GENERAL MATHEMATICS/MATHEMATICS (CORE)

WAEC SYLLABUS ON GENERAL MATHEMATICS/MATHEMATICS (CORE)

AIMS AND OBJECTIVES

The aims of the syllabus are to test candidates:
(1) mathematical competency and computational skills;
(2) understanding of mathematical concepts and their relationship to theacquisition of entrepreneurial skills for everyday living in the global world;
(3) ability to translate problems into mathematical language and solve themusing appropriate methods;
(4) ability to be accurate to a degree relevant to the problem at hand;
(5) logical, abstract and precise thinking.This syllabus is not intended to be used as a teaching syllabus. Teachers are advised touse their own National teaching syllabuses or curricular for that purpose.

EXAMINATION SCHEME


There will be two papers, Papers 1 and 2, both of which must be taken.

PAPER 1:

will consist of fifty multiple-choice objective questions, drawn from the commonareas of the syllabus, to be answered in 1½ hours for 50 marks.

PAPER 2:

will consist of thirteen essay questions in two sections – Sections A and B, to beanswered in 2½ hours for 100 marks. Candidates will be required to answer tenquestions in all.

Section A:

Will consist of five compulsory questions, elementary in nature carrying atotal of 40 marks. The questions will be drawn from the common areas ofthe syllabus.

Section B:

will consist of eight questions of greater length and difficulty.
Thequestions shall include a maximum of two which shall be drawn fromparts of the syllabuses which may not be peculiar to candidates’ homecountries.
Candidates will be expected to answer five questions for60marks.


DETAILED SYLLABUS

The topics, contents and notes are intended to indicate the scope of the questions whichwill be set. The notes are not to be considered as an exhaustive list ofillustrations/limitations.
TOPICSUB-TOPICSCONTENTSNOTES

A. NUMBER ANDNUMERATION

(a) Number bases

(i) conversion of numbersfrom one base toanother

Conversion from one baseto base 10 and vice versa.Conversion from one baseto another base

(ii) Basic operations onnumber bases

Addition, subtraction andmultiplication of numberbases.

(b) Modular Arithmetic

(i) Concept of ModuloArithmetic.

(ii) Addition, subtraction andmultiplication operationsin modulo arithmetic.

Interpretation of moduloarithmetic e.g.6 + 4 = k(mod7),3 x 5 = b(mod6),m = 2(mod 3), etc.

(iii) Application to daily life

Relate to market days,clock,shift duty, etc.

(c) Fractions, Decimals andApproximations

(i) Basic operations onfractions and decimals.

(ii) Approximations andsignificant figures.

Approximations should berealistic e.g. a road is notmeasured correct to thenearest cm.

(d) Indices

(i) Laws of indices

e.g. ax x ay = ax + y,
ax ÷ ay= ax – y,
(ax)y = axy,
etcwhere x , y are realnumbers and a ≠ 0.
Include simple examplesof negative and fractionalindices.

(ii) Numbers in standardform(scientific notation)

Expression of large andsmall numbers in standardform
e.g. 375300000 = 3.753 x108
0.00000035 = 3.5 x 10-7
Use of tables of squares,square roots andreciprocals is accepted.

(e) Logarithms

(i) Relationship betweenindices and logarithmse.g. y = 10k implieslog10y = k .
(ii) Basic rules of logarithmse.g.log10(pq) = log10p +log10q
log10(p/q) = log10p –log10q
log10pn = nlog10p.

(iii) Use of tables oflogarithms andantilogarithms.

Calculations involvingmultiplication, division,powers and roots.

(f) Sequence and Series

(i) Patterns of sequences.

Determine any term of agiven sequence. Thenotation Un = the nthtermof a sequence may beused.

(ii) Arithmetic progression(A.P.)
Geometric Progression(G.P.)

Simple cases only,including word problems.(Include sum for A.P. andexclude sum for G.P).

(g) Sets

(i) Idea of sets, universalsets, finite and infinitesets, subsets, empty setsand disjoint sets.

Notations: ∈, ⊂, ꓴ, ꓵ, { }, ∅, P’ (the compliment ofP).

Idea of and notation forunion, intersection andcomplement of sets.

properties e.g.commutative, associativeand distributive

(ii) Solution of practicalproblems involvingclassification using Venndiagrams.

