@

CARIBBEAN EXAMINATIONS COUNCIL

Caribbean AdvancedProficiency examination

Pure MathematicsSyllabus

Effective for examinations from Mav/June 2008

Correspondence related to the syllabus should be addressed ro:

The Pro-RegistrarCaribbean Examinations Council

Caenwood Centre37 Amold Road, Kingston 5, Jamaica, W.l.

Telephone Number: (876) 920-6714Facsimile Number: (876) 967-4912

E-mail address: cxcwzo@ cxc.orgWebsite: www.cxc.org

Copyright @ 2007, by Caribbean Examinations CouncilThe Carrison, St. Michael BB 11158, Barbados

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A6N2/07

o uxtr | - tlglBnl EI0ilHnVilD llil,GUlUSMODULE l: BASIC ATGEBRA AND FUNCTTONS

GENERAL OBJECTIVES

On completion of rhis Module, students should;

I. understand the concept ofnumber;

Z. develop the ability to construct simple proofs of mathematical assertions;

3. understand the concept ofa function;

4' be confident in the manipulation of algebraic expressions and the solutions of equations andinequalities;

ll 5 deuelop the ability to use concepts to model and solue real-world problems.t l

SPECIFIC OBJECTIVES

(a) The Real Number System - R

Students should be able to:

L use subsets of R:

2. use the propert ies of the inclusionchain NcWc Zcec R, IcR;

3. use the concepts of identity, closure, inverse, cornmutativiry, associativity, distributivity ofaddition and multiplication of real numbers;

4. demonstrate that the real numbers are ordered:

ll 5 . perform operations inuoluing surds;t l

6. construct simple proofs, specifically direct proofs, or proof by the use of counter examples;

7 . use the surrunarion notation (l );

8. establish simple proofs by using the principle of mathematical induction.

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UNIT IMODUTE l: BASIC ATGEBRA AND FUNCTIONS (coni'd)

CONTENT

(a) The Real Number System - R

(i) Axioms of the system - including commutative, associative and distributive laws; non-existence of the multiplicative inverse of zero.

f l

-L^ ^-)^-!-^a^i. .^- l lfl f;r) Tlrc order properties. rl

i l l ti l_t llf (ttt) Operatiots inuoluingsurds. ll

(iv) Methods of proof - direct, counter-examples.

(u) Simple applications of mathematical induction.

SPECIFIC OBJECTIVES

(b) AlgebraicOperations

Students should be able to:

1 L a11ly real numher axiams to carry out operations of addition, subtraction, multiplication illl

"nd division of polynomial and rational expressions; ll

2. factorize quadratic polynomial expressions leading to real linear factors (real coefficients

only);

3. use the Remainder Theorem;

4, use the Factor Theorem to ftnd factors and to evaluate unknown coefficients;

ll 5. extract all factors of cn- b" for positive integers ns 6; lltl

6. use the concept of identity of polynomial expressions.

CONTENT

(b) Algebraic Operations

(i) Addition, subtraction, multiplication, division and factorization of algebraic expressions.

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A,N,'T

UNIT IMODULE l: BASIC ATGEBRA AND FUNCTTONS (cont'd)

(ii) Fac:<.irTlworan.

(iii) RentainlerTlworen.

SPECIFIC OBJECTIVES

(c) IndicesndLogorithms

Students should be able to:

l. use the laws of indices to simplifr expressions(including expressions involving negative

z.

3.

and rational indices);

use tlla fact tlwthgob = a Q qc = $;

4.

5.

sirnplib expressroru b ^rn1 tlw laws of lagaritlmu, such as;

(i) Ioc eQ) : IosP + bsQ,

(ii) bs?lQ) : losP -loeQ,

(i i l IogY = abgP;

use logaithms to solue e4wtiorrs of tlw form a' = b;

solue problenr inuoluing changing of ilwbase of alogaitlm.

CONTENT

(c) Indices ondlogarithms

(0 Laws of indices, including negative and rational exponents.

(i) l-aws of logarithms appliad to problems.

(iii) Solutianof eqntbrc of.tlwformd = b.

(iv) Clange of base.

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A6N,07

UNIT IMODUTE l: BASIC ALGEBRA AND FUNCTIONS (cont 'd)

SPECIFIC OBJECTIVES

(d) Functions

Students should be able to:

I .

5.

6.

1t.

8.

9,

10.

z.

use the terms: function, domain, range, open interval, half open interval, closed interval,one-to-one function (injective function), onto function (surjective function), one-ro-oneand onto function (bijective function), inverse and composition of functions;

show that there are functions which are deftned as a set of ordered pairs and not by.asingle formula;

plot and sketch functions and their inverses (if they exist);

snte tlw geometrbal ''elatbrchip betwem tlw furctian y-- f(x) and its inuerse [reflection in tlwlirw y = ,1'

interpret grdpls of simple polyrcmialfurctbrc;

show that, if g is the inverse function of i then ftg(*)l = x, for allx, jn the domain of g;

perform c alculations involving given functions ;

showgraphicalsolutions of f(x) : g(x), f(*) < g(x), f(x) > g(x);

identify an increasing or decreasing function, using the sign o, f @) - {(b) when a + b;a-b

illustrate by means of graphs, the relationship between the function y = f(x) given ingraphical form and y=af(x) ; y=f(x+a); y= f (x) la; y=af(xf b) ; y=f(ax),y = lf(x)1, where a, b are real numbers, and, where it is invertible, y --f ',(x).

3.

4.

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A^N,,T l0

UNIT IMODULE l: BASIC ALGEBRA AND FUNCTTONS (cont,d)

CONTENT

(d) Functions

(i) Domain, range, composition.

(ii) Injecrive, surjective, bijective funcrions, inverse function.

(iii) Graphicalsolutionsofproblemsinvolvingfunctions.

(iv) Simple transformarions.

(v) Transformationof the graph)=f(x) to y=af(x); l=fk t a); I =fe) t a;J = af(x t b); y = f(erx);y = I /(x) | and, if appropriate, to 1 =f't(x).

