WASSCE / WAEC Further Maths Syllabus
AIMS OF THE SYLLABUS
The aims of the syllabus are to test candidates’
(i)development of further conceptual and manipulative skills in Mathematics;
(ii)understanding of an intermediate course of study which bridges the gap between Elementary Mathematics and Higher Mathematics;
(iii)acquisition of aspects of Mathematics that can meet the needs of potential Mathematicians, Engineers, Scientists and other professionals.
(iv)ability to analyse data and draw valid conclusion
(v)logical, abstract and precise reasoning skills.
EXAMINATION SCHEME
There will be two papers, Papers 1 and 2, both of which must be taken.
PAPER 1: will consist of forty multiplechoice objective questions, covering the entire syllabus. Candidates will be required to answer all questions in 1hours for 40 marks. The questions will be drawn from the sections of the syllabus as follows:
Pure Mathematics 
– 
30 questions 
Statistics and probability 
– 
4 questions 
Vectors and Mechanics 
– 
6 questions 
PAPER 2: will consist of two sections, Sections A and B, to be answered in 2 hours for 100 marks.
Section A will consist of eight compulsory questions that areelementary in type for 48 marks. The questions shall be distributed as follows:
Pure Mathematics 
– 
4 questions 
Statistics and Probability 
– 
2 questions 
Vectors and Mechanics 
– 
2 questions 
Section B will consist of seven questions of greater length and difficulty put into three parts:Parts I, II and III as follows:
Part I: Pure Mathematics 
– 
3 questions 
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Part II: Statistics and Probability 
– 
2 questions 
Part III: Vectors and Mechanics 
– 
2 questions 
Candidates will be required to answer four questions with at least one from each part for 52 marks.
DETAILED SYLLABUS
In addition to the following topics, more challenging questions may be set on topics in the General Mathematics/Mathematics (Core) syllabus.
In the column for CONTENTS, more detailed information on the topics to be tested is given while the limits imposed on the topics are stated under NOTES.
Topics which are marked with asterisks shall be tested in Section B of Paper 2 only.
KEY:
*Topics peculiar to Ghana only. ** Topics peculiar to Nigeria only
Topics 


Content 



Notes 







I. Pure Mathematics 




















(1) Sets 
(i) Idea of a set defined by a 
(x : x is real), ∪, ∩, { },∉, ∈, 


property, Set notations and their 
⊂, ⊆, 








meanings. 


















(ii) Disjoint sets, Universal set and 
U (universal set) and 








complement of set 



A’ (Complement of set A). 


(iii) Venn diagrams, Use of sets 
More challenging problems 


And Venn diagrams to solve 
involving union, intersection, 


problems. 





the universal set, subset and 










complement of set. 








(iv) Commutative and Associative 
Three set problems. Use of De 


laws, Distributive properties 
Morgan’s laws to solve related 


over union and intersection. 
problems 



























(2) Surds 




√ 
All the four operations on surds 

Surds of the form 
√ 
, a and 











Rationalising the denominator 


a+b√ where a is rational, b is a 













of surds such as 

, 
√ 

, 


positive integer and n is not a 













√ 
√ 


perfect square. 













































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(3) 
Binary Operations 



Properties: 


Closure, Commutativity, 


Associativity and Distributivity, 


Identity elements and inverses. 
(4) 
Logical Reasoning 
(i) Rule of syntax: 


true or false statements, 


rule of logic applied to 


arguments, implications and 


deductions. 


(ii) The truth table 
(5) 
Functions 
(i) Domain and codomain of a 


function. 


(ii) Onetoone, onto, identity and 


constant mapping; 


(iii) Inverse of a function. 


