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GENERAL MATHEMATICS/MATHEMATICS (CORE), 1. AIMS OF THE SYLLABUS, The aims of the syllabus are to test candidates’:, (1) mathematical competency and computational skills;, (2) understanding of mathematical concepts and their relationship to the, acquisition of entrepreneurial skills for everyday living in the global world;, (3) ability to translate problems into mathematical language and solve them, using 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 to, use their own National teaching syllabuses or curricular for that purpose., 1. 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 common, areas 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 be, answered in 2½ hours for 100 marks. Candidates will be required to answer ten, questions in all., , Section A -, , Will consist of five compulsory questions, elementary in nature carrying a, total of 40 marks. The questions will be drawn from the common areas of, the syllabus., , Section B -, , will consist of eight questions of greater length and difficulty. The, questions shall include a maximum of two which shall be drawn from, parts of the syllabuses which may not be peculiar to candidates’ home, countries. Candidates will be expected to answer five questions for, 60marks.
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2. DETAILED SYLLABUS, The topics, contents and notes are intended to indicate the scope of the questions which, will be set. The notes are not to be considered as an exhaustive list of, illustrations/limitations., , TOPICS, , CONTENTS, , NOTES, , A. NUMBER AND, NUMERATION, ( a ) Number bases, , ( i ) conversion of numbers, from, one base to, another, , ( ii ) Basic operations on, number, bases, (b) Modular Arithmetic, , (i) Concept of Modulo, Arithmetic., (ii) Addition, subtraction and, multiplication operations, in, modulo arithmetic., (iii) Application to daily life, , ( c ) Fractions, Decimals and, Approximations, , (i) Basic operations on, fractions, and decimals., (ii) Approximations and, significant figures., , ( d ) Indices, , ( i ) Laws of indices, , Conversion from one base, to base 10 and vice versa., Conversion from one base, to another base ., Addition, subtraction and, multiplication of number, bases., Interpretation of modulo, arithmetic e.g., 6 + 4 = k(mod7),, 3 x 5 = b(mod6),, m = 2(mod 3), etc., Relate to market days,, clock,shift duty, etc., , Approximations should be, realistic e.g. a road is not, measured correct to the, nearest cm., e.g. ax x ay = ax + y , ax÷ay, = ax – y, (ax)y = axy, etc, where x, y are real, numbers and a ≠0.
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( ii ) Numbers in standard, form, ( scientific notation), , ( e) Logarithms, , ( f ) Sequence and Series, , ( i ) Relationship between, indices, and logarithms, e.g. y = 10k, implies, log10y = k., ( ii ) Basic rules of logarithms, e.g., log10(pq) = log10p +, log10q, log10(p/q) = log10p –, log10q, log10pn = nlog10p., (iii) Use of tables of, logarithms, and, antilogarithms., (i) Patterns of sequences., , (ii) Arithmetic progression, (A.P.), Geometric Progression, (G.P.), ( g ) Sets, , (i) Idea of sets, universal, sets,, finite and infinite, sets,, subsets, empty sets, and, disjoint sets., Idea of and notation for, union,, intersection and, complement, of sets., , Include simple examples, of negative and fractional, indices., Expression of large and, small numbers in standard, form, e.g. 375300000 = 3.753 x, 108, 0.00000035 = 3.5 x 10-7, Use of tables of squares,, square roots and, reciprocals is accepted., , Calculations involving, multiplication, division,, powers and roots., , Determine any term of a, given sequence. The, notation Un = the nth, termof a sequence may be, used., Simple cases only,, including word problems., (Include sum for A.P. and, exclude sum for G.P)., Notations: ℰ, ⊂, ∪, ∩, { },, ∅, P’( the compliment of, P)., • properties e.g., commutative, associative, and distributive