SPECIFIC OBJECTIVES

(e) The Modulus Fzncion

Scudenr slwuld be able to:

1. def inethemodulusfurrct ion, forexample, l r | : {_: . i | : : I } , [s l =5,' [ - x l t x < u ) ' | | - '

2. use tlw fact tlwt | * | ir the positive square root of I ;

3, usethefact that l r l . I Vl i iandonly i f ,xr . (yz;

4. solue eEtatiarc inuoluing tlv nndulus furctions.

CONTENT

x k) The Modulus Flnctiont ltlll

(i) Definition and properties of the modulus fwrction.

(ii) The triangle inequality.

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A6N,o7 l t

UNIT IMODUTE l: BASIC ATGEBRA AND FUNCTIONS (conl'd)

SPECIFIC OBJECTIVES

fl (0 Quailratic ot:dCubic Frnctions cnd Equations

Students should be able to:

tlll t. expresstlwEndratbfunction m2 + bx+c intlwform a(x+h)'+ k;

CONTENT

ll tO Quadratic otdCubicFunctioru andEquations

Z. sketch the graph of the quadratic function, including maximum or minimum points;

3. determine the nature of the roots of a quadratic equation;

4. ftnd the roots of a cubic equation;

5. use the relationship between the sums and products of the roou anA tfu coeffbien* of:

( i ) a l +bx +c: 0,( i i ) ax3+bf *cx*d= 0,

(i) Quadratic equations in one unknown.

(ii) Tlw ruture of tlw roo* of qudratic eEntiotrs.

(iii) Sketching graph,s of quadratic functions.

(iv) Roo* of cubic eE;za,tioru.

(u) Sums and products, with applicatrbru, of the roots of quadratb anA cubic equatbrc.

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A^N,,T l2

UNII IMoDUtEl:BAS|CAIGEBRAANDFUNCI|oNS(con| 'd)

SPECIFIC OFBJECTIVES

(g) Inequalities

students should be able a we algebrab and graphical nrctlnds to ftnd the solution se$ of:

t. lhwar hwqnlities;

2. Eudratb tttcEnlities;

3. hwqrntities of ttuform #ro,

4. irrcqtnlitics of dle form lax +bl<lcx + dl'

CONIENT

(g) Inequalities

(0 Linear inequalities'

(i0 Quadratic inequalities'

( i iOInequal i t iesinvolvingsimplerat ionalandmodulusfunct ions.

Suggested Teaching and Learning Activities

To facilitate srudenrs, artainmenr of the objectives of this Module, teachers are advised to engage students

in the teaching and leaming activities listed below'

l. The Real Number Svstem

The teacher should encourage students to practise different methods of proof by constructing

simpleproofsofelementaryassertionsaboutrealnumbers,suchas:

( i ) - ( -2) :2;

(ii) For anY real number a, 0'a=0;

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A6N,o7l3

UNIT IMODUTE

(ii i)

(iv)

l: BASIC ATGEBRA AND FUNCTIONS (conf'd)

( - l ) ( - l ) = l ;

The statement "For all real x and 1, lr * y l= lr l+ ly | " i, false (by counter-example).

Proof bv Mathematical Induction (MI)

TvpicalQuestion

Prove that some formula or /statement P is true for all positive integers n 2 k, where k is somepositive integer; usually k : l.

Procedure

Step l: Verifo that when k : l: P is true for n : k : L This establishes that P is true for n :l .

Step2: Assume P is true for n: k, where k is a positive integer ) l. At this point, thestatement k replaces n in the statement P and is taken as rrue.

Step3: ShorvthatPistrueforn: k + I usingthe truestarement instep2withnreplacedbyk.

Step 4: At the end of step 3, it is stated that statement P is true for allpositive integers n > k.

Summarv

Proof by MI: For k > l, verifu step I for k and proceed through to step 4.

Observation

Most users of MI do not see how this proves that P is true. The reason for this is that there is amassive gap between Steps 3 and 4 which can only be filled by becoming aware that Step 4 onlyfollows because Steps I to 3 are repeated an infinity of times to g.rr.r"i. the set of all positiveintegers' The focal point is the few words "for all positive integers n ) k" which points to thedetermination of the set S of all positive integers for which p is true.

2.

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IiI

UNIT 1MODUIE l: BASIC ATGEBRA AND FUNCTIONS (cont'd)

Step I says that le S fork : L

Step3 says that k + I e S wheneverk e S, soimmediately 2 eS since I e S.

I terat ingonStep3saysthat3essince2esandsoon,sothatS={1,2,3. . , } , that is,Sis ' thesetofal lpositive integers when k : I which brings us to Step 4.

When k > l, the procedure starts at a different positive integer, but the execution of steps is the same.Thus, it is necessary to explain what happens between Steps 3 and 4 to obtain a full appreciation of themethod.

Example l: Use Mathematical Induction to prove that n3 - n is divisible by 3, whenever n is apositive integer.

Solution: Let P (n) be the proposition that "nr - n is divisible by 3".

Basic Step: P(l) is rrue, since 13 - I = 0 which is divisible by 3.

Inductive Step: Assume P(n) is true: that is, n3 - n is divisible by 3.We must show that P(n + l) is true, if P(n) is true. That is,(n + l)r - (n + l) is divisible by 3.

Now, (n+1) ' - (n+ l ) = (n3 +3n2 * 3n* 1) - (n+ l )=(n3-n)+3(n2+n)

Both terms are divisible by 3 since (n'- n) is divisible by 3 by theassumption and 3(n2 * n) is a multiple of 3. Hence, P (n+ 1) istrue whenever P (n) is true.

Thus, nr - n is divisible by 3 whenever n is a positive inreger.

Example 2: Prove by Mathematical Induction that the sum S. of the ftrst n odd positiveintegers is n2.

Solution: Let P (n) be the proposition that the sum Sn of the ftrst n odd positive integer isn2.

Basic Step: For n= I the first one odd positive integer is l, so Sr : 1, that isSr = I : lz, hence P(1) is nue.

