FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)
WAEC SYLLABUS ON FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)
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 multiple-choice objective questions, covering the entire
syllabus. Candidates will be required to answer all questions in 1½ hours 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 2½ hours for 100 marksSection 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
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 |
CONTENTS |
NOTES |
|---|---|---|
I. Pure Mathematics(1) Sets |
(i) Idea of a set defined by a
property, Set notations and
their
meanings. (ii) Disjoint sets, Universal set and complement of set (iii) Venn diagrams, Use of sets And Venn diagrams to solve problems. (iv) Commutative and Associative laws, Distributive properties over union and intersection. |
(x : x is real), ꓴ, ꓵ, { }, ∉, ∈, ⊂, ⊆, U (universal set) and A’ (Complement of set A). More challenging problems involving union, intersection, the universal set, subset and complement of set. Three set problems. Use of De Morgan’s laws to solve related problems |
| (2) Surds | Surds of the he form a/√a , a√𝑏 and a+b√𝑛 where a is rational, b is a positive integer and n is not a perfect square. | All the four operations on
surds
Rationalising the denominator
of surds such as a/√a, (a+√b)/(c-√d), (a+√b)/(√c-√d) |
| (3) Binary Operations | Properties: Closure, Commutativity, Associativity and Distributivity, Identity elements and inverses. | Use of properties to solve related problems. |
| (4) Logical Reasoning |
(i) Rule of syntax:
true or false statements,
rule of logic applied to
arguments, implications and
deductions. (ii) The truth table |
Using logical reasoning to
determine the validity of
compound statements
involving implications and
connectivities. Include use of
symbols: ~p, pꓥq, pꓦq, p⇒q Use of Truth tables to deduce conclusions of compound statements. Include negation. |
| (5) Functions | (i) Domain and co-domain of a function. | The notation e.g. f : x → 3x+4; g : x → x2 ; where x ∈ R . |
| (ii) One-to-one, onto, identity and constant mapping; | Graphical representation of a function ; Image and the range. | |
| (iii) Inverse of a function. | Determination of the inverse
of a one-to-one function e.g. If f: x →sx + 4/3, the inverse relation f-1: x → (1/3)x - 4/9 is also a function. |
|
| (iv) Composite of functions. | Notation: fog(x) =f(g(x)) Restrict to simple algebraic functions only. | |
| (6) Polynomial Functions | (i) Linear Functions, Equations and Inequality | 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; y/a + x/b = k. Parallel and Perpendicular lines. Linear Inequalities e.g. 2x + 5y ≤ 1, x + 3y ≥ 3 Graphical representation of linear inequalities in two variables. Application to Linear Programming. |
|
(ii) Quadratic Functions, Equations and Inequalities |
Recognition and sketching
graphs of quadratic functions
e.g. f: x → ax2 +bx + c, where a, b and c Є R. Identification of vertex, axis of symmetry, maximum and minimum, increasing and decreasing parts of a parabola. Include values of x for which f(x) >0 or f(x) < 0. Solution of simultaneous equations: one linear and one quadratic. Method of completing the squares for solving quadratic equations. Express f(x) = ax2 + bx + c in the form f(x) = a(x + d)2 + k, where k is the maximum or minimum value. Roots of quadratic equations – equal roots (b2 - 4ac = 0), real and unequal roots (b2 – 4ac > 0), imaginary roots (b2 – 4ac < 0); sum and product of roots of a quadratic equation e.g. if the roots of the equation 3x2 + 5x + 2 = 0 are 𝛼 and β, form the equation whose roots are 1/𝛼 and 1/β Solving quadratic inequalities. |
|
| (iii) Cubic Functions and Equations | Recognition of cubic functions
e.g. f: x → ax3 + bx2 + cx + d. Drawing graphs of cubic functions for a given range. Factorization of cubic expressions and solution of cubic equations. Factorization of a3 ± b3. Basic operations on polynomials, the remainder and factor theorems i.e. the 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). |
|
| (7) Rational Functions |
(i) Rational functions of the form
Q(x) = f(x)/g(x) ,g(x) ≠ 0. where g(x) and f(x) are polynomials. e.g. f:x → (ax + b)/(px2 + qx + r) |
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). |
| (ii) Resolution of rational functions into partial fractions. | The four basic operations. Zeros, domain and range, sketching not required. |
|
| (8) Indices and Logarithmic Functions | (i) Indices | Laws of indices.