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(ii) Solution of practical, problems, involving, classification using, Venn, diagrams., Simple statements. True and, false statements. Negation of, statements, implications., The four basic operations on, rational numbers., , ( h ) Logical Reasoning, (i) Positive and negative, integers, rational numbers, , ( j ) Surds (Radicals), , Simplification and, rationalization of simple, surds., , Use of Venn diagrams, restricted to at most 3, sets., Use of symbols: ⟹, ⇐, use, of Venn diagrams., Match rational numbers, with points on the number, line. Notation: Natural, numbers (N), Integers ( Z, ), Rational numbers ( Q )., Surds of the form, , 𝑎, √𝑏, , , a√𝑏, , and a ±√𝑏where a is a, rational number and b is a, positive integer., Basic operations on surds, (exclude surd of the form, 𝑎, )., 𝑏+𝑐, √𝑑, , ( i ) Identification of order,, notation and types of, matrices., , • ( k ) Matrices and, Determinants, , ( ii ) Addition, subtraction,, scalar multiplication and, multiplication of, matrices., ( iii ) Determinant of a matrix, , ( l ) Ratio, Proportions and, , Rates, , Ratio between two similar, quantities., Proportion between two or, more similar quantities., Financial partnerships, rates, of work, costs, taxes, foreign, exchange, density (e.g., population), mass, distance,, time and speed., , Not more than 3 x 3, matrices. Idea of columns, and rows., Restrict to 2 x 2 matrices., , Application to solving, simultaneous linear, equations in two variables., Restrict to 2 x 2 matrices., , Relate to real life, situations., Include average rates,, taxes e.g. VAT,, Withholding tax, etc
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( m ) Percentages, , Simple interest, commission,, discount, depreciation, profit, and loss, compound interest,, hire purchase and, percentage error., , Limit compound interest, to a maximum of 3 years., , ( n) Financial Arithmetic, , ( i ) Depreciation/, Amortization., , Definition/meaning,, calculation of depreciation, on fixed assets,, computation of, amortization on capitalized, assets., , ( ii ) Annuities, , Definition/meaning, solve, simple problems on, annuities., , (iii ) Capital Market, Instruments, , ( o ) Variation, , Direct, inverse, partial and, joint variations., , Shares/stocks,, debentures, bonds, simple, problems on interest on, bonds and debentures., Expression of various, types of variation in, mathematical symbols e.g., direct (z ∝n ), inverse (z ∝, 1, ), etc., 𝑛, , Application to simple, practical problems., B. ALGEBRAIC PROCESSES, ( a ) Algebraic expressions, , ( b ) Simple operations on, algebraic expressions, , (i) Formulating algebraic, expressions from given, situations, , e.g. find an expression for, the cost C Naira of 4 pens, at x Naira each and 3, oranges at y naira each., Solution: C = 4x + 3y, , ( ii ) Evaluation of algebraic, expressions, , e.g. If x =60 and y = 20,, find, C., C = 4(60) + 3(20) = 300, naira., , ( i ) Expansion, , e.g. (a +b)(c + d),, (a + 3)(c - 4), etc., , (ii ) Factorization, , factorization of
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• (iii) Binary Operations, , expressions of the form ax, + ay,, a(b + c) + d(b + c), a2 –, b2 ,, ax2 + bx + c where a, b, c, are integers., Application of difference, of two squares e.g. 492 –, 472 =, (49 + 47)(49 – 47) = 96 x, 2 = 192., Carry out binary, operations on real, numbers such as: a*b =, 2a + b – ab, etc., , ( c ) Solution of Linear, Equations, , ( i ) Linear equations in one, variable, , Solving/finding the truth, set (solution set) for linear, equations in one variable., , ( ii ) Simultaneous linear, equations in two, variables., , Solving/finding the truth, set of simultaneous, equations in two variables, by elimination,, substitution and graphical, methods. Word problems, involving one or two, variables, , ( d ) Change of Subject of a, Formula/Relation, , ( i ) Change of subject of a, formula/relation, (ii) Substitution., , e.g. if =, , ( e ) Quadratic Equations, , ( i ) Solution of quadratic, equations, , Using factorization i.e. ab, = 0 ⇒ either a = 0 or b =, 0., •By completing the, square and use of formula, , (ii) Forming quadratic, equation, with given, roots., (iii) Application of solution of, quadratic equation in, , 1, 𝑓, , 1, 𝑢, , 1, 𝑣, , + , find v., , Finding the value of a, variable e.g. evaluating v, given the values of u and, f., , Simple rational roots only, e.g. forming a quadratic, equation whose roots are, 5, 5, -3 and 2 ⇒ (x + 3)(x - 2), = 0.
