yf −1 inverse function of ff g composite function of f and g which is defined by f g(x) = f[g(x)]ex exponential function of xloga x logarithm to base a of xln x natural logarithm of x, loge xlg x logarithm to base 10 of x, log10 xsin, cos, tan, trigonometric functionscsc, sec, cotsin−1, cos−1, tan−1, inverse trigonometric functionscsc−1, sec−1, cot−1sinh, cosh, tanh, hyperbolic functionscsch, sech, cothsinh−1, cosh−1, tanh−1, inverse hyperbolic functionscsch−1, sech−1, coth−1 17
  22. 22. Derivatives and integralslim f ( x) limit of f(x) as x tends to ax→ady first derivative of y with respect to xdxf ( x) first derivative of f(x) with respect to xd2 y second derivative of y with respect to xdx 2f ( x) second derivative of f(x) with respect to xdn y nth derivative of y with respect to xdx nf (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 bVector-valued functionsκ curvatureT unit tangent vectorN unit normal vectorPartial derivatives∂y partial derivative of y with respect to x∂x ∂ ∂ ∂∇ del operator, ∇ = i +j +k ∂x ∂y ∂z 18
  23. 23. Electronic CalculatorsDuring the written paper examination, candidates are advised to have standard scientific calculatorswhich must be silent. Programmable and graphic display calculators are prohibited.Reference BooksDiscrete 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 5. Dolan, S., Neill, H. and Quadling, D., 2009. Further Mathematics for the IB Diploma: Standard Level. United Kingdom: Cambridge University Press. 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 11. Dolan, S., Neill, H. and Quadling, D., 2009. Further Mathematics for the IB Diploma: Standard Level. United Kingdom: Cambridge University Press. 12. Gaulter, B. and Gaulter, M., 2001. Further Pure Mathematics. United Kingdom: Oxford 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: Brooks/Cole, Cengage Learning. 15. Tan, S. T., 2011. Calculus: Early Transcendentals. California: Brooks/Cole, Cengage Learning. 19
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  25. 25. 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 ineither English or Bahasa Malaysia. All necessary working should be shown clearly. Scientific calculators may be used. Programmable and graphic display calculators areprohibited. A list of mathematical formulae is provided on page of this question paper.Arahan kepada calon: JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUATDEMIKIAN. 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 grafiktidak 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 21
  26. 26. 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 areintegers. [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 thesame colour are identical. (a) Find the number of ways in which all the balls can be arranged in a row so that all the whiteballs 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 onereceives 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 thesame colour. [2 marks]5 Let an be the number of ways (where the order is significant) the natural number n can be writtenas 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]956/1 22
  27. 27. 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, bagisemua 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 bentuk2501m + 2173n, dengan m dan n integer yang perlu ditentukan. [6 markah] Dengan yang demikian, cari integer positif terkecil p yang sebegitu rupa sehinggakan9977 + 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 berwarnahitam. Bola yang berwarna sama adalah secaman. (a) Cari bilangan cara semua bola itu boleh disusun dalam satu baris supaya semua bola putihbersama-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 daripadabola merah dan bola kuning. [3 markah] (c) Cari bilangan cara semua bola itu boleh diagihkan kepada 4 orang supaya setiap orangmenerima sekurang-kurangnya satu bola bagi setiap warna. [4 markah] (d) Tentukan bilangan bola yang mesti dipilih untuk memperoleh sekurang-kurangnya 4 bolayang berwarna sama. [2 markah]5 Katakan an ialah bilangan cara (tertib adalah bererti) nombor asli n boleh ditulis sebagai hasiltambah 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
  28. 28. 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 anycolumn of this incidence matrix? [2 marks]956/1 24
  29. 29. 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 hasiltambah kemasukan sebarang lajur matriks insidens ini? [2 markah]956/1 25
  30. 30. 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 positiveintegers 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
  31. 31. 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 positifyang 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 yangmungkin. [4 markah] (b) Jika n = 11 dan m = 46, tunjukkan bahawa G adalah berkait. [7 markah]956/1 27
  32. 32. 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
  33. 33. SPECIMEN PAPER 956/2 STPM FURTHER MATHEMATICS (MATEMATIK LANJUTAN) PAPER 2 (KERTAS 2) 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 ineither English or Bahasa Malaysia. All necessary working should be shown clearly. Scientific calculators may be used. Programmable and graphic display calculators areprohibited. A list of mathematical formulae is provided on page of this question paper.Arahan kepada calon: JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUATDEMIKIAN. 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 grafiktidak dibenarkan. Senarai rumus matematik dibekalkankan 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 956/2 29