Application of the laws of
indices to evaluating products,
quotients, powers and nth
root. Solve equations involving indices. |
| (ii) Logarithms | Laws of Logarithms. Application of logarithms in calculations involving product, quotients, power (log an), nth roots (log √𝑎, 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; 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 |
|
| (9) Permutation And Combinations. | (i) Simple cases of arrangements | Knowledge of arrangement
and selection is expected. The notations: nCr, (nr) 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!/ (n-r)! nCr= n!/ r!(n-r)! |
| (ii) Simple cases of selection of objects. | ||
| (10) Binomial Theorem |
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 |
Use of the binomial theorem
for positive integral index only. Proof of the theorem not required. |
| (11) Sequences and Series | (i) Finite and Infinite sequences. | e.g. (i) U1, U2,…, Un. (ii) U1, U2,…. |
|
(ii) Linear sequence/Arithmetic
Progression (A.P.) and Exponential sequence/Geometric Progression (G.P.) |
Recognizing the pattern of a
sequence. e.g.
(i) Un = U1 + (n-1)d, where
d is the common difference.
(ii) Un= U1rn-1 where r is the common ratio. |
|
| (iii) Finite and Infinite series. | ||
|
(iv) Linear series (sum of A.P.) and |
(i) U1 + U2 + U3 + … + Un (ii)U1 + U2 + U3 + …. |
|
| exponential series (sum of G.P.) | (i) Sn = n/2(U1 + Un) (ii) Sn = 1/2 [2a + (n – 1)d] (iii) Sn = U1 (1-rn)/(1 - r), for r < 1 (iv) Sn = U1(rn - 1)/r – 1, for r > l. (v) Sum to infinity (S) = a/(1 - r) for r < 1 |
|
| *(v) Recurrence Series | Generating the terms of a recurrence series and finding an explicit formula for the sequence e.g. 0.9999 = 9/10 + 9/102 + 9/103 + 9/104 + .... | |
| (12) Matrices and Linear Transformation | (i) Matrices | 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) |
| (ii) Determinants | Evaluation of determinants of
2 x 2 matrices. **Evaluation of determinants of 3 x 3 matrices. |
|
| (iii) Inverse of 2 x 2 Matrices | Application of determinants to solution of simultaneous linear equations. | |
| (iv) Linear Transformation | e.g. If A = (𝑎𝑐 𝑑𝑏) , then A-1 = 1/𝑎𝑐 - 𝑑𝑏 (𝑑-𝑐 -𝑏𝑎) Finding the images of points under given linear transformation 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) (10 0-1) reflection in the x - axis (ii) (-10 01) reflection in the y - axis (iii) (01 10) reflection in the line y = x (iv) (cos θsin θ -sin θcos θ ) for anticlockwise rotation through θ about the origin. (v) (cos 2θsin 2θ sin 2θ-cos 2θ ) the general matrix for reflection in a line through the origin making an angle θ with the positive x-axis. *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, bearing (negative and positive angles) including use of sine and cosine rules, etc, Simple cases only. sin (A ± B), cos (A ± B), tan(A ± B). |
| (ii) Compound and Multiple Angles. | Use of compound angles in
simple identities and solution
of trigonometric ratios e.g. finding sin 75o, cos 150o etc, 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 ± 𝛼) or Rsin (x ± 𝛼) for 0o ≤ 𝛼 ≤ 90o and use the result to calculate the minimum and maximum points of a given functions. |
|
| (14) Co-ordinate Geometry | (i) Straight Lines | Mid-point of a line segment
Coordinates of points which
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 = (m1 - m2)/(1 + m1m2). 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 = |(ax1 + by1 + c)/√(a2 + b2) |. 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 = √(𝑎2 + 𝑏2 - 𝑐). Tangents and normals to circles |
| (ii) Conic Sections | 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 of the axis of symmetry. |
|
| (15) Differentiation | (i) The idea of a limit | (i) Intuitive treatment of limit.