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practical problems., (f) Graphs of Linear and, functions., , Quadratic, , (i) Interpretation of graphs,, coordinate of points, table, of, values, drawing, quadratic, graphs and, obtaining roots, from, graphs., , Finding:, (i) the coordinates of, maximum and minimum, points on the graph., (ii) intercepts on the axes,, identifying axis of, symmetry, recognizing, sketched graphs., , ( ii ) Graphical solution of a, pair, of equations of the, form:, y = ax2 + bx + c and y = mx, +k, , Use of quadratic graphs to, solve related equations, e.g. graph of y = x2 +, 5x + 6 to solve x2 + 5x +, 4 = 0., Determining the gradient, by drawing relevant, triangle., , (iii) Drawing tangents to, curves to determine the, gradient at a given point., ( g ) Linear Inequalities, , (i) Solution of linear, inequalities, in one, variable and, representation on the, number, line., (ii) Graphical solution of, linear, inequalities in two, variables., , ( h ) Algebraic Fractions, , Truth set is also required., Simple practical problems, , (iii) Graphical solution of, simultaneous linear, inequalities in two, variables., , Maximum and minimum, values. Application to real, life situations e.g., minimum cost, maximum, profit, linear, programming, etc., , Operations on algebraic, fractions with:, ( i ) Monomial denominators, , Simple cases only e.g., +, , ( ii ) Binomial denominators, , 1, 𝑦, , =, , 𝑥+𝑦, (, 𝑥𝑦, , 1, 𝑥, , x≠0, y≠ 0)., , Simple cases only e.g., 1, 1, 2𝑥−𝑎−𝑏, + 𝑥−𝑏 = (𝑥−𝑎)(𝑥−𝑏), 𝑥−𝑎, , where a andb are, constants and x≠a or b.
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Values for which a fraction, 1, is undefined e.g. 𝑥+3is not, •(i) Functions and Relations, , Types of Functions, , defined for x = -3., One-to-one, one-to-many,, many-to-one, many-tomany., Functions as a mapping,, determination of the rule, of a given, mapping/function., , C. MENSURATION, ( a ) Lengths and, Perimeters, , ( b ) Areas, , (i) Use of Pythagoras, theorem,, sine and, cosine rules to, determine, lengths and, distances., (ii) Lengths of arcs of, circles,, perimeters of, sectors and, segments., (iii) Longitudes and, Latitudes., ( i ) Triangles and special, quadrilaterals –, rectangles,, parallelograms and, trapeziums, (ii) Circles, sectors and, segments, of circles., (iii) Surface areas of cubes,, cuboids, cylinder,, pyramids,, righttriangular, prisms, cones, andspheres., , ( c ) Volumes, , (i) Volumes of cubes,, cuboids,, cylinders, cones,, right, pyramids and, spheres., ( ii ) Volumes of similar solids, , D. PLANE GEOMETRY, , No formal proofs of the, theorem and rules are, required., , Distances along latitudes, and Longitudes and their, corresponding angles., Areas of similar figures., Include area of triangle =, ½ base x height and, ½absinC., Areas of compound, shapes., Relationship between the, sector of a circle and the, surface area of a cone., , Include volumes of, compound shapes.
