math t ++ guideline for assignment 2013

7/27/2019 Math T ++ guideline for assignment 2013

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STPM/S(E)954

PEPERIKSAAN

SIJIL TINGGI PERSEKOLAHAN MALAYSIA(MALAYSIA HIGHER SCHOOL CERTIFICATE EXAMINATION)

MATHEMATICS (T)

Syllabus and Specimen Papers

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

MAJLIS PEPERIKSAAN MALAYSIA

(MALAYSIAN EXAMINATIONS COUNCIL)


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NATIONAL EDUCATION PHILOSOPHY

Education in Malaysia is an on-going effort towards further

developing the potential of individuals in a holistic andintegrated manner, so as to produce individuals who are

intellectually, spiritually, emotionally and physically

balanced and harmonious, based on a belief in and devotion

to God. Such effort is designed to produce Malaysian

citizens who are knowledgeable and competent, who possess

high moral standards, and who are responsible and capableof achieving a high level of personal well-being as well as

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

society and the nation at large.


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FOREWORD

This revised Mathematics (T) 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 firstexamination will also be held the same year. The revision of the syllabus takes into account the

changes made by the Malaysian Examinations Council (MEC) to the existing STPM examination.Through the new system, the form six study will be divided into three terms, and candidates will sitfor 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 teachingand learning orientation of form six so as to be in line with the orientation of teaching and learning incolleges and universities.

The Mathematics (T) syllabus is designed to provide a framework for a pre-university course that

enables candidates to develop the understanding of mathematical concepts and mathematical thinking,and acquire skills in problem solving and the applications of mathematics related to science andtechnology. The assessment tools of this syllabus consist of written papers and coursework.Coursework offers opportunities for candidates to conduct mathematical investigation and

mathematical modelling that enhance their understanding of mathematical processes and applicationsand provide a platform for them to develop soft skills.

The syllabus contains topics, teaching periods, learning outcomes, examination format, gradedescription and specimen papers.

The design of this syllabus was undertaken by a committee chaired by Professor Dr. Abu Osman binMd Tap from International Islamic University Malaysia. Other committee members consist of

university lecturers, representatives from the Curriculum Development Division, Ministry ofEducation Malaysia, and experienced teachers who are teaching Mathematics. On behalf of MEC, Iwould like to thank the committee for their commitment and invaluable contribution. It is hoped thatthis syllabus will be a guide for teachers and candidates in the teaching and learning process.

Chief ExecutiveMalaysian Examinations Council


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CONTENTS

Syllabus 954 Mathematics (T)

Page

Aims 1

Objectives 1

Content

First Term: Algebra and Geometry 25

Second Term: Calculus 68

Third Term: Statistics 912

Coursework 13

Scheme of Assessment 14

Performance Descriptions 15

Mathematical Notation 1619

Electronic Calculators 20

Reference Books 20

Specimen Paper 1 2126



Specimen Assignment Paper 4 4546


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SYLLABUS

954 MATHEMATICS (T)[May not be taken with 950 Mathematics (M)]

Aims

The Mathematics (T) syllabus is designed to provide a framework for a pre-university course thatenables candidates to develop the understanding of mathematical concepts and mathematical thinking,and acquire skills in problem solving and the applications of mathematics related to science andtechnology.

Objectives

The objectives of the syllabus are to enable candidates to:

(a) use mathematical concepts, terminology and notation;

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

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

(d) use appropriate mathematical models in different contexts;

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

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

(g) interpret the significance and reasonableness of results;

(h) present mathematical explanations, arguments and conclusions.