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A6Nz^7 l5

Now, Sn*1 : I +3 + 5 +. . .+ (2n-1) + (2n+ l ): [ I + 3 + 5 +. . .+ (2n- l ) ] + (2n+ 1): n2 + (2n + l), by the assumption,:(n * l )2

Thus, P(n* l),is true whenever P(n) is true.

Since P(l) is true and P(n) --+ P(n * l), the proposition P(n) istrue for all positive integers n.

3. Functions (Iniective, suriective, biiective) - Inverse Function

Teacher and students should explore the mapping properties of quadratic functions which:

(i) will, or will not, be injective, depending on which subset of the real line is chosen as thedomain;

(ii) will be surjective if its range is taken as the co-domain (completion of the square is useful,here);

(ii| if both injective and surjective, will have an inverse function which can be constructed bysolving a quadratic equation.

Example: Use the function f :A -+ B given bv f(x) =3x2 + 6x+5, where the domain A is

altematively the whole of the real line, or the set {xeR l* > - 1}, and rhe co-domain B is R or the set { y€R lv>2}.

UNIT IMODUTE l: BASIC ATGEBRA AND FUNCTIONS (conf'd)

RESOURCES

Aub, M. R.

Bostock, L. and Chandler, S.

Cadogan, C.

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A6N,'T

InductiveStep: Assume P(n) is true. Thatis, Sn : I + 3 + 5 + .,.. + (2n - 1)= n2.

Tlw ReaI Number System, Barbados: Caribbean ExaminationsCouncil, 1997.

Core Matlwrwtics for A-Levels, United Kingdom: StanleyThomes Publishing Limited, 1997,

Proof b1 Mathematical Inductbn (MI), Barbados: CaribbeanEmmirwtiors Courcil, 2 00 4,

t6

UNIT IMODULE l: BASIC ALGEBRA AND FUNCTTONS (cont'd)

Greaves, Y. Solution of Simulareous Lirvar Waations b Rou, Reiluction,Barbados: Caribbean Examinarions Council, 1998.

erutical Modelling in tlw Secorfury School Cunianlum, Arce Guide of Classroorn Erercses, Vancouver, Uniteds of America: National Council of Teachers of,ematics, Incorporated Reston, 1991.

Hutchinson, C. Injectiue and Surjectiue Functrbru, Barbados: CaribhanExaminations Council, I 998.

Martin, A., Brown, K., Rigby, P. and Aduanred l*uel Matlwrutics Tutoriak Pwe Matl:rritwdcr CD-Ridley, S. ROM scrnple (TradeEdition), Cheltenham, United Kingdom:

Stanley Thomes (Publishers) Limited, Multi-user versionand Single-user version, 2000.

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A6N,07 t7

UNIT II ODUIE 2: TRIGONOMFTRV AND PTANE GEOMETRY

GENERAI OBJECTIVES

On completion of this Module, students should:

l .

2,

3.

4.

5.

6.

develop the ability to represent and deal with objects in the plane through the use of coordinaregeometry, vectors;

understand that the altemative descriptions of objects are equivalent;

develop the ability to manipulate and describe the behaviour of trigonometric functions;

develop the ability to establish trigonometric identities;

develop skills to solve trigonometric equations;

deuebp tlw abilitl to use con epts to nwdel and solue real-worW problenx.

SPECIFIC OBJECTIVES

(a) Trigonorrctric Flnctions, Idertities an"il Eqlio;tiorres (all angles utill be assumed b be in radiansrmless othswise sutcil)

Students should be able to:

1. graph the functions sin kx, cos kx, tan kx, k e R;

2. relate the periodicity, syrnmetries and amplitudes of the functions in Specific Objective Iaboue to their graphs;

3. use tl,fact rhatr*f+tr)=cosx;' \ .z )

4.

) .

6.

t ,

use tlw formulae for sin(A t B), cos(A t B) and tan (A t B);

deriue the multiple angle identities for sin kA, cos kA, tan kA, for keQ;

deriue rhe identity cos'0 + sinz 0 = l;

use tle reciprocalfurrctiorc sec x, cosec x and cot x;

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A^N,,T l8

UNIT 1MODUTE 2:

8.

9,

10.

I l .

TR|GONOMETRY AND ptANE GEOiiETRy (cont,d)

derive the corresponding identities for tanzx, cotzx, seczx and cosec2x;

develop and use the expressions for sin A t sin B, cos A t cos B;

use Specifb Objectiues 3, 4, 5, 6,7, g andg aboue ta proue simplc idmtities;

express a cos 0 * bsin 0 in the form r cos (0 t cr) and rsin(gt c) where r is positive

o< a <| ;

find the general solution of equations of the form

(i) sin lcO =c,

(it cos k0 =c,

(iil tank} =c,

( i r ) asin0*bsin0:c,

fot a, b, c, lc, e R;

find tlw solutbru of tlw eEntiors in 12 abue for a gium range;

obtain maxirnum or minimum values of f(0) for 0 S 0 <Zn ,

12.

13,

t4.

CONTENT

(a) Trigonometric Functions,'Identities and Equations (all angles will be assumed to be radians)

(i) The circle, radian measure, length of an arc and area of a sector.

(ii) Sine rule, cosine rule.

(iii) Area of a triangle, using Area = j ab sin C.

(iv) The functions sin x, cos x, tan x, cot x, sec x, cosec x.

(u) compound-angle formulae for sin (AtB), cos (AtB), tan (AtB).

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A6Nz/07 l9

UNIT Il,lODUtE 2: TRIGONOMFIRY AND PTANE GEOMETRY (cont'd)

(vi) Multiple-angle formulae.

(vii) Formulae for sin A t sin B, cos A t cos B.

(viii) Use of appropriate formulae to prove identities.

(ix) Expression of a sin e + b cos 0 in the forms r sin (Otcr) and r cos (Oto), where r is

posit ive, Osc< L'2

(x) Ceneral solutfon of simple trigonometric equations, including graphical interpretation.

(xi) Trigonometricidentitiescos20 + sin20 = l, I +cot20 = cosec20, l+tanzg = seczg.

(xii) Maximum and minimum values of functions of sin 0 and cos 0.