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(a) Angles, , (i) Angles at a point add up, to, 360o., (ii) Adjacent angles on a, straight, line are, supplementary., (iii) Vertically opposite angles, are, equal., , (b) Angles and intercepts on, parallel lines., , (i) Alternate angles are, equal., ( ii )Corresponding angles, are, equal., ( iii )Interior opposite angles, are, supplementary, (iv) Intercept theorem., , (c) Triangles and Polygons., , The degree as a unit of, measure., Consider acute, obtuse,, reflex angles, etc., , Application to proportional, division of a line segment., , (i) The sum of the angles of, a, triangle is 2 right, The formal proofs of, angles., those underlined may be, (ii) The exterior angle of a, required., triangle equals the sum of, the two interior opposite, angles., (iii) Congruent triangles., , Conditions to be known, but proofs not required, e.g. SSS, SAS, etc., , ( iv ) Properties of special, triangles, Isosceles,, equilateral,, right-angled, etc, , Use symmetry where, applicable., , (v) Properties of special, quadrilaterals –, parallelogram, rhombus,, square, rectangle,, trapezium., ( vi )Properties of similar, triangles., ( vii ) The sum of the angles, of a, polygon, , Equiangular properties, and ratio of sides and, areas., Sum of interior angles =, (n - 2)180o or (2n –, 4)right angles, where n is, the number of sides
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(viii) Property of exterior, angles, of a polygon., (ix) Parallelograms on the, same, base and between, the same, parallels are, equal in area., ( d ) Circles, , (i) Chords., , (ii) The angle which an arc of, a, circle subtends at the, centre, of the circle is, twice that, which it, subtends at any, point on, the remaining part, of the, circumference., , Angles subtended by, chords in a circle and at, the centre. Perpendicular, bisectors of chords., , the formal proofs of, those underlined may be, required., , (iii) Any angle subtended at, the, circumference by a, diameter, is a right angle., (iv) Angles in the same, segment, are equal., (v) Angles in opposite, segments, are, supplementary., ( vi )Perpendicularity of, tangent, and radius., (vii )If a tangent is drawn to, a, circle and from the, point of, contact a chord, is drawn,, each angle, which this chord, makes, with the tangent is, equal to the angle in the, alternate segment., ( e ) Construction, , ( i ) Bisectors of angles and, line, segments, (ii) Line parallel or, perpendicular, to a given, line., , Include combination of
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( iii )Angles e.g. 90o, 60o,, 45o,, 30o, and an angle, equal to a, given angle., (iv) Triangles and, quadrilaterals, from, sufficient data., ( f ) Loci, , E. COORDINATE, GEOMETRY OF, STRAIGHT LINES, , Knowledge of the loci listed, below and their intersections, in 2 dimensions., (i) Points at a given distance, from, a given point., (ii) Points equidistant from, two, given points., ( iii)Points equidistant from, two, given straight lines., (iv)Points at a given distance, from a given, straight line., (i) Concept of the x-y plane., (ii) Coordinates of points on, the x-y plane., , these angles e.g. 75o,, 105o,135o, etc., , Consider parallel and, intersecting lines., Application to real life, situations., , Midpoint of two points,, distance between two, points i.e. |PQ| =, √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 ,, where P(x1,y1) and Q(x2,, y2), gradient (slope) of a, 𝑦 −𝑦, line m= 𝑥2 − 𝑥1 , equation, 2, , 1, , of a line in the form y =, mx + c and y – y1 = m(x, – x1), where m is the, gradient (slope) and c is a, constant., F. TRIGONOMETRY, (a) Sine, Cosine and Tangent, an angle., , of, , (i) Sine, Cosine and Tangent, of, acute angles., , Use of right angled, triangles, , (ii) Use of tables of, trigonometric, ratios., (iii) Trigonometric ratios of, 30o,, 45o and 60o., , Without the use of tables., , (iv) Sine, cosine and tangent, of angles from 0o to 360o., , Relate to the unit circle.
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( v )Graphs of sine and, cosine., , 0o≤ x ≤ 360o., e.g.y = asinx, y = bcosx, , (vi)Graphs of trigonometric, ratios., , Graphs of simultaneous, linear and trigonometric, equations., e.g. y = asin x + bcos x,, etc., , ( b ) Angles of elevation and, depression, , (i) Calculating angles of, elevation and depression., (ii) Application to heights and, distances., , Simple problems only., , ( c ) Bearings, , (i) Bearing of one point from, another., , Notation e.g. 035o, N35oE, , (ii) Calculation of distances, and angles, , Simple problems only. Use, of diagram is, required.Sine and, cosine rules may be used., , (i) Differentiation of algebraic, functions., , Concept/meaning of, differentiation/derived, dy, function,, , relationship, dx, between gradient of a, curve at a point and the, differential coefficient of, the equation of the curve, at that point. Standard, derivatives of some basic, dy, function e.g. if y = x2, dx, , G. INTRODUCTORY, CALCULUS, , = 2x. If s = 2t3 + 4,, , (ii) Integration of simple, Algebraic functions., , ds, dt, , =, , v = 6t2, where s =, distance, t = time and v =, velocity. Application to, real life situation such as, maximum and minimum, values, rates of change, etc., Meaning/ concept of, integration, evaluation of, simple definite algebraic, equations.
