# 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.*

TOPIC | SUB-TOPICS | CONTENTS | NOTES |
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## 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. | ||

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## (b) Modular Arithmetic | ## (i) Concept of ModuloArithmetic. | 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. | Approximations should berealistic e.g. a road is notmeasured correct to thenearest cm. | |

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## (d) Indices | ## (i) Laws of indices | e.g. a^{x} x a^{y} = a^{x + y},a ^{x} ÷ ^{a}y= a^{x – y},(a ^{x})^{y} = a^{xy},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 x10 ^{8}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 = 10 | Calculations involvingmultiplication, division,powers and roots. | |

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## (f) Sequence and Series | ## (i) Patterns of sequences. | Determine any term of agiven sequence. Thenotation U_{n} = the nthtermof a sequence may beused. | |

## (ii) Arithmetic progression(A.P.) | Simple cases only,including word problems.(Include sum for A.P. andexclude sum for G.P). | ||

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## (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)). | |

## We provide educational resources/materials, curriculum guide, syllabus, scheme of work, lesson note & plan, waec, jamb, O-level & advance level GCE lessons/tutorial classes, on various topics, subjects, career, disciplines & department etc. for all the Class of Learners | |||

## (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. | ||

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## (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. | |

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## (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. | |

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## 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. | ||

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## (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), a ^{2} –b^{2},ax 2 + bx + c where a , b , care integers. | ||

## (iii) Binary Operations | Application of differenceof two squares e.g. 49^{2} –47^{2} =(49 + 47)(49 – 47) = 96 x2 = 192.Carry out binaryoperations on realnumbers such as: a*b =2a + b – ab , etc. | ||

## We provide educational resources/materials, curriculum guide, syllabus, scheme of work, lesson note & plan, waec, jamb, O-level & advance level GCE lessons/tutorial classes, on various topics, subjects, career, disciplines & department etc. for all the Class of Learners | |||

## (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 | ||

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## (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 = ax | Use of quadratic graphs tosolve related equationse.g. graph of y = x^{2} +5x + 6 to solve x^{2} + 5x +4 = 0. | ||

## (iii) Drawing tangents tocurves to determine thegradient at a given point. | Determining the gradientby drawing relevanttriangle. | ||

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## (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: | 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. | |

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## C. MENSURATION | ## (a) Lengths andPerimeters | ## (i) Use of Pythagorastheorem, sine andcosine rules to determinelengths and distances. | 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 | |||

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## D. PLANE GEOMETRY | ## (a) Angles | ## (i) Angles at a point add upto 360 | The degree as a unit ofmeasure.Consider acute, obtuse,reflex angles, etc. |

## (b) Angles and intercepts onparallel lines. | ## (i) Alternate angles areequal. | Application to proportionaldivision of a line segment. | |

## We provide educational resources/materials, curriculum guide, syllabus, scheme of work, lesson note & plan, waec, jamb, O-level & advance level GCE lessons/tutorial classes, on various topics, subjects, career, disciplines & department etc. for all the Class of Learners | |||

## (c) Triangles and Polygons. | ## (i) The sum of the angles ofa triangle is 2 rightangles. | 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)180^{o} 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. | |||

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## (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 | Include combination of these angles e.g. 75^{o}, 105^{o}, 135^{o}, etc. | |

## (f) Loci | ## Knowledge of the loci listedbelow and their intersectionsin 2 dimensions. | Consider parallel andintersecting lines.Application to real lifesituations. | |

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## E. COORDINATE GEOMETRY OFSTRAIGHT LINES | ## Connrdinate Geometry of Straight Lines | ## (i) Concept of the x-y plane. | Midpoint of two points,distance between twopoints i.e. |PQ| = √[(𝑥 _{2} - 𝑥_{1})^{2} + (𝑦_{2} - y_{1})^{2}],where P(x1,y1) and Q(x2,y2), gradient (slope) of aline m= (y _{2} - y_{1})/(x_{2} - x_{1}),equationof a line in the form y =mx + c and y – y _{1} = m(x– x_{1}),where m is thegradient (slope) and c is aconstant. |

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## F. TRIGONOMETRY | ## (a) Sine, Cosine and Tangent ofan angle. | ## (i) Sine, Cosine and Tangentof acute angles. | Use of right angledtriangles |

## (iii) Trigonometric ratios of30 | Without the use of tables. | ||

## (iv) Sine, cosine and tangentof angles from 0 | Relate to the unit circle. | ||

## (v)Graphs of sine andcosine. | 0^{o}≤ x ≤ 360^{o}.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. | Simple problems only. | |

## (c) Bearings | ## (i) Bearing of one point fromanother. | Notation e.g. 035^{o}, N35^{o}E | |

## (ii) Calculation of distancesand angles | Simple problems only. Useof diagram isrequired. Sine andcosine rules may be used. | ||

## 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 = x^{2}, dy/dx= 2x.If s = 2t ^{3} + 4, ds/dt =v = 6t^{2}, 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. | ||

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## 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 | ||

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## (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. | ||

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## I. VECTORS ANDTRANSFORMATION | ## (a) Vectors in a Plane | ## Vectors as a directed linesegment. | (5, 060^{o}) |

## 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. | ||

### 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.*