(iv) Composite of functions. 
(6) 
Polynomial 
(i) Linear Functions, Equations and 

Functions 
Inequality 



√.√√
Use of properties to solve related problems.
Using logical reasoning to determine the validity of compound statements involving implications and connectivities. Include use of symbols: ~P
p ν q, p ∧ q, p ⇒ q
Use of Truth tables to deduce conclusions of compound statements. Include negation.
The notation e.g. f : x →3x+4;
g: x → x2 ; where x ∈R.
Graphical representation of a function ; Image and the range.
Determination of the inverse of a onetoone function e.g. If f: x →sx + , the inverse
relation f1: x →x – is also a function.
Notation: fog(x) =f(g(x)) Restrict to simple algebraic functions only.
Recognition and sketching of graphs of linear functions and equations.
Gradient and intercepts forms
of linear equations i.e.
ax + by + c = 0; y = mx + c; + = k. Parallel and
Perpendicular lines. Linear Inequalities e.g. 2x + 5y ≤ 1, x + 3y ≥ 3
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(ii) Quadratic Functions, Equations
and Inequalities
(ii) Cubic Functions and Equations
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(7) Rational Functions
(8)Indices and Logarithmic Functions
(i) Rational functions of the form Q(x) = !()() ,g(x) ≠ 0.
where g(x) and f(x) are
polynomials. e.g.
f:x →“#$%
(ii)Resolution of rational functions into partial
fractions.
(i)Indices
(ii)Logarithms
remainder when f(x) is divided by f(x – a) = f(a). When f(a) is zero, then (x – a) is a factor of f(x).
g(x) may be factorised into linear and quadratic factors (Degree of Numerator less than that of denominator which is less than or equal to 4).
The four basic operations. Zeros, domain and range, sketching not required.
Laws of indices. Application of the laws of indices to evaluating products, quotients, powers and nth root. Solve equations involving indices.
Laws of Logarithms. Application of logarithms in calculations involving product, quotients, power (log an), nth roots (log √&, log a1/n).
Solve equations involving logarithms (including change of base).
Reduction of a relation such as y = axb, (a,b are constants) to a linear form:
log10y = b log10x+log10a. Consider other examples such
as
log abx = log a + x log b;
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(9) Permutation And Combinations.
(10)Binomial Theorem
(11)Sequences and Series
(i)Simple cases of arrangements
(ii)Simple cases of selection of objects.
Expansion of (a + b)n.
Use of (1+x)n ≈1+nx for any rational n, where x is sufficiently small. e.g (0.998)1/3
(i)Finite and Infinite sequences.
(ii)Linear sequence/Arithmetic Progression (A.P.) and
Exponential sequence/Geometric
Progression (G.P.)
(iii)Finite and Infinite series.
(iv)Linear series (sum of A.P.) and exponential series (sum of G.P.)
log (ab)x = x(log a + log b)
=x log ab
*Drawing and interpreting
graphs of logarithmic functions e.g. y = axb. Estimating the values of the constants a and b from the graph
Knowledge of arrangement and selection is expected. The notations: nCr, ‘(%) and nPr for selection and arrangement respectively should be noted and used. e.g. arrangement of students in a row, drawing balls from a box with or without
replacements. npr = n!
(nr)!
nCr= n! r!(nr)!
Use of the binomial theorem for positive integral index only. Proof of the theorem not required.
e.g. (i) u1, u2,…, un.(ii) u1, u2,….
Recognizing the pattern of a sequence. e.g.
(i)Un = U1 + (n1)d, where d is the common difference.
(ii)Un= U1 rn1 where r is the common ratio.
(i)U1 + U2 + U3 + … + Un(ii)U1 + U2 + U3 + ….
(i)Sn = ((U1+Un)
(ii)Sn = ([2a + (n – 1)d]
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*(v) Recurrence Series
(12)Matrices and 
(i) Matrices 
Linear 