  34. 34. Section A [45 marks] Answer all questions in this section.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 allpolynomials 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 astangents 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 segmentYX produced meets the line segment CA produced at a point Q. The line segment YZ produced meetsthe 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 transformationrepresented by BA4B. [6 marks]956/2 30
  35. 35. Bahagian A [45 markah] Jawab semua soalan dalam bahagian ini.1 Katakan S ialah set semua nombor perdana yang kurang daripada 20 dan R ialah hubungan pada Syang 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 Haialah 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 setsemua polinomial berbentuk ax 3 + bx 2 − bx + a . Tunjukkan bahawa W ialah subruang P3, dan carisatu asas bagi W. [8 markah]5 Gambar rajah di bawah menunjukkan satu bulatan yang terterap dalam segitiga ABC, dengan sisiAB, 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 dilanjurkandi titik P. Tembereng garis YX yang dilajurkan bertemu dengan tembereng garis CA yang dilanjurkandi titik Q. Tembereng garis YZ yang dilanjurkan bertemu dengan tembereng garis BA yang dilanjurkandi 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 penjelmaantunggal yang diwakili oleh BA4B. [6 markah]956/2 31
  36. 36. Section B [15 marks] Answer any one question in this section.7 The set G = {e, a, a2, a3, b, ab, a2b, a3b} and the operation are such that (G, ) is a group, wheree 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 isan addition modulo 8. Justify your conclusion. [3 marks] 3 38 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]956/2 32
  37. 37. Bahagian B [15 markah] Jawab mana-mana satu soalan dalam bahagian ini.7 Set G = {e, a, a2, a3, b, ab, a2b, a3b} dan operasi adalah sebegitu rupa sehinggakan (G, ) ialahsatu 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 +8penambahan modulo 8. Justifikasikan kesimpulan anda. [3 markah] 3 38 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
  38. 38. MATHEMATICAL FORMULAE (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
  39. 39. SPECIMEN PAPER 956/3 STPM FURTHER MATHEMATICS (MATEMATIK LANJUTAN) PAPER 3 (KERTAS 3) 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 ineither English or Bahasa Malaysia. All necessary working should be shown clearly. Scientific calculators may be used. Programmable and graphic display calculators areprohibited. A list of mathematical formulae is provided on page of this question paper.Arahan kepada calon: JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUATDEMIKIAN. 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 grafiktidak 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 956/3 35
  40. 40. Section A [45 marks] Answer all questions in this section.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] ∞ 13 Using a comparison test, show that the series ∑ 3k 2 + k is convergent. [4 marks] k =1 dy4 It is given that = x( x 2 + y + 2) with the initial condition y = 1 when x = 0. Using the Euler dxformula y1 ≈ y0 + hf ( x0 , y0 ), obtain an estimate of y at x = 0.1 in five steps correct to four decimalplaces. [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 x6 Show that lim exists, and find its value. [6 marks] ( x , y ) → (1, 0 ) ( x − 1) 2 + y 2956/3 36
  41. 41. Bahagian A [45 markah] Jawab semua soalan dalam bahagian ini.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] ∞ 13 Dengan menggunakan ujian bandingan, tunjukkan bahawa siri ∑ 3k 2 + k adalah menumpu. k =1 [4 markah] dy4 Diberikan bahawa = x( x 2 + y + 2) dengan syarat awal y = 1 apabila x = 0. Dengan dxmenggunakan rumus Euler y1 ≈ y0 + hf ( x0 , y0 ), dapatkan satu anggaran y di x = 0.1 dalam limalangkah 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 x6 Tunjukkan bahawa had wujud, dan cari nilainya. [6 markah] ( x , y ) →(1,0) ( x − 1) 2 + y 2956/3 37
  42. 42. Section B [15 marks] Answer any one question in this section.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 2integer. [3 marks] (b) Show that, whatever the initial conditions, the limiting solution as t → ∞ may be expressed inthe form y = k sin(2t + α ), where k is a positive integer and α an acute angle which are to bedetermined. [4 marks]8 A right pyramid has a rectangular base of length 2x and width 2y. The slant edges are each oflength 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 from4.00 to 4.04 and y changes from 3.00 to 2.94. [6 marks]956/3 38
  43. 43. Bahagian B [15 markah] Jawab mana-mana satu soalan dalam bahagian ini.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 → ∞ bolehdiungkapkan dalam bentuk y = k sin(2t + α ), dengan k integer positif dan α sudut tirus yang perluditentukan. [4 markah]8 Satu piramid tegak mempunyai tapak segiempat dengan panjang 2x dan lebar 2y. Setiap tepisendeng 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 seluruhjika x berubah dari 4.00 ke 4.04 dan y berubah dari 3.00 ke 2.94. [6 markah]956/3 39
  44. 44. MATHEMATICAL FORMULAE (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

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