Use of Venn diagramsrestricted to at most 3sets.

(h) Logical Reasoning

Simple statements. True andfalse statements. Negation ofstatements, implications.

Use of symbols:⇒, ⇐, useof Venn diagrams.

(i) Positive and negativeintegers, rational numbers

The four basic operations onrational numbers.

Match rational numberswith points on the numberline. Notation: Naturalnumbers (N), Integers (Z), Rational numbers (Q).

(j) Surds (Radicals)

Simplification andrationalization of simplesurds.

Surds of the form a/√b, a√𝑏and a ± √𝑏 where a is arational number and b is apositive integer.Basic operations on surds(exclude surd of the form a/(b + c√a)).

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(k) Matrices andDeterminants

(i) Identification of order,notation and types ofmatrices.

Not more than 3 x 3matrices. Idea of columnsand rows.

(ii) Addition, subtraction,scalar multiplication andmultiplication ofmatrices.

Restrict to 2 x 2 matrices.

(iii) Determinant of a matrix

Application to solvingsimultaneous linearequations in two variables.Restrict to 2 x 2 matrices.

(l) Ratio, Proportions and Rates

Ratio between two similarquantities.Proportion between two ormore similar quantities.

Relate to real lifesituations.

Financial partnerships, ratesof work, costs, taxes, foreignexchange, density (e.g.population), mass, distance,time and speed.

Include average rates,taxes e.g. VAT,Withholding tax, etc

(m) Percentages

Simple interest, commission,discount, depreciation, profitand loss, compound interest,hire purchase andpercentage error.

Limit compound interestto a maximum of 3 years.

(n) Financial Arithmetic

(i) Depreciation/Amortization.

Definition/meaning,calculation of depreciationon fixed assets,computation ofamortization on capitalizedassets.

(ii) Annuities

Definition/meaning, solvesimple problems onannuities.

(iii) Capital MarketInstruments

Shares/stocks,debentures, bonds, simpleproblems on interest onbonds and debentures.

(o) Variation

Direct, inverse, partial andjoint variations.

Expression of varioustypes of variation inmathematical symbols e.g.
direct (z α n),
inverse (z α 1/n), etc.Application to simplepractical problems.

B. ALGEBRAIC PROCESSES

(a) Algebraic expressions

(i) Formulating algebraicexpressions from givensituations

e.g. find an expression forthe cost C Naira of 4 pensat x Naira each and 3oranges at y naira each.Solution: C = 4x + 3y

(ii) Evaluation of algebraicexpressions

e.g. If x =60 and y = 20,find C.
C = 4(60) + 3(20) = 300naira.

(b) Simple operations onalgebraic expressions

(i) Expansion

e.g. (a +b )(c + d ),
(a + 3)(c - 4), etc.

(ii) Factorization

factorization of expressions of the form
ax + ay,
a(b + c) + d (b + c),
a2 –b2,
ax 2 + bx + c where a , b , care integers.

(iii) Binary Operations

Application of differenceof two squares e.g. 492 –472 =(49 + 47)(49 – 47) = 96 x2 = 192.

Carry out binaryoperations on realnumbers such as: a*b =2a + b – ab , etc.

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(c) Solution of LinearEquations

(i) Linear equations in onevariable

Solving/finding the truthset (solution set) for linearequations in one variable.

(ii) Simultaneous linearequations in twovariables.

Solving/finding the truthset of simultaneousequations in two variablesby elimination,substitution and graphicalmethods.
Word problemsinvolving one or twovariables

(d) Change of Subject of aFormula/Relation

(i) Change of subject of aformula/relation

e.g. if 1/f = 1/u+ 1/v, find v.

(ii) Substitution.

Finding the value of avariable e.g. evaluating vgiven the values of u andf

(e) Quadratic Equations

(i) Solution of quadraticequations

Using factorization i.e. ab= 0 ⇒ either a = 0 or b =0.

(ii) Forming quadraticequation with givenroots.

Simple rational roots onlye.g. forming a quadraticequation whose roots are-3 and 5/2⇒ (x + 3)(x - 5/2)= 0.

(iii) Application of solution ofquadratic equation in practical problems.

(f) Graphs of Linear and Quadraticfunctions.