Relate to the gradient of
a curve. e.g. f1x = limh→o{[f(x + h) - f(x)]/h} |
| (iii) Differentiation of polynomials | (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 ∈ I , k is a constant. |
|
| (iv) Differentiation of trigonometric Functions | e.g. sin x, y = a sin x ± b cos x. Where a, b are constants. | |
| (v) Product and quotient rules. Differentiation of implicit functions such as ax2 + by2 = c | including polynomials of the form (a + bxn)m. | |
| **(vi) Differentiation of Transcendental Functions | e.g. y = eax, y = log 3x, y = ln x | |
|
(vii) Second order derivatives and
Rates of change and small
changes (Δx), Concept of Maxima and Minima |
(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. |
|
| (16) Integration | (i) Indefinite Integral | (i) Integration of polynomials
of
the form axn; n ≠ -1. i.e. ∫xn dx = (xn + 1)/(n + 1) + c, n ≠ -1. |
| (ii) Definite Integral | (ii) Integration of sum and
difference of polynomials. e.g. ∫(4x3+3x2-6x+5) dx |
|
| (iii) Applications of the Definite Integral | **(iii)Integration of
polynomials
of the form axn; n = -1. i.e. ∫x-1 dx = ln x Simple problems on integration by substitution. Integration of simple trigonometric functions of the form ∫ab sin𝑥 𝑑𝑥. (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. |
|
| II. Statistics and
Probability (17) Statistics |
(i) Tabulation and Graphical representation of data | Frequency tables. Cumulative frequency tables. Histogram (including unequal class intervals). Cumulative frequency curve (Ogive) for grouped data. |
| (ii) Measures of location | 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). |
|
| (iii) Measures of Dispersion | Determination of: (i) Range, Inter- Quartile and Semi inter-quartile range from an Ogive. (ii) Mean deviation, variance and standard deviation for grouped and ungrouped data. Using an assumed mean or true mean. |
|
| (iv)Correlation | 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). |
|
| (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. |
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. |
| III. Vectors and
Mechanics (19) Vectors |
(i) Definitions of scalar and vector
Quantities. |
(i) Binomial distribution
P(x=r)= nCrprqn-r , 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) = (e-λλx)/ x! , where λ = np, n is large and p is small. |
| (ii) Representation of Vectors. | Representation of vector (ab) in the form ai + bj. | |
| (iii) Algebra of Vectors. | Addition and subtraction,
multiplication of vectors by
vectors, scalars and equation
of
vectors. Triangle, Parallelogram and polygon Laws. |
|
| (iv) Commutative, Associative and Distributive Properties. | Illustrate through diagram,
Illustrate by solving problems
in
elementary plane geometry
e.g con-currency of medians and diagonals. |
|
| (v) Unit vectors. | The notation:
i for the unit vector (10) and j for the unit vector (01) along the x and y axes respectively. Calculation of unit vector (â) along a i.e. â = a/|a|. |
|
|
(vi) Position Vectors. |
Position vector of A relative to
O is OA→ Position vector of the midpoint of a line segment. Relate to coordinates of mid-point of a line segment. *Position vector of a point that divides a line segment internally in the ratio (λ : μ). |
|
| (vii) Resolution and Composition of Vectors. | 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. |
|
|
(viii) Scalar (dot) product and its
application. **(ix) Vector (cross) product and its application. |
Finding angle between two
vectors. Using the dot product to 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) (sin A)/a = (sin B)/b = (sinC)/c |
|
| (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. |
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. |
| (21) Dynamics | (i) The concepts of motion | The definitions of displacement, velocity, acceleration and speed. Composition of velocities and accelerations. |
| (ii) Equations of Motion | 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 at2; v2 = u2 + 2as. | |
| (iii) The impulse and momentum equations: | Conservation of Linear
Momentum(exclude coefficient
of restitution). Distinguish between momentum and impulse. |
|
| **(iv) Projectiles. | 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
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.