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FIRST TERM: ALGEBRA AND GEOMETRY

TopicTeachingPeriod

Learning Outcome

1 Functions 28 Candidates should be able to:

1.1 Functions 6 (a) state the domain and range of a function, andfind composite functions;

(b) determine whether a function is one-to-one,and find the inverse of a one-to-one function;

(c) sketch the graphs of simple functions,including piecewise-defined functions;

1.2 Polynomial andrational functions

8 (d) use the factor theorem and the remaindertheorem;

(e) solve polynomial and rational equations andinequalities;

(f) solve equations and inequalities involvingmodulus signs in simple cases;

(g) decompose a rational expression into partial

fractions in cases where the denominator hastwo distinct linear factors, or a linear factor

and a prime quadratic factor;

1.3 Exponential and

logarithmic functions

6 (h) relate exponential and logarithmic functions,

algebraically and graphically;

(i) use the properties of exponents and logarithms;

(j) solve equations and inequalities involvingexponential or logarithmic expressions;

1.4 Trigonometricfunctions

8 (k) relate the periodicity and symmetries of thesine, cosine and tangent functions to their

graphs, and identify the inverse sine, inversecosine and inverse tangent functions and theirgraphs;

(l) use basic trigonometric identities and theformulae for sin (AB), cos (AB) andtan (AB), including sin 2A, cos 2A andtan 2A;

(m) express a sin + b cos in the forms

rsin ( ) and rcos ( );

(n) find the solutions, within specified intervals,of trigonometric equations and inequalities.


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TopicTeachingPeriod

Learning Outcome

2 Sequences and Series 18 Candidates should be able to:

2.1 Sequences 4 (a) use an explicit formula and a recursive formulafor a sequence;

(b) find the limit of a convergent sequence;

2.2 Series 8 (c) use the formulae for the nth term and for thesum of the first n terms of an arithmetic series

and of a geometric series;

(d) identify the condition for the convergence of ageometric series, and use the formula for thesum of a convergent geometric series;

(e) use the method of differences to find the nthpartial sum of a series, and deduce the sum ofthe series in the case when it is convergent;

2.3 Binomial expansions 6 (f) expand (a + b)n

, where n ;

(g) expand (1 +x)n , where n , and identify the

condition x < 1 for the validity of thisexpansion;

(h) use binomial expansions in approximations.

3 Matrices 16 Candidates should be able to:

3.1 Matrices 10 (a) identify null, identity, diagonal, triangular andsymmetric matrices;

(b) use the conditions for the equality of two

matrices;

(c) perform scalar multiplication, addition,subtraction and multiplication of matrices withat most three rows and three columns;

(d) use the properties of matrix operations;

(e) find the inverse of a non-singular matrix usingelementary row operations;

(f) evaluate the determinant of a matrix;

(g) use the properties of determinants;


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TopicTeachingPeriod

Learning Outcome

3.2 Systems of linearequations

6 (h) reduce an augmented matrix to row-echelonform, and determine whether a system of linear

equations has a unique solution, infinitelymany solutions or no solution;

(i) apply the Gaussian elimination to solve a

system of linear equations;

(j) find the unique solution of a system of linearequations using the inverse of a matrix.

4 Complex Numbers 12 Candidates should be able to:

(a) identify the real and imaginary parts of acomplex number;

(b) use the conditions for the equality of twocomplex numbers;

(c) find the modulus and argument of a complex

number in cartesian form and express thecomplex number in polar form;

(d) represent a complex number geometrically bymeans of an Argand diagram;

(e) find the complex roots of a polynomialequation with real coefficients;

(f) perform elementary operations on two

complex numbers expressed in cartesian form;

(g) perform multiplication and division of twocomplex numbers expressed in polar form;

(h) use de Moivres theorem to find the powersand roots of a complex number.

5 Analytic Geometry 14 Candidates should be able to:

(a) transform a given equation of a conic into thestandard form;

(b) find the vertex, focus and directrix of a

parabola;

(c) find the vertices, centre and foci of an ellipse;

(d) find the vertices, centre, foci and asymptotesof a hyperbola;

(e) find the equations of parabolas, ellipses andhyperbolas satisfying prescribed conditions(excluding eccentricity);


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TopicTeachingPeriod

Learning Outcome

(f) sketch conics;

(g) find the cartesian equation of a conic definedby parametric equations;

(h) use the parametric equations of conics.