Transformation 

(ii) Determinants
(iii)Inverse of 2 x 2 Matrices
(iv)Linear Transformation
(iii)Sn= U1(1rn) , r<1
l – r
(iv)Sn=U1(rn1) , r>l. r – 1
(v)Sum to infinity (S) =% r < 1
Generating the terms of a recurrence series and finding an explicit formula for the sequence e.g. 0.9999 =
* + *# + *+ + *, + ….
Concept of a matrix – state the order of a matrix and indicate the type.
Equal matrices – If two matrices are equal, then their corresponding elements are equal. Use of equality to find missing entries of given matrices
Addition and subtraction of matrices (up to 3 x 3 matrices). Multiplication of a matrix by a scalar and by a matrix (up to 3 x 3 matrices)
Evaluation of determinants of 2 x 2 matrices.
**Evaluation of determinants of 3 x 3 matrices.
Application of determinants to solution of simultaneous linear equations.
e.g. If A = & 
, then 



– . 
− 

1 

. 

A 
= 

& 


− 
Finding the images of points under given linear transformation
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Determining the matrices of given linear transformation. Finding the inverse of a linear transformation (restrict to 2 x 2 matrices).
Finding the composition of linear transformation. Recognizing the Identity transformation.

(i) 
1 
0 
reflection in the 


0 
−1 


x – axis 



(ii) 
−1 
0 reflection in the 



0 
1 


y – axis 



(iii) 
0 
1 
reflection in the line 


1 
0 



y = x 



(iv) 
cos 5 
− sin 5for anti 



sin 5 
cos 5 


clockwise rotation through θ 


about the origin. 


(v) 
8925 
sin 25 , the 



9;25 
−8925 


general matrix for reflection in 


a line through the origin 


making an angle θ with the 


positive xaxis. 


*Finding the equation of the 


image of a line under a given 


linear transformation 

(13)Trigonometry 
(i) Trigonometric Ratios and Rules 




Sine, Cosine and Tangent of 


general angles (0o≤θ≤360o). 


Identify trigonometric ratios of 


angles 30O, 45O, 60o without 


use of tables. 


Use basic trigonometric ratios 


and reciprocals to prove given 


trigonometric identities. 


Evaluate sine, cosine and 


tangent of negative angles. 


Convert degrees into radians 


and vice versa. 


Application to real life situations 


such as heights and distances, 


perimeters, solution of 


triangles, 
angles of elevation 


and depression, 
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bearing(negative and positive 

angles) including use of sine 

and cosine rules, etc, 

Simple cases only. 
(ii) Compound and Multiple 

Angles. 
sin (A± B),cos (A ± B), 

tan(A ± B). 

Use of compound angles in 

simple identities and solution of 

trigonometric ratios e.g. finding 

sin 75o, cos 150oetc, finding tan 

45o without using mathematical 

tables or calculators and 

leaving your answer as a surd, 

etc. 

Use of simple trigonometric 

identities to find trigonometric 

ratios of compound and 

multiple angles (up to 3A). 
(iii)Trigonometric Functions and Equations
Relate trigonometric ratios to Cartesian Coordinates of points (x, y) on the circle x2 + y2 = r2. f:x →sin x,
g: x → a cos x + b sin x = c. Graphs of sine, cosine, tangent and functions of the form asinx + bcos x. Identifying maximum and minimum point, increasing and decreasing portions. Graphical solutions of simple trigonometric equations e.g. asin x + bcos x = k. Solve trigonometric equations
up to quadratic equations e.g. 2sin2x – sin x – 3 =0, for 0o ≤ x ≤ 360o.
*Express f(x) = asin x + bcos x
in the form Rcos (x ±) or Rsin (x ± ) for 0o ≤ ≤ 90oand use
the result to calculate the minimum and maximum points of a given functions.
(14)Coordinate 
(i) Straight Lines 
Geometry 

Midpoint of a line segment
Coordinates of points which
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divides a given line in a given 



ratio. 













Distance between two points; 



Gradient of a line; 







Equation of a line: 







(i) Intercept form; 






(ii) Gradient form; 






Conditions for parallel and 




perpendicular lines. 






Calculate the acute angle 




between two intersecting lines 



e.g. if m1 and m2 are the 




gradients of two intersecting 



lines, then tan θ = 
=> = 
# 
. If 



= 
> 
= 


























# 




m1m2 = 1, then the lines are 



perpendicular. 