SPECIFIC OBJECTIVES

(b) Co-ordinate Geometry

Students should be able to:

use the gradient of the line segment;

use the relationships.between the gradients of parallel and mutually perpendicular lines;

find the point of intersection of two lines;

write the equation of a circle with given cenrre and radius;

ftnd the centre and radius of a circle from its general equation;

find equations of tangents and normals to clrcles;

ftnd the points of intersection of a curve with a straight line;

finn thc fuin* of intersection of two curues;

obtain the Uartesian equation of a curve given its parametric representation.

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A^N,,T 20

UNIT I

il MoDULE 2: TRlGoNoMErRv AND ptANE GEOMEIRy (cont,d)

CONTENT

(b) Co-ordinatc Geometry

O Properties of tfu circlc.

(it Tangents anl rwrnwk,

(iii) Intersectbns betwean lhles and cutues,

(iu) cartesian eEutioru and parametric represmntioru of'anv.'*,

SPECIFIC OBJECTIVES

(c) Vectors

Students should be able to:

l. express a vector in the form [-l * xi+yj;ty/

Z. define equality of two vectors;

3. add and subtract vectors;

4, multiply a vector by a scalar quantity;

5. derive and use unit vectorsi

6. find displacement vectors;

7 . find the magnitude and direction of a vector;

8, apply properties of paraler vectors and perpendicular uecaors;

9. de{ine the scalar product of two vectors:

(i) in terms of their componenrs,

(ii) in terms of their magnitudes and the angre berween rhem;

10. find the angle between two given vectors.

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A6N,07

ll

2l

UNIT II ODUIE 2: TRIGONOMEIRy AND PI,ANE GEOMEfRY (conf'd)

CONTENI

(c) Vectors

(0

(ii)

(iii)

(iv)

(v)

Expression ofa given vector in the f"r- |...) or xi + yj.(vi

Equality, addition and subtraction of vectors; multiplication by a scalar.

Position vectors, unit vectors, displacement vectors.

Length (magnitude/modulus) and direction of a vector.

Scalar (Dot) Product.

Suggested Teaching and Learning Activities

To facilitate students' attainment of the objectives of this Module, teachers are advised to engage studentsin the teaching and leaming activities listed below.

l. Trigonometric Identities

Much practice is required to master proofs of Trigonometric ldentities using identities such as theformulae for:

sin (A * B), cos (A t B), tan (A t B), sin 2A, cos 24, tanZA

Example: The identity I -'cos ld = 6n 20 can be established by realizing that

' sin 40cos 40 : | - 2 sin2 20 and sin 40 = 2 sin 20 cos 20,

Deriue tlw tigornnwtrb functians sin x and cos x for anglzs x of arry ualue (irrcluding negatiue ualues),rlsing tlw coorditwtes of points on tlw unit circle.

RESOURCE

Bostock, L. and Chandler, S. Matlwnatics - Tfu Core Course for A-LeveI, United Kingdom: StanleyThomes (Publishers) Limited, 1997.

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A^N,.T 22

UNIT IMODUTE 3: CALCUIUS I

GENERAL OBJECTIVES

On completion of this Module, students should:

L understand the concept of continuity of a function and its graph;

2. appreciate that functions need not be continuous;

3, develop the ability to {ind the limits (when they exist) of functions in simple cases;

4. know the relationships between the derivative of a function at a point and the behaviour of thefunction and its tangent at that point;

5, be confident in differentiating given functions;

6. know the relationship between integration and differentiation;

7 . know the relationship between integration and the area under the graph of the function;

8. know the properties of the integral and the differential;

ll ,. develop the ability to use concep* n npdeland solve real-ivorld problems.tl

SPECIFIC OBJECTIVES

(a) Limits

Students should be able to:

1 I . use grdplu to determirw tlre continuity and continuity of functions;tltl

2, describe the behaviour of a function f(x) as r gets arbitrarily close to some given ffxednumber, using a descriptive approach;

3. usethelimitnotation l im f(x)=L,f(x)- L as x+ o i

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"*" A6N2to7 23

UNIT IMODUTE 3: CATCUIUS | (conf'd)

4. use the simple limit theorems:

If lim /(x)=F, JgS(x)=C and k is a constant,

then *lim.kf(x)=pp, *limrf(x)g(x)

=FG, *tg.f{*)+

g(*)} =F+G,

and, providedG*0, 6- Ilt =f,x-+a g(x) G

use limit theorems in simple problems, including cases in which the limit of f(x) ar a is nor

f(a), for example, t,*^{fu} ,,r. use of L'Hopital's Rule is not ailowed);x+z L *_, )

use tlv fact tlwt 1irn- stM

= 1 , dernaflstrated by a geometric approach (tlrc use of L'Hopial's' r+Oy

Rule u rctallowed);

solve simple problems involving limits and requiring algebraic manipulation (the use ofL' Hopial's Ruh nq be allwed) i

) .

6.

t .

8.

9,

t0.

identifr the region over which a function is continuous;

identifu the points where a function is discontinuous and describe the nature of itsdiscontinuity;

use the concept of left handed or right handed continuity, and conrinuity on a closedinterval.

Concept of limit of a function.

Limit Theorems.

Continuity and Discontinuity.

CONIENT

(a) Limits

(i)

(ii)

(iii)

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A^N,.T 24

UNIT IMODUTE 3: CATCULUS I (conf'd)

SPECIFIC OBJECTIVES

(b) Differentiation I

Students should be able to:

l. demonstrate understanding of the concept of the derivative at a point x : c as thegradient ofthe tangent to the graph at x = c;

Z, deffne the derivative at a point as a limit;

3. use the f '(x) notation for theJtrst derivative at x;

4, differentiate, from ftrst principles, such functions as:

( i) f(x)=k wlvrek eR,

( i t f (x) : xn, ut l laren e { '3, '2, '1, 'y2, h, 1,2,3},

( i i i ) f (x): sinx;

5, demonstrate an understanding of how to obtain the derivative of xn, where n is anynumber;

6. demonstrate understanding of simple theorems about derivatives of y = c f(x),Y = f(x) +g(x); where c is a constant;

7. use 5 and 6 abwerepeatcdly n cabulatc tlrc deriuatiues of:

(i) po$rwnials,

(ii) tigorwnetrb fwrctions;

8. use the product and quotient rules for differentiation;

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A6N,,T 25

UNIT IMODUIE 3:

9.