6 Vectors 20 Candidates should be able to:

6.1 Vectors in two andthree dimensions

8 (a) use unit vectors and position vectors;

(b) perform scalar multiplication, addition andsubtraction of vectors;

(c) find the scalar product of two vectors, anddetermine the angle between two vectors;

(d) find the vector product of two vectors, and

determine the area a parallelogram and of atriangle;

6.2 Vector geometry 12 (e) find and use the vector and cartesian equationsof lines;

(f) find and use the vector and cartesian equationsof planes;

(g) calculate the angle between two lines, betweena line and a plane, and between two planes;

(h) find the point of intersection of two lines, andof a line and a plane;

(i) find the line of intersection of two planes.


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SECOND TERM: CALCULUS

TopicTeachingPeriod

Learning Outcome

7 Limits and Continuity 12 Candidates should be able to:

7.1 Limits 6 (a) determine the existence and values of the left-hand limit, right-hand limit and limit of afunction;

(b) use the properties of limits;

7.2 Continuity 6 (c) determine the continuity of a function at a

point and on an interval;

(d) use the intermediate value theorem.

8 Differentiation 28 Candidates should be able to:

8.1 Derivatives 12 (a) identify the derivative of a function as a limit;

(b) find the derivatives ofxn (n ), ex, lnx,

sinx, cosx, tanx, sin1x, cos

1x, tan

1x, with

constant multiples, sums, differences,products, quotients and composites;

(c) perform implicit differentiation;

(d) find the first derivatives of functions definedparametrically;

8.2 Applications ofdifferentiation

16 (e) determine where a function is increasing,decreasing, concave upward and concavedownward;

(f) determine the stationary points, extremumpoints and points of inflexion;

(g) sketch the graphs of functions, includingasymptotes parallel to the coordinate axes;

(h) find the equations of tangents and normals tocurves, including parametric curves;

(i) solve problems concerning rates of change,including related rates;

(j) solve optimisation problems.

9 Integration 28 Candidates should be able to:

9.1 Indefinite integrals 14 (a) identify integration as the reverse of

differentiation;

(b) integratexn

(n ), ex, sinx, cosx, sec

2x, with

constant multiples, sums and differences;


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TopicTeachingPeriod

Learning Outcome

(c) integrate rational functions by means ofdecomposition into partial fractions;

(d) use trigonometric identities to facilitate the

integration of trigonometric functions;

(e) use algebraic and trigonometric substitutionsto find integrals;

(f) perform integration by parts;

9.2 Definite integrals 14 (g) identify a definite integral as the area under acurve;

(h) use the properties of definite integrals;

(i) evaluate definite integrals;(j) calculate the area of a region bounded by a

curve (including a parametric curve) and linesparallel to the coordinate axes, or between twocurves;

(k) calculate volumes of solids of revolution aboutone of the coordinate axes.

10 Differential Equations 14 Candidates should be able to:

(a) find the general solution of a first order

differential equation with separable variables;(b) find the general solution of a first order linear

differential equation by means of an integratingfactor;

(c) transform, by a given substitution, a first orderdifferential equation into one with separablevariables or one which is linear;

(d) use a boundary condition to find a particularsolution;

(e) solve problems, related to science and

technology, that can be modelled by differentialequations.

11 Maclaurin Series 12 Candidates should be able to:

(a) find the Maclaurin series for a function and theinterval of convergence;

(b) use standard series to find the series expansionsof the sums, differences, products, quotientsand composites of functions;


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TopicTeachingPeriod

Learning Outcome

(c) perform differentiation and integration of apower series;

(d) use series expansions to find the limit of a

function.