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H. STATISTICS AND, PROBABILITY., ( A ) Statistics, , (i) Frequency distribution, , ( ii ) Pie charts, bar charts,, histograms and frequency, polygons, (iii) Mean, median and mode, for both discrete and, grouped data., , (iv) Cumulative frequency, curve (Ogive)., (v) Measures of Dispersion:, range, semi interquartile/inter-quartile range,, variance, mean deviation and, standard deviation., , ( b ) Probability, , (i) Experimental and, theoretical probability., , Construction of frequency, distribution tables,, concept of class intervals,, class mark and class, boundary., Reading and drawing, simple inferences from, graphs, interpretation of, data in histograms., Exclude unequal class, interval., Use of an assumed mean, is acceptable but not, required. For grouped, data, the mode should be, estimated from the, histogram while the, median, quartiles and, percentiles are estimated, from the cumulative, frequency curve., Application of the, cumulative frequency, curve to every day life., Definition of range,, variance, standard, deviation, inter-quartile, range. Note that mean, deviation is the mean of, the absolute deviations, from the mean and, variance is the square of, the standard deviation., Problems on range,, variance, standard, deviation etc., Standard deviation of, grouped data, Include equally likely, events e.g. probability of, throwing a six with a fair
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die or a head when, tossing a fair coin., , (ii) Addition of probabilities, for, mutually exclusive and With replacement., independent events., without replacement., (iii) Multiplication of, probabilities, for, independent events., , Simple practical problems, only. Interpretation of, “and” and “or” in, probability., , I. VECTORS AND, TRANSFORMATION, (a) Vectors in a Plane, , (b) Transformation in the, Cartesian Plane, , Vectors as a directed line, segment., , (5, 060o), , Cartesian components of a, vector, , e.g., , Magnitude of a vector, equal, vectors, addition and, subtraction of vectors, zero, vector, parallel vectors,, multiplication of a vector by, scalar., , Knowledge of graphical, representation is, necessary., , Reflection of points and, shapes in the Cartesian, Plane., , Rotation of points and, shapes in the Cartesian, Plane., , 𝑜, , sin 60, (55𝑐𝑜𝑠60, 𝑜 )., , Restrict Plane to the x and, y axes and in the lines x =, k, y = x and y = kx,, where k is an integer., Determination of mirror, lines (symmetry)., Rotation about the origin, and a point other than the, origin., Determination of the, angle of rotation (restrict, angles of rotation to -180o, to 180o)., , Translation of points and, shapes in the Cartesian, Plane., , Translation using a, translation vector., , Enlargement, , Draw the images of plane, figures under enlargement, with a given centre for a
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given scale factor.Use, given scales to enlarge or, reduce plane figures., 3. 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, , –, , Ghana, , -, , 100 bututs (b) = 1 Dalasi (D), 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., 4. OTHER IMPORTANT INFORMATION, ( 1) Use of Mathematical and Statistical Tables, Mathematics and Statistical tables, published or approved by WAEC may be used, in the examination room. Where the degree of accuracy is not specified in a
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question, the degree 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, calculators must, however not have the capability to print out nor to receive, or send any 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, required for papers of the subject. They will not be allowed to borrow such, instruments and any other material from other candidates in the examination, hall., Graph papers ruled in 2mm squares will be provided for any paper in which it is, required., ( 4) Disclaimer, In spite of the provisions made in paragraphs 4 (1) and (2) above, it should be, noted that some questions may prohibit the use of tables and/or calculators.