(i) Interpretation of graphs,coordinate of points, tableof values, drawingquadratic graphs andobtaining roots fromgraphs.

Finding:
(i) the coordinates ofmaximum and minimumpoints on the graph.
(ii) intercepts on the axes,identifying axis ofsymmetry, recognizingsketched graphs.

(ii) Graphical solution of apair of equations of theform:y = ax2 + bx + c and y = mx+ k

Use of quadratic graphs tosolve related equationse.g. graph of y = x2 +5x + 6 to solve x2 + 5x +4 = 0.

(iii) Drawing tangents tocurves to determine thegradient at a given point.

Determining the gradientby drawing relevanttriangle.

(g) Linear Inequalities

(i) Solution of linearinequalities in onevariable andrepresentation on thenumber line.

Truth set is also required.Simple practical problems

(ii) Graphical solution oflinear inequalities in twovariables.

Maximum and minimumvalues. Application to reallife situations e.g.minimum cost, maximumprofit, linearprogramming, etc.

(iii) Graphical solution ofsimultaneous linearinequalities in twovariables.

(h) Algebraic Fractions

Operations on algebraicfractions with:
(i) Monomial denominators

Simple cases only e.g. 1/x+ 1/y= (x + y)/xy
( x ≠ 0, y ≠ 0).

(ii) Binomial denominators

Simple cases only e.g. 1/(x - a) + 1/(x-b) = (2x - a - b)/(x - a)(x - b)
where a andb areconstants and x 􀵍a or b .
Values for which a fractionis undefined e.g. 1/(x + 3) is notdefined for x = -3.

(i) Functions and Relations

Types of Functions

One-to-one, one-to-many,many-to-one, many-tomany.Functions as a mapping,determination of the ruleof a givenmapping/function.

C. MENSURATION

(a) Lengths andPerimeters

(i) Use of Pythagorastheorem, sine andcosine rules to determinelengths and distances.

(ii) Lengths of arcs ofcircles, perimeters ofsectors and segments.

No formal proofs of thetheorem and rules arerequired.

(iii) Longitudes andLatitudes.

Distances along latitudesand Longitudes and theircorresponding angles.

(b) Areas

(i) Triangles and specialquadrilaterals –rectangles,parallelograms andtrapeziums

Areas of similar figures.
Include area of triangle =½ base x height and
½absinC.

(ii) Circles, sectors andsegments of circles.

Areas of compoundshapes.

(iii) Surface areas of cubes,cuboids, cylinder,pyramids, righttriangularprisms, conesandspheres.

Relationship between thesector of a circle and thesurface area of a cone.

(c) Volumes

(i) Volumes of cubes,cuboids, cylinders, cones,right pyramids andspheres.

Include volumes ofcompound shapes.

( ii ) Volumes of similar solids

D. PLANE GEOMETRY

(a) Angles

(i) Angles at a point add upto 360 o.

(ii) Adjacent angles on astraight line aresupplementary.

(iii) Vertically opposite anglesare equal.

The degree as a unit ofmeasure.Consider acute, obtuse,reflex angles, etc.

(b) Angles and intercepts onparallel lines.

(i) Alternate angles areequal.

( ii )Corresponding anglesare equal.

( iii )Interior opposite anglesare supplementary

(iv) Intercept theorem.

Application to proportionaldivision of a line segment.

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(c) Triangles and Polygons.

(i) The sum of the angles ofa triangle is 2 rightangles.

(ii) The exterior angle of atriangle equals the sum ofthe two interior oppositeangles.

The formal proofs ofthose underlined may berequired.

(iii) Congruent triangles.

Conditions to be knownbut proofs not requirede.g. SSS, SAS, etc.

(iv) Properties of specialtriangles -Isosceles, equilateral,right-angled, etc

Use symmetry whereapplicable.

(v) Properties of specialquadrilaterals –parallelogram, rhombus,square, rectangle,trapezium.

(vi)Properties of similartriangles.

Equiangular propertiesand ratio of sides andareas.

(vii) The sum of the anglesof a polygon

Sum of interior angles =(n - 2)180o or (2n –4)right angles,
where n isthe number of sides

(viii) Property of exteriorangles of a polygon.

(ix) Parallelograms on thesame base and betweenthe same parallels areequal in area.