12 Numerical Methods 14 Candidates should be able to:

12.1 Numerical solution of

equations

10 (a) locate a root of an equation approximately by

means of graphical considerations and bysearching for a sign change;

(b) use an iterative formula of the form

1 f ( )n nx x to find a root of an equation to a

prescribed degree of accuracy;

(c) identify an iteration which converges ordiverges;

(d) use the Newton-Raphson method;

12.2 Numerical integration 4 (e) use the trapezium rule;

(f) use sketch graphs to determine whether thetrapezium rule gives an over-estimate or an

under-estimate in simple cases.


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THIRD TERM: STATISTICS

TopicTeachingPeriod

Learning Outcome

13 Data Description 14 Candidates should be able to:

(a) identify discrete, continuous, ungrouped andgrouped data;

(b) construct and interpret stem-and-leaf diagrams,box-and-whisker plots, histograms and

cumulative frequency curves;

(c) state the mode and range of ungrouped data;

(d) determine the median and interquartile rangeof ungrouped and grouped data;

(e) calculate the mean and standard deviation ofungrouped and grouped data, from raw data

and from given totals such as1

( )n

ii

x a

and

1

2( )

n

ii

x a

;

(f) select and use the appropriate measures ofcentral tendency and measures of dispersion;

(g) calculate the Pearson coefficient of skewness;

(h) describe the shape of a data distribution.

14 Probability 14 Candidates should be able to:

(a) apply the addition principle and themultiplication principle;

(b) use the formulae for combinations andpermutations in simple cases;

(c) identify a sample space, and calculate theprobability of an event;

(d) identify complementary, exhaustive and

mutually exclusive events;

(e) use the formula

P(AB) = P(A) + P(B) P(A B);

(f) calculate conditional probabilities, and identifyindependent events;

(g) use the formulae

P(A B) = P(A) P(B|A) = P(B) P(A|B);

(h) use the rule of total probability.


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TopicTeachingPeriod

Learning Outcome

15 Probability Distributions 26 Candidates should be able to:

15.1 Discrete randomvariables

6 (a) identify discrete random variables;

(b) construct a probability distribution table for adiscrete random variable;

(c) use the probability function and cumulativedistribution function of a discrete randomvariable;

(d) calculate the mean and variance of a discreterandom variable;

15.2 Continuous random

variables

6 (e) identify continuous random variables;

(f) relate the probability density function andcumulative distribution function of acontinuous random variable;

(g) use the probability density function andcumulative distribution function of acontinuous random variable;

(h) calculate the mean and variance of acontinuous random variable;

15.3 Binomial distribution 4 (i) use the probability function of a binomial

distribution, and find its mean and variance;

(j) use the binomial distribution as a model forsolving problems related to science and

technology;

15.4 Poisson distribution 4 (k) use the probability function of a Poissondistribution, and identify its mean andvariance;

(l) use the Poisson distribution as a model forsolving problems related to science andtechnology;

15.5 Normal distribution 6 (m) identify the general features of a normaldistribution, in relation to its mean andstandard deviation;

(n) standardise a normal random variable and usethe normal distribution tables;

(o) use the normal distribution as a model forsolving problems related to science andtechnology;

(p) use the normal distribution, with continuitycorrection, as an approximation to the

binomial distribution, where appropriate.


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TopicTeachingPeriod

Learning Outcome

16 Sampling and Estimation 26 Candidates should be able to:

16.1 Sampling 14 (a) distinguish between a population and a sample,and between a parameter and a statistic;

(b) identify a random sample;

(c) identify the sampling distribution of a statistic;

(d) determine the mean and standard deviation ofthe sample mean;

(e) use the result that Xhas a normal distribution ifXhas a normal distribution;

(f) use the central limit theorem;

(g) determine the mean and standard deviation ofthe sample proportion;

(h) use the approximate normality of the sampleproportion for a sufficiently large sample size;

16.2 Estimation 12 (i) calculate unbiased estimates for the populationmean and population variance;

(j) calculate an unbiased estimate for thepopulation proportion;

(k) determine and interpret a confidence interval

for the population mean based on a samplefrom a normally distributed population with

known variance;

(l) determine and interpret a confidence intervalfor the population mean based on a largesample;

(m) find the sample size for the estimation of

population mean;

(n) determine and interpret a confidence interval

for the population proportion based on a large

sample;(o) find the sample size for the estimation of

population proportion.