10.

11.

tz.

13.

14.

15.

16.

17,

18.

19.

CALCUTUS | (cont'd)

differentiate products and quotients ofi

(i) polynomials,

(ii) trigonometric functions;

apply the chain rule in the differentiation of composite functions (substitution);

demonstrate an understanding of the concept of the derivative as a rate of change;

use the sign of the derivative to investigate where I function is increasing or decreasing;

demonstrate the concept of stationary (critical) points;

determine the nature of stationary points;

Iocate stationary points, maxima and minimaderivative:

by considering sign changes of the

calculate second derivatives;

interpret the significance of the sign of the second derivative;

use the sign of the second derivative to determine the nature of stationary points;

sketch graphs of polynomials, rational functions and trigonometric functions using thefeatures of the function and its first and second derivatives;

describe the behaviour ofsuch graphs for large values ofthe independent variable;

obtain equations of tangents and normals to curves.

20.

zt.

CONTENT

(b) Differentiation I

(i) The Gradient.

(ii) The Derivative as a limit.

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A6N,o7 26

UNIT IMODUTE 3: CAICULUS | (cont'd)

(iii) Rates of change.

(iv) Differentiation from ffrst principles.

(v) Differentiation of simple funcrions, product, quotients.

(vi) Stationary points and chain rule.

(vii) Second derivatives of functions.

(viii) Curvesketching.

(ix) Tangents and Normals to curves.

SPECIFIC OBJECTIVES

(c) Integration I

Students should be able to:

l. define integration as rhe inverse of differentiation;

2. demonstrate an understanding of the inde{inite integral and the use of the integrationnotation i f(x) dx ;

3. show that the indefinite integral represents a family of functions which differ byconstants;

4, demonstrate use of the following integration theorems:

(i) J cf(x) dx = cl f(x) dx , where c is a consranr,

(ii) I {f(*)+ e(x) } dx = i f(x) dx + J g(x) dx;

ll s. fhd:

(i) indeftnite integrals using integration rheorems,

(ii) integrals of polynomial functions,

(iii) integrals of simple trigonometric functions;

qft- I "ro"

A6N,,T 27

UNIT IMODUTE 3: CALCULUS | (cont'd)

6. define and calculat- lbo f f*> dx = F(b) - F(a), where F(x) is an indeftnite integral of f(x)and integrate, using substitution;

7. use the results:

(i) 1"br1*; a* = 1"bf1g dt ,(ii) i6 fl*l a* = 16 f(a - x) dx, for a >0 ;

8.' apply integration to:

(i) ftnding areas under the curve,

(ii) finding volumes of revolution by rotating regions about both the x and y axes;

9, formulate and solve differential equations of the form y' = f(x) where f is a polynomial ora trigonometric function.

CONTENT

(c) Integration I

(i) Integration as the inverse of differentiation.

(ii) Linearity of integration.

(iii) Indefinite integrals (concept and use).

(iv) Deffnite integrals.

(u) Applications of integration - areas, volumes and solutions to elementary differentialequations.

(vi) Integration of polynomials.

(vii) Integration of simple trigonometric functions.

(viii) Use of I f <-'rdr = F(b) - F(a), where F'(x) =/(x).

(i*) First order differential equations.

qfu" | "*"

A6N,o7 28

UNIT IMODULE 3: CALCULUS | (cont'd)

Suggested Teaching and Learning Activities

To facilitate students' attainment of the objectives of this Module, reachers are advised to engage studentsin the teaching and learning activities listed below.

The Area under the Graph of a Continuous Function

Class discussion should play a major role in dealing with this topic. Activities such as that which followsmay be performed to motivate the discussion.

Example of classroom activity:

Consideratr iangleof area.quul to luni ts,boundedbythegraph,sofy: x,y:0andx: l .

(i) Sketch the graphs u.ra ia.r,,-iry the triangular region enclosed.

(ii) Subdivide the interval [0, 1] into n equal subintervals.

(iii) Evaluate the sum, s(n), of the areas of the inscribed rectangles and S(n), of the circumscribedrectangles, erected on each subinterval.

(iv) By using different values of n, for example, for n : 5, 10,7,5,50, 100, show that both s(n) andS(n) get closer to the required area of the given region.

RESOURCES

Aub, M. R.

Bostock, L., and Chandler, S,

Ragnathsingh, S.

Differentiation from First Prhrciples: TIrc Power Functian, Barbados:Caribbean Examinations Council, 1998.

Marlwmatics - Tlw Core Course for A-Leuel, IJnited Kingdun:Stanley Thornes Publishing Limited, (Chapters 5, 8 and 9),r99t.

Area uriler tltn Graph of a Continuous Furrctrbn, Barbados:Caribbean Examinations Council, 1998.

qfu' | "*"

A6N,07 29

o 0!ilru0]$tl$rIfl

Pafur 01(l lwur 30 minutcs)

Popt 02(2lwurs 30 minuus)

INTERNAL ASSESSMENT

Each Unit of the syllabus is assessed separately. The scheme of assessment for each Unit is the same. Acandidate's performance on each Unit is reported as an overall grade and a grade on each Module of theUnit. The assessment comprises rwo components, one extemal and one intemal.

EXTERNAL ASSESSMENT

The candidate is required to sit two written papers for a total of 4 hrs,

(80o/o)

Thts Pa\er cotnprises forty-fiue,c ompuls ory muhiple - clwic e items .

Th* DaPer cornprises srx, compukorlexnde d - r e sporls e que stbtu,

3Oo/o

5Oo/o

(2Oo/o)

Intemal Assessment in respect of each Unit will contribute 20o/o to the total assessment of a candidate'sperformance on that Unit.