(d) Circles

(i) Chords.

Angles subtended bychords in a circle and atthe centre. Perpendicularbisectors of chords.

(ii) The angle which an arc ofa circle subtends at thecentre of the circle istwice that which itsubtends at any point onthe remaining part of thecircumference.

the formal proofs ofthose underlined may berequired.

(iii) Any angle subtended atthe circumference by adiameter is a right angle.

(iv) Angles in the samesegment are equal.

(v) Angles in oppositesegments aresupplementary.

(vi) Perpendicularity oftangent and radius.

(vii)If a tangent is drawn toa circle and from thepoint of contact a chordis drawn, each anglewhich this chord makeswith the tangent isequal to the angle in thealternate segment.

(e) Construction

(i) Bisectors of angles andline segments

(ii) Line parallel orperpendicular to a givenline.

(iii)Angles e.g. 90o, 60o,45o, 30o, and an angleequal to a given angle.

(iv) Triangles andquadrilaterals fromsufficient data.

Include combination of these angles e.g. 75o, 105o, 135o, etc.

(f) Loci

Knowledge of the loci listedbelow and their intersectionsin 2 dimensions.

(i) Points at a given distancefrom a given point.

(ii) Points equidistant fromtwo given points.

(iii)Points equidistant fromtwo given straight lines.

(iv)Points at a given distancefrom a givenstraight line.

Consider parallel andintersecting lines.Application to real lifesituations.

E. COORDINATE GEOMETRY OFSTRAIGHT LINES

Connrdinate Geometry of Straight Lines

(i) Concept of the x-y plane.

(ii) Coordinates of points onthe x-y plane.

Midpoint of two points,distance between twopoints i.e.
|PQ| = √[(𝑥2 - 𝑥1)2 + (𝑦2 - y1)2],
where P(x1,y1) and Q(x2,y2), gradient (slope) of aline
m= (y2 - y1)/(x2 - x1),
equationof a line in the form
y =mx + c and
y – y1 = m(x– x1),
where m is thegradient (slope) and c is aconstant.

F. TRIGONOMETRY

(a) Sine, Cosine and Tangent ofan angle.

(i) Sine, Cosine and Tangentof acute angles.

(ii) Use of tables oftrigonometric ratios.

Use of right angledtriangles

(iii) Trigonometric ratios of30o, 45o and 60o.

Without the use of tables.

(iv) Sine, cosine and tangentof angles from 0o to 360o.

Relate to the unit circle.

(v)Graphs of sine andcosine.

0o≤ x ≤ 360o.e.g.y = a sinx , y = b cosx

(vi) Graphs of trigonometricratios.

Graphs of simultaneouslinear and trigonometricequations.e.g. y = asin x + bcos x,etc.

(b) Angles of elevation anddepression

(i) Calculating angles ofelevation and depression.

(ii) Application to heights anddistances.

Simple problems only.

(c) Bearings

(i) Bearing of one point fromanother.

Notation e.g. 035o, N35oE

(ii) Calculation of distancesand angles

Simple problems only. Useof diagram isrequired.
Sine andcosine rules may be used.

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G. INTRODUCTORYCALCULUS

Calculus

(i) Differentiation of algebraicfunctions.

Concept/meaning ofdifferentiation/derivedfunction, , dy/dx , relationshipbetween gradient of acurve at a point and thedifferential coefficient ofthe equation of the curveat that point. Standardderivatives of some basicfunction e.g. if y = x2, dy/dx= 2x.
If s = 2t3 + 4, ds/dt =v = 6t2, where s =distance, t = time and v =velocity. Application toreal life situation such asmaximum and minimumvalues, rates of changeetc.

(ii) Integration of simpleAlgebraic functions.

Meaning/ concept ofintegration, evaluation ofsimple definite algebraicequations.

H. STATISTICS ANDPROBABILITY.

(a) Statistics

(i) Frequency distribution

Construction of frequencydistribution tables,concept of class intervals,class mark and classboundary.

(ii) Pie charts, bar charts,histograms and frequencypolygons

Reading and drawingsimple inferences fromgraphs, interpretation ofdata in histograms.

(iii) Mean, median and modefor both discrete andgrouped data.