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TopicTeachingPeriod

Learning Outcome

17 Hypothesis Testing 14 Candidates should be able to:

(a) explain the meaning of a null hypothesis andan alternative hypothesis;

(b) explain the meaning of the significance level

of a test;

(c) carry out a hypothesis test concerning thepopulation mean for a normally distributedpopulation with known variance;

(d) carry out a hypothesis test concerning thepopulation mean in the case where a largesample is used;

(e) carry out a hypothesis test concerning thepopulation proportion by direct evaluation ofbinomial probabilities;

(f) carry out a hypothesis test concerning thepopulation proportion using a normalapproximation.

18 Chi-squared Tests 14 Candidates should be able to:

(a) identify the shape, as well as the mean andvariance, of a chi-squared distribution with a

given number of degrees of freedom;

(b) use the chi-squared distribution tables;

(c) identify the chi-squared statistic;

(d) use the result that classes with small expectedfrequencies should be combined in a chi-squared test;

(e) carry out goodness-of-fit tests to fit prescribed

probabilities and probability distributions withknown parameters;

(f) carry out tests of independence in contingencytables (excluding Yates correction).


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Coursework

The Mathematics (T) coursework is intended to enable candidates to carry out mathematicalinvestigation and mathematical modelling, so as to enhance the understanding of mathematicalprocesses and applications and to develop soft skills.

The coursework comprises three assignments set down by the Malaysian Examinations Council.The assignments are based on three different areas of the syllabus and represent two types of tasks:mathematical investigation and mathematical modelling.

A school candidate is required to carry out one assignment in each term under the supervision ofthe subject teacher as specified in the Teachers Manual for Mathematics (T) Coursework which canbe downloaded from MECs Portal (http://www.mpm.edu.my) by the subject teacher during the firstterm of form six. The assignment reports are graded by the subject teacher in the respective terms. Aviva session is conducted by the teacher in each term after the assessment of the assignment reports.

An individual private candidate is required to carry out one assignment in each term as specified

in the Individual Private Candidates Manual for Mathematics (T) Coursework which can bedownloaded from MECs Portal (http://www.mpm.edu.my) by the candidate during the first term ofform six. The assignment reports are graded by an external examiner in the respective terms. A vivasession is conducted by the examiner in each term after the assessment of the assignment reports.

A repeating candidate may use the total mark obtained in the coursework for the subsequentSTPM examination. Requests to carry forward the moderated coursework mark should be madeduring the registration of the examination.
http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/http://www.mpm.edu%2Cmy/


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Scheme of Assessment

Term ofStudy

Paper Codeand Name

Type of TestMark

(Weighting)Duration Administration

First

Term

954/1

Mathematics (T)Paper 1

Written test 60

(26.67%)

1 hours Central assessment

Section AAnswer all 6 questions ofvariable marks.

45

Section BAnswer 1 out of 2questions.

All questions are based ontopics 1 to 6.

15

SecondTerm

954/2Mathematics (T)

Paper 2

Written test 60(26.67%)


Section AAnswer all 6 questions of

variable marks.

45



15

Third

Term


Paper 3

Written test 60

(26.67%)


Section AAnswer all 6 questions ofvariable marks.

45



15

First,

Second

and

Third

Terms


Paper 4

Coursework

3 assignments, each based

on topics 1 to 6, topics 7 to12 and topics 13 to 18.