*The distance from an external 



point P(x1, y1) to a given line 



ax + by + c using the formula 



> 
> 









d =  





. 


















√# 
# 




















(ii) Conic Sections 














Loci of variable points which 



move under given conditions 



Equation of a circle: 






(i) Equation in terms of 






centre, (a, b), and 







radius, r, 










(x – a)2+(y – b)2 = r2; 



(ii) The general form: 




x2+y2+2gx+2fy+c = 0, where 



(g, –f) is the centre and radius, 





. 







r = √& + − – 







Tangents and normals to circles 



Equations of parabola in 





rectangular Cartesian 






coordinates (y2 = 4ax, include 



parametric equations (at2, at)). 



Finding the equation of a 




tangent and normal to a 




parabola at a given point. 




*Sketch graphs of given 





parabola and find the equation 

(15)Differentiation 

of the axis of symmetry. 



(i) The idea of a limit 














(i) Intuitive treatment of limit. 
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(ii) The derivative of a function
(iii)Differentiation of polynomials
(iv)Differentiation of trigonometric Functions
(v)Product and quotient rules. Differentiation of implicit
functions such as ax2 + by2 = c
**(vi) Differentiation of Transcendental Functions
(vii)Second order derivatives and Rates of change and small changes (∆x), Concept of Maxima and Minima
(16)Integration
(i) Indefinite Integral
Relate to the gradient of a curve. e.g. f1(x) =
limC→* (C ) ().
C
(ii)Its meaning and its determination from first
principles (simple cases only).
e.g. axn + b, n ≤ 3, (n ∈I )
e.g. axm – bxm – 1+ …+k, where m Є I , k is a constant.
e.g. sin x, y = a sin x ± b cos x. Where a, b are constants.
including polynomials of the form (a + bxn)m.
e.g. y = eax, y = log 3x, y = ln x
(i)The equation of a tangent to a curve at a point.
(ii)Restrict turning points to maxima and minima.
(iii)Include curve sketching (up to cubic functions) and linear
kinematics.
(i) Integration of polynomials of the form axn; n ≠ 1. i.e.
∫xn dx = 
E> 
+ c, n ≠ 1. 

( 



(ii)Integration of sum and
difference of polynomials. e.g. ∫(4x3+3x26x+5) dx
**(iii)Integration of polynomials of the form axn; n = 1.
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II. Statistics and
Probability
(17)Statistics
(ii)Definite Integral
(iii)Applications of the Definite Integral
(i)Tabulation and Graphical representation of data
(ii)Measures of location
(iii) Measures of Dispersion
i.e. ∫ x 1 dx = ln x
Simple problems on integration by substitution.
Integration of simple trigonometric functions of the
form F sin G .G.
(i) Plane areas and Rate of Change. Include linear
kinematics.
Relate to the area under a curve.
(ii)Volume of solid of revolution
(iii)Approximation restricted to trapezium rule.
Frequency tables. Cumulative frequency tables. Histogram (including unequal class intervals).
Cumulative frequency curve (Ogive) for grouped data.
Central tendency: mean, median, mode, quartiles and percentiles.
Mode and modal group for grouped data from a histogram. Median from grouped data. Mean for grouped data (use of an assumed mean required).
Determination of:
(i) Range, Inter Quartile and Semi interquartilerange
from an Ogive.
(ii)Mean deviation, variance and standard deviation for grouped and ungrouped
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(iv)Correlation
(18)Probability
(i)Meaning of probability.
(ii)Relative frequency.
(iii)Calculation of Probability using simple sample spaces.
(iv)Addition and multiplication of probabilities.
(v)Probability distributions.
III. Vectors and
Mechanics
(19)Vectors
(i)Definitions of scalar and vector Quantities.
(ii)Representation of Vectors.
data. Using an assumed mean or true mean.
Scatter diagrams, use of line of best fit to predict one variable from another, meaning of correlation; positive, negative and zero correlations,. Spearman’s Rank coefficient. Use data without ties. *Equation of line of best fit by least square method. (Line of regression of y on x).
Tossing 2 dice once; drawing from a box with or without replacement.
Equally likely events, mutually exclusive, independent and conditional events.
Include the probability of an event considered as the probability of a set.
(i)Binomial distributionP(x=r)=nCrprqnr ,where
Probability of success = p, Probability of failure = q,
p + q = 1 and n is the number of trials. Simple problems only.
**(ii) Poisson distribution
P(x) = 
HIJK 
, where λ = np, 