Paper 03A

This paper is intended for candidatbs registered through a school or other approved educationalinstitution.

The Intemal Assessment comprises three class tests designed and assessed internally by the teacher andextemally by CXC, The duration of each test is I to lVz hours. The tests must span, individually orcollectively, the three Modules, and must irrcIule mathematical modelling.

Paper 03B (Alternative to Paper 03A)

This paper is an altemative to Paper 03A and is intended for private candidates.

The paper comprises three questions. The duration of the paper is I % hours.

qft.f cxc A6N,o750

MODERATION OF INTERNAL ASSESSMENT (PAPER O3A)

Each year an Intemal Assessment Record Sheet will be sent to each school submitting candidates for theexaminatiors.

All Intemal Assessment Record Sheets and samples of tests from the school must be submitted to CXC byMay 3 I of the year of the examination. A sample of tests must be submitted to CXC for moderation purposes.The tests will be re-assessed by CXC Examiners who moderate the Intemal Assessment. The teachers'marksmay be adjusted as a result of the moderation. The Examiners'comments will be sent to the teacher.

Copies of the candidates' assignments must be retained by the school until th,ree months after publication byCXC of the examination resuhs.

ASSESSMENT DETAILS FOR EACH UNIT

External Assessment by Written Papers (80% of Total Assessment)

PaDer Ol (1 hour 30 minutes - 300,6 of Total Assesnrmtl

l. Composition of the Paper

(i) This paper consists of forty-fiue multiplc-clwbe items, with fiftcn itens based on eachModule.

(ii) All rrems are compulsory.

2. Syllabus Coverage

(i) Knowledge of the entire syllabus is required.

(ii) The paper is designed to test a candidate's knowledge across the breadth of the syllabus.

3. Question Type

Questions may be presented using words, symbols, tables, diagrams or a combination of these.

4. Mark Allocation

(i) Eachitemis albcatcd l nark.

(it Each Module is allocated 15 marks.

(ii) '[he nnl marks auailable for this paper is 45.

(iu) This paper contributes 30olo towards the ffnal assessmenr.

%.f cxc A6,o2/075l

5. Award of Marks

Marks will be awarded for reasoning, algorithmic knowledge and conceptual knowledge,

Reasoning:

Algorithmic Knowledee:

Conceptual Knowledge:

Selection of appropriate stratag), evidence of clear thinking,explanation and/or logical argument.

Evidence of knowledge, ability to apply concepts and skills, andto analyse a problem in a logicalmanner.

Recall or selection of facts or principles; computational skill,numerical accuracy, and acceptable tolerance limits in drawingdiagrams.

6. Use of Calculators

(i) Each candidate is required to have a silent, non-progranunable calcularor for the durationof the examinarion, and is entirely responsible for its functioning.

(ii) The use of calculators with graphical displays will not be permitted.

(iii) Answers found by using a calculator, without relevant working shown, may not beawarded full marks.

(iv) Calculators must not be shared during the examination.

7. Use of MatlwwticalTables

A bool<Iet of nwtlwrutical formulae willbe prouided,

Paper 02 (2 hours 30 minutes - 5096 of Total Assessment)

l. Composition of Paper

(i) The paper consists of six questioru two questions are based on each Module (Module l,Module 2 and Module 3).

(ii) All questions are compulsory.

2, Syllabus Coverage

(i) Each ques..on may be based on one or more than one topic in the Module from which thequesrion is taken.

(ii) Each question may develop a single theme or unconnected themes,

%.1 cxc A6N2to752

3. Question Type

(i) Questions may require an extended response.

(ii) Questions may be presented using words, symbols, tables, diagrams or a combination ofthese.

4. Mark Allocation

(i) Each question is worth 25 narl<s.

(ii) The number of marks allocated to each sub-question will appear in brackets on theexamination paper.

(iii) Each Module is allocated 50 marks.

(iv) The totalmarks available for this paper is 150.

(u) This paper contributes 50olo rowards the final assessmenr.

5. Award of Marks

(i) Marks will be awarded for reasoning, algorithmic knowledge and conceptual knowledge.

Reasonins: Selecrion d appropriate stratcg!, euida.rce of clear kirrki.r{,,explanation and/or logical argument.

pw vt vw' u'|iw''Et ll

Alsorithmic Knowledge: Evidence of knowledge, ability to apply concepts andskills, and to analyse a problem in a logical manner.

Conceptual Knowledse: Recall or selection of facts or principles; computationalskill, numerical accuracy, and acceptable tolerance limitsin drawing diagrams.

(ii) Full marks will be awarded for correct answers and presence of appropriate working

(iii) Where an incorrect answer is given, credit may be awarded for correct method providedthat the working is shown.

(iv) Ifan incorrect answer in a previous question or part-question is used later in a section or aquestion, then marks may be awarded in the latter part even though the original answer isincorrect. In this way, a candidate is not penalised twice for the same mistake.

(v) A correct answer given with no indication of the merhod used (in the form of writtenworking) will receive no marks. Candidates are, therefore, advised to show all relevantworking.

%.1 cxc A6N2to7

6. Use of Calculators

Each candidate is required to have a silent, non-programmable calculator for the durationof the examination, and is responsible for its functioning.

The use of calculators with graphical displays will not be permitted.

Answers found by using a calculator, without relevant working shown, may not beawarded full marks.

(iv) Calculators must not be shared during the examination.

7, Ux of tvlatlwwticolTablcs

Abooldct of nwtlenwtical fornwlae wil|be prouided.

INTERNAL ASSESSMENT

Intemal Assessment is an integral part of student assessment in the course covered by this syllabus. It isintended to assist students in acquiring certain knowledge, skills, and attitudes that are associated withthe subject. The activities for the Intemal Assessment are linked to the syllabus and should form part ofthe leaming activities to enable the student to achieve the objectives of the syllabus.

During the course of study for the subject, students obtain marks for the competence they develop anddemonstrate in undertaking their Intemal Assessment assignments. These marks contribute to the finalmarks and grades that are awarded to students for their performance in the examination.