180

to bescaled to

45(20%)

Throughoutthe three

terms

Assessment byschool teachers for

candidates fromgovernment and

government-aidedschools

Assessment by

appointed assessorsfor candidates fromprivate schools and

individual privatecandidates


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Performance Descriptions

A grade A candidate is likely able to:

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

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

(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 whereappropriate;

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


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Mathematical Notation

Miscellaneous symbols

= is equal to

is not equal to

is identical to or is congruent to

is approximately equal to

< is less than

is less than or equal to

> is greater than

is greater than or equal to

infinity

therefore

Operationsa + b a plus b

a b a minus b

a b, ab a multiplied by b

a b,a

b a divided by b

a : b ratio ofa to b

an nth power ofa

1

2 ,a a positive square root ofa

1

, nna a positive nth root ofa

|a| absolute value of a real numbera

1

n

i

i

u

u1 + u2 + + un

n! n factorial forn

n

r

binomial coefficient!

!( )!

n

r n rforn, r , 0 r n

Set notation

is an element of

is not an element of

empty set

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

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

set of integers

set of positive integers

set of rational numbers set of real numbers

>

<


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Bahagian A [45 markah]

Jawab semua soalan dalam bahagian ini.

1 Fungsi f dan g ditakrifkan oleh

2

2

f : e , ;

g : (ln ) , 0.

xx x

x x x

(a) Cari f1

dan nyatakan domainnya. [3 markah]

(b) Tunjukkan bahawa 12g = g(2), dan nyatakan, dengan satu sebab, sama ada g mempunyaisongsangan. [4 markah]

2 Satu jujukan ditakrifkan oleh ur = e(r1)

er

bagi semua integerr 1. Cari1

,

n

r

r

u

dalam sebutan

n, dan deduksikan nilai1

.rr

u

[5 markah]

3 Matriks P =

2 2 0

0 0 2

a b c

dan Q =

1 1 0

0 0 1

0 2 2

adalah sebegitu rupa sehinggakan PQ = QP.

(a) Tentukan nilai a, b, dan c. [5 markah]

(b) Cari nombor nyata m dan n supaya P = mQ + nI, dengan I matriks identiti 3 3. [5 markah]

4 Ungkapkan nombor kompleks z= 1 3i dalam bentuk kutub. [4 markah]

Dengan yang demikian, cari5

5 1

zz dan 5

5

1.z

z [4 markah]

5 Persamaan satu hiperbola ialah 4x29y

224x18y9 = 0.

(a) Dapatkan bentuk piawai bagi persamaan hiperbola itu. [3 markah]

(b) Cari bucu dan persamaan asimptot hiperbola itu. [6 markah]

6 Cari persamaan satah yang mengandung garis lurus2

1

3

43

zyx dan serenjang dengan

satah 3x + 2yz= 3. [6 markah]

954/1

>


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Section B [15 marks]

Answer anyonequestioninthissection.

7 Express cos x + sin x in the form rcos (x), where r> 0 and 0 < < 12

. Hence, find the

minimum and maximum values of cos x + sin x and the corresponding values ofx in the interval0 x 2. [7 marks]

(a) Sketch the graph ofy = cosx + sinx for 0 x 2. [3 marks]

(b) By drawing appropriate lines on your graph, determine the number of roots in the interval

0 x 2of each of the following equations.

(i) cosx + sinx = 12

[1 mark]

(ii) cosx + sinx = 2 [1 mark]

(c) Find the set of values ofx in the interval 0 x 2for which cos sin 1.x x [3 marks]

8 The position vectors a, b and c of three pointsA,B and Crespectively are given by

a = i +j + k,

b = i + 2j + 3k,

c = i3j +2k.

(a) Find a unit vector parallel to a + b + c. [3 marks]

(b) Calculate the acute angle between a and a + b + c. [3 marks]

(c) Find the vector of the form i + j + kperpendicular to both a and b. [2 marks]

(d) Determine the position vector of the point D which is such that ABCD is a parallelogramhaving BD as a diagonal. [3 marks]

(e) Calculate the area of the parallelogramABCD. [4 marks]

954/1

math t ++ guideline for assignment 2013

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