! 



n is large and p is small.
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(iii)Algebra of Vectors.
(iv)Commutative, Associative and Distributive Properties.
(v)Unit vectors.
(vi) Position Vectors.
(vii)Resolution and Composition of Vectors.
(viii) Scalar (dot) product and its
Representation of vector ‘)in the form ai + bj.
Addition and subtraction, multiplication of vectors by vectors, scalars and equation of vectors. Triangle, Parallelogram and polygon Laws.
Illustrate through diagram, Illustrate by solving problems in elementary plane geometry e.g concurrency of medians and diagonals.
The notation:
i for the unit vector 1 and 0
j for the unit vector 0 1 along the x and y axes
respectively. Calculation of unit vector (â) along a i.e. â =  .
Position vector of A relative to
Ois PPPPPQNO.
Position vector of the midpoint of a line segment. Relate to coordinates of midpoint of a line segment.
*Position vector of a point that divides a line segment internally in the ratio (λ : μ).
Applying triangle, parallelogram and polygon laws to composition of forces acting at a point. e.g. find the resultant of two forces (12N, 030o) and (8N, 100o) acting at a point.
*Find the resultant of vectors by scale drawing.
Finding angle between two vectors.
Using the dot product to application.
**(ix) Vector (cross) product and its application.
(20)Statics
(i)Definition of a force.
(ii)Representation of forces.
(iii)Composition and resolution of coplanar forces acting at a
point.
(iv)Composition and resolution of general coplanar forces on
rigid bodies.
(v)Equilibrium of Bodies.
(vi)Determination of Resultant.
(vii)Moments of forces.
(viii)Friction.
(21)Dynamics
establish such trigonometric formulae as
(i) Cos (a ± b) =
cos a cos b ∓ sin a sin b
(ii) sin (a ± b)=
sin a cos b ± sin b cosa
(iii)c2 = a2 + b2 – 2ab cos C
(iv)STU V =STU W = STU Y.
Apply to simple problems e.g. suspension of particles by strings.
Resultant of forces, Lami’s theorem
Using the principles of moments to solve related problems.
Distinction between smooth and rough planes. Determination of the coefficient of friction.
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(i)The concepts of motion
(ii)Equations of Motion
(iii)The impulse and momentum equations:
**(iv) Projectiles.
The definitions of displacement, velocity, acceleration and speed.
Composition of velocities and accelerations.
Rectilinear motion. Newton’s laws of motion. Application of Newton’s Laws Motion along inclined planes (resolving a force upon a plane into normal and frictional forces).
Motion under gravity (ignore air resistance).
Application of the equations of
motions: V = u + at, S = ut + ½ at 2; v2 = u2 + 2as.
Conservation of Linear Momentum(exclude coefficient of restitution).
Distinguish between momentum and impulse.
Objects projected at an angle to the horizontal.
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
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1000 cubic centimeters (cm3) = 1 litre (l)
(4 ) Mass
1000milligrammes (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 the examination room. Where the degree of accuracy is not specified in a question, the degree of accuracy expected will be that obtainable from the mathematical tables.
(2)Use of calculators
The use of nonprogrammable, silent and cordless calculators is allowed. The calculators must, however not have a paper print out nor be capable of receiving/sending 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 2 (1) and (2) above, it should be noted that some questions may prohibit the use of tables and/or calculators.