The guidelines provided in this syllabus for selecting appropriate tasks are intended to assist teachers andstudents in selecting assignments that are valid for the purpose of Intemal Assessment. In order to ensurethat the scores awarded by teachers are in line with the CXC standards, the Council undertakes themoderation of a sample of the Intemal Assessment assignments marked by each teacher.

Intemal Assessment provides an opportunity to individualise a part of the curriculum to meet the needs ofstudents. It facilitates feedback to the students at various stages of their experience. This helps to buildthe self-conftdence of students as they proceed with their studies. Intemal Assessment also facilitates thedevelopment of the critical skills and abilities emphasised by this CAPE subject and enhance the validityof the examination on which candidate performance is reported. Intemal assessment, therefore, makes asignificant and unique contribution, to both the development of relevant skills and the testing andrewarding of students for the development of those skills.

The Caribbean Examinations Council seeks to ensure that the Intemal Assessment scores are valid andreliable estimates of accomplishment. The guidelines provided in this syllabus are intended to assist indoing so.

(i)

(ii)

(iii)

qfr.f cxc A'N,'T54

Paper 03A (20olo of Total Assessment)

This paper comprises three tests. The tests, designed and assessed bymoderated by CXC. The duration of each test is I to I % hours.

l. Composition of the Tests

The three tests of which the Intemal Assessment is comprisedcollectively, the three Modules anl irrclude natlwmntical nwdelling.marl<s must be albcated to mathamatbal modelling.

the teacher, are externally

must span, individually. orAt least thirty per catt of tlv

7

3,

Question Type

Paper 03B may be used as a prototype but teachers are encouraged to be creative and original.

Mark Allocation

(i) There is a maximum of 20 marks for each test.

(ii) There is a maximum of 60 marks for the Intemal Assessmenr.

(iii) The candidate's mark is the total mark for the three tests. One-third of the total marksfor the three tests is allocated to each of the three Modules. (See 'General Guidelines forTeachers' below.)

(iu) For each test, marks should be allocated for the skills outlined on page 3 of this Syllabus.

Award of Marks

(i) Marks will be awarded for reasoning, algorithmic knowledge and conceptualknowledge.

For each test, the 20 marks should be awarded as follows:

Reasoning: Selection of appropriatc. strdteg!, euiderrce of clearthi"tking, explanation and/or logical argument.

(3 - 5 marks)

Evidence of knowledge, ability to apply conceprs andskills, and to analyse a problem in a logical rnanner.

(10 - 14 marks)

Recall or selection of facts or principles;computational skill, numerical accuracy, andacceptable tolerance limits in drawing diagrams.

(3 - 5 marks)

55

Algorithmic Knowledge:

Conceptual Knowledge:

4.

%"f cxc A6N,07

2,

(ii) If an incorrect answer in an earlier question or part-question is used later in a section or aquestion, then marks may be awarded in the later part even though the original answer isincorrect. In this way, a candidate is not penalised twice for the same mistake.

(iii) A correct answer given with no indication of the method used (in the form of writtenworking) will receive no marks. Candidates should be advised to show qll relevantworking.

PoDq O3B (20% of Tonl Assesnvntl

l . Compositioor of Popu

(i) This paper consrsts of three questians, eachbased on onc of tlw threeModules.

(ii) All questions are compulsory.

QtnstionTyfr-,

(0 Each question nwy require an extended resporce.

(ii) A part of or an entire question may focus on matlvmatbal modeling,

(iii) A qucstion nwy be presmted using words, slmbols, nbles, diagrans or d combirwtion of these.

Mor|r Albcation

(i) Each qucstbn catries a nlaximum of 20 nwrl<s.

(ii) Thn. Paper caries a maximum of 60 nwrl<s.

(iii) For each question, marl<s slnuld be allocated for tlw skilk outlined on page 3 of this Syllabus.

Awordof Morls

(i) Marks will be awarded for reasoning, algoithmic krwwledge and conceptual knowledge.

For each test, tlw 20 narl<s slwuldbe awarded as follows:

Reasoning: Selection of appropriate strateg), euidence of clearr easoning, explanatian and I or lo gic al ar gument.

(3 - 5 nwrl<s)

Euilence of luwwledge, ability to app$ corrcepts anlskills, anl to atwlyse a problan in a logical TrnnTlEr.

(10 - 14 nwrl,s)

AlgoithmrcKwwledge:

3.

4.

%.1 cxc A6N,0756

CorcepunlKwuledge: Recall or selectbn of facs or prfuiplcs; cotnpuationalskiJl, nwrcrical accuracy, and accep:;a,blc nlsance limitsin drawing diag1anu.

(3 - 5 rruil<s)

(ii) If an irconect answer in a preuiaus questian or Wrt-qucstbn k used htcr in a section or aquestian, thcn marlcs nwy be awarded in tltc later pdrt eucrl tlwugh tlw orighvl atuwer ishrcorrect. In this way, a candidate is rct penalised twbe for tlw sanw mistal<e.

(iii) A conect answer given with rc indication of tlw metlwd wed (in tfu form of writtcn workinf,will receiue rc nwrl<s. Candidates slwuld be aduised n slwut aU releuant worlang.

GENERAL GUIDELINES FOR TEACIIERS

1. Marks must be submitted to CXC on a yearly basis on the Intemal Assessment forms provided.The forms should be despatched through the Local Registrar for submission to CXC by May 31 inYear I and May 3l in Year 2.

Z. The Internal Assessment frrr each year should be completed in duplicate. The original should besubmitted to CXC and the copy retained by the school.

3, CXC will require a sample of the tests for extemal moderation. These tests must be retained by ,the school for at least three months after publication of examination results.

4. Teachers should note that the reliabiliry of marks awarded is a signiffcant factor in IntemalAssessrnent, and has far-reaching implications for the candidate's ffnal grade.

5, Candidates who do not fulfil the requirements of the Intemal Assessment will be consideredabsent from the whole examination.