Exclude unequal classinterval.
Use of an assumed meanis acceptable but notrequired.
For groupeddata, the mode should beestimated from thehistogram while themedian, quartiles andpercentiles are estimatedfrom the cumulativefrequency curve.

(iv) Cumulative frequencycurve (Ogive).

Application of thecumulative frequencycurve to every day life.

(v) Measures of Dispersion:range, semi interquartile/inter-quartile range,variance, mean deviation andstandard deviation.

Definition of range,variance, standarddeviation, inter-quartilerange. Note that meandeviation is the mean ofthe absolute deviationsfrom the mean andvariance is the square ofthe standard deviation.
Problems on range,variance, standarddeviation etc.
Standard deviation ofgrouped data

(b) Probability

(i) Experimental andtheoretical probability.

Include equally likelyevents e.g. probability ofthrowing a six with a fair die or a head whentossing a fair coin.

(ii) Addition of probabilitiesfor mutually exclusive andindependent events.

With replacement.without replacement.

(iii) Multiplication ofprobabilities forindependent events.

Simple practical problemsonly. Interpretation of“and” and “or” inprobability.

I. VECTORS ANDTRANSFORMATION

(a) Vectors in a Plane

Vectors as a directed linesegment.

(5, 060o)

Cartesian components of avector

e.g. (5 sin60o 5 cos60o )

Magnitude of a vector, equalvectors, addition andsubtraction of vectors, zerovector, parallel vectors,multiplication of a vector byscalar.

Knowledge of graphicalrepresentation isnecessary.

(b) Transformation in theCartesian Plane

Reflection of points andshapes in the CartesianPlane.

Restrict Plane to the x andy axes and in the lines x =k, y = x and y = kx ,where k is an integer.Determination of mirrorlines (symmetry).

Rotation of points andshapes in the CartesianPlane.

Rotation about the originand a point other than theorigin.
Determination of theangle of rotation (restrictangles of rotation to -180oto 180o).

Translation of points andshapes in the CartesianPlane.

Translation using atranslation vector.

Enlargement

Draw the images of planefigures under enlargementwith a given centre for a given scale factor.Usegiven scales to enlarge orreduce plane figures.

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1. UNITS

Candidates should be familiar with the following units and their symbols.

( 1 ) Length

1000 millimetres (mm) = 100 centimetres (cm) = 1 metre(m).
1000 metres = 1 kilometre (km)

( 2 ) Area

10,000 square metres (m2) = 1 hectare (ha)

( 3 ) Capacity

1000 cubic centimeters (cm3) = 1 litre (l)

( 4 ) Mass

1000 milligrammes (mg) = 1 gramme (g)
1000 grammes (g) = 1 kilogramme( kg )
1000 ogrammes (kg) = 1 tonne.

( 5) Currencies

The Gambia – 100 bututs (b) = 1 Dalasi (D)
Ghana - 100 Ghana pesewas (Gp) = 1 Ghana Cedi ( GH¢)
Liberia - 100 cents (c) = 1 Liberian Dollar (LD)
Nigeria - 100 kobo (k) = 1 Naira (N)
Sierra Leone - 100 cents (c) = 1 Leone (Le)
UK - 100 pence (p) = 1 pound (£)
USA - 100 cents (c) = 1 dollar ($)
French Speaking territories 100 centimes (c) = 1 Franc (fr)
Any other units used will be defined.

2. OTHER IMPORTANT INFORMATION

( 1) Use of Mathematical and Statistical Tables

Mathematics and Statistical tables, published or approved by WAEC may be used in theexamination room. Where the degree of accuracy is not specified in a question, thedegree of accuracy expected will be that obtainable from the mathematical tables.

(2) Use of calculators

The use of non-programmable, silent and cordless calculators is allowed. The calculatorsmust, however not have a paper print out nor be capable of receiving/sendingany information. Phones with or without calculators are not allowed.

(3) Other Materials Required for the examination

Candidates should bring rulers, pairs of compasses, protractors, set squares etc requiredfor papers of the subject. They will not be allowed to borrow such instruments and anyother material from other candidates in the examination hall.Graph papers ruled in 2mm squares will be provided for any paper in which it isrequired.

( 4) Disclaimer

In spite of the provisions made in paragraphs 2 (1) and (2) above, it should be notedthat some questions may prohibit the use of tables and/or calculators.

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