6. Teachers are asked to note the following:

(i) the relationship between the marks for the assignment and those submitted to CXC onthe internal assessment form should be clearly shown;

(i0 the teacher is required to allocate one-third of the total score for the Intemal Assessmentto each Module. Fractional marks should not be awarded. In cases where the mark is notdivisible by three, then:

(a) when the remainder is I rnark, the mark should be allocated to Module 3;

(b) when the remainder is 2, then a mark should be allocated io Module 3 and theother mark to Module 2;

%"1 cxc A6,u2to757

II

I

for example, 35 marks would be allocated as follows:

3513 : ll remainder 2 so 1l marks to Module I and 12 marks to each of Modules 2and 3.

nt,

(iii) the standard of marking should be consistent.

Teachers are required to submit a copy of EACH tesr, rhethe sample.

(a) recall, select and use appropriate facts, concepts and principles in a variety of contexts;

(b) manipulate mathematical expressions and procedures using appropriate symbols andlanguage, logical deduction and inferences;

select and use a simple mathematicalmodel to describe a real-world situation;

simplifo and solve mathematical models;

interpret mathematical results and their application in a real-world problem.

T NTEUUNO$ ruN NT{|I GITDIDfiTSCandidates, who have earned a moderated score of at least 50o/o of the total marks for the InternalAssessment component, may elect not [o repeat this component, provided they re-write the examinationno later than TWO years following their {irst attempt. These resit candidates must complete Papers 0land 02 of the examination for the year in which they register.

Resit candidates must be entered through a school or other approved educational institution.

Candidates who have obtained less than 50o/o of the marks for the Internal Assessmenr component mustrepeat the component at any subsequent sitting or write Paper 03B.

O NTGUUNOffi ]ON PNflTIT GilDilIfiESCandidates who are registered privately will be required to sit Paper 01, Paper 02Paper03B will be 1% hours'duration and will consist of three questions, each worthquestion will be based on the objectives and content of one of the three Modules of thewill contribute 20o/o of the total assessment of a candidate's performance on that Unit.

Paper O3B (lr/, hours)

The paper consists of three questions. Each question is based on the topics contained intests candidates'skills and abilities to;

(c)

(d)

(e)

solutions and the mark schemes with

and Paper O3B.20 marks. EachUnit. Paper 038

one Module and

%"1 cxc AGN,0758

$$$rrfrEnillThe Assessment Grid for each Unit contains marks assigned to papers and to Modules and percentagecontributions ofeach paper to total scores.

Units I and?

PaDers Module I Module 2 Module 3 Total (%)

External AssessmentPaper 0l(l hour 30 minutes)

t5(30 weichtcd)

I5(30 weighted)

15(30 weishted)

45(90 weishtad)

(30)

Paper 02(2 hours 30 minutes)

50 50 50 150 (s0)

Internal AssessmentPaper 03A orPaper 038(1 hour 30 minutes)

z0 z0 z0 60 (20)

Total r00 100 100 300 (100)

qfu. I cxc A6N,o759

o ttmiltl|tillil]Itil0lThe following list summarises the notation used in the Mathematics papers of the Caribbean AdvancedProficiency Examinations.

Set Notation

ee{x, . . . }n(A)aUA'wNz0Y

Rcce

cq

L/

n[a, b](a, b)Ia, b)(a, bl

Logic

--)o+c)

is an element ofis not an element ofthe set ofall x such that ...the number of elements in set Athe empty setthe universal setthe complement of the set Athe set of whole numbers {0, l, 2, 3, ... }the set of naturalnumbers {1, 2,3, . . . }the set of integersthe set of rational numbers

tltc set of inatiorwlnumbersthe set of real numbersthe set of complex numbersis a proper subset ofis not a proper subset ofis not a proper subset ofis a subset of

is not a subset of

unionintersectionthe closed interval {x e R: a Sx Sb}the open interval {x e R: a < x <b}the interval {x e Rr a <x < b}the interval {x e R: a < x <bl

conJuncnon(inclusive) disjunctionexclusive disjunction

negationconditionalitybi-conditionalityimplicationequivalence

%.f cxc A6N,o7

Miscellaneous Symbols

= is identical toN is approximately equal tocc is proportional to@ infinitv

Operations

n

Zr, x1* x2 * . . . * xni=1

I x the positive square root of the real number xl" I the modulus of the real number x

n! nfactor ia l , lx lx. . .xnforneN (0! =l)

nr\ fn) - r r . , , tA. n l"U,, | | thebinomialcoeff ic ient , , forn,r€ 'W,0<r<n' \ r / '@-r) l r ! "" '

nP, nl '

(n -r)l

Functions

f the function ff (*) the value of the funcrion f at xf: A -+ B the function f under which each element of the set A has an image in the set Bf; x -+ ) the function f maps the element x to the element If-' the inverse of the function ffg the composire function f(g(x))lim f(x) the limit of f(x) as x tends to ax -)a

Ax, 6x an increment of xdv' . v' the frsr derivative of y with respect to xdx' 'A n,,", | , y (n)

the nth derivative of y with respecr to xdxn ' '

f '(x), f"(x), " ', ttn)1x) the first, second, ..., nth derivatives of f(x) with respecr to x* , X the first and second derivatives of x with respect to time te the exponential constantIn x the natural logarithm of x (to base e)lg t the logarithm of x to base l0

%"f cxc A6N2/076l

Complex Numbers

I

1RezImrlz largT7, z*

a complex number, z :x + 1i where x, 1 e Rthe real part of1the imaginary part of 1the modulus of zthe argument of 1, where -n 1 arg:-3nthe complex conjugate of 1

Vectors

+ara, ABel " la.bi , j ,k

fx)t llv I\z)

vectorsa unit vector in the direction of the vector athe magnitude of the vector athe scalar product of the vectors a and bunit vectors in the directions of the positive Caftesian coordinate axes

xi+yj+zk

Probability

sA,B,. . .P(A)

Matrices

M("- ' )

the sample spacethe events A, B, .. . ithe probability that the event A does not occur

Mt,Mrdet M, lM I

a matrix Minverse of the non-singular square matrix M

transpose of the matrix Mdeterminant of the square matrix M

WestsnTnleO{lice2007/M/23

%.f cxc A6N2to762