WAEC Syllabus for Further Mathematics/Mathematics (Elective)

(1)  Sets

(2)  Surds

(3)  Binary Operations

(4)  Logical Reasoning

(5)  Functions

(6)  Polynomial Functions

(7)  Rational Functions

(8)  Indices and Logarithmic Functions

(9)  Permutation

And Combinations.

(10) Binomial          Theorem

(11) Sequences

and Series

(12)Matrices and Linear Transformation

(13)Trigonometry

(14)Co-ordinate

Geometry

(15)Differentiation

(16)Integration

II. Statistics and

Probability

(17)Statistics

(18)Probability

III. Vectors and        Mechanics

(19)Vectors

(20)Statics

(21)Dynamics

(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.

Surds of the form  , a and a+b where a is rational, b is a positive integer and n is not a perfect square.

Properties:

Closure, Commutativity, Associativity and Distributivity, Identity elements and inverses.

(i) Rule of syntax:

true or false statements,

rule of logic applied to arguments, implications and deductions.

(ii) The truth table

(i) Domain and co-domain of a

function.

(ii) One-to-one, onto, identity and       constant mapping;

(iii) Inverse of a function.

(iv) Composite of functions.

(i) Linear Functions, Equations and

Inequality

(ii) Quadratic Functions, Equations      and Inequalities

(ii) Cubic Functions and Equations

(i) Rational functions of the form

Q(x) =  ,g(x) ¹ 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

(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.)

*(v) Recurrence Series

(i) Matrices

(ii) Determinants

(iii) Inverse of 2 x 2 Matrices

(iv) Linear Transformation

(i) Trigonometric Ratios and Rules

(ii)  Compound and Multiple       Angles.

(iii) Trigonometric Functions and       Equations

(i) Straight Lines

(ii) Conic Sections

(i) The idea of a limit

(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

(i) Indefinite Integral

(ii) Definite Integral

(iii) Applications of the Definite

Integral

(i)  Tabulation and Graphical

representation of data

(ii) Measures of location

(iii) Measures of Dispersion

(iv)Correlation

(i) Meaning of probability.

(ii) Relative frequency.

(iii) Calculation of Probability using        simple sample spaces.

(iv) Addition and multiplication of        probabilities.

(v) Probability distributions.

(i) Definitions of scalar and vector

Quantities.

(ii) Representation of Vectors.

(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

application.

**(ix) Vector (cross) product and

its application.

(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.

(i) The concepts of motion

(ii) Equations of Motion

(iii) The impulse and momentum

equations:

**(iv) Projectiles.

(x : x is real), È, Ç, { },Ï, Î, , ,

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

All the four operations on surds

Rationalising the denominator of surds such as  ,  ,  .

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  q, p Ù q, p Þ 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 ÎR.

Graphical representation of a function ; Image and the range.

Determination of the inverse of a one-to-one function e.g. If

f: x →sx + , the inverse relation f^-1: 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

Graphical representation of linear inequalities in two variables. Application to Linear Programming.

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  and β, form the equation whose roots are and . Solving quadratic inequalities.

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).

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 a^1/n).

Solve equations involving logarithms (including change of base).

Reduction of a relation such as     y = ax^b, (a,b are constants) to a linear form:

log₁₀y = b log₁₀x+log₁₀a.

Consider other examples such as

log ab^x = 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 = ax^b. Estimating the values of the constants a and b from the graph

Knowledge of arrangement and selection is expected. The notations: ⁿC_r,  and ⁿP_r 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.

ⁿp_r =  n!

(n-r)!

ⁿC_r=    n!

r!(n-r)!

Use of the binomial theorem for positive integral index only.

Proof of the theorem not required.

e.g. (i) u₁, u₂,…, u_n.

(ii) u₁, u₂,….

Recognizing the pattern of a sequence. e.g.

(i)  U_n = U₁ + (n-1)d, where

d is the common difference.

(ii) U_n= U₁ r^n-1 where r is the

common ratio.

(i) U₁ + U₂ + U₃ + … + U_n

(ii)U₁ + U₂ + U₃ + ….

(i) S_n = (U₁+U_n)

(ii) S_n = [2a + (n – 1)d]

(iii) S_n= U₁(1-rⁿ) , r<1

l – r

(iv)    S_n=U₁(rⁿ-1)  , 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

A^-1 =

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)    reflection in the           x – axis

(ii)   reflection in the                y – axis

(iii)    reflection in the line       y = x

(iv)  for anti-clockwise rotation through θ about the origin.

(v)   , 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

Sine, Cosine and Tangent of general angles (0^o≤θ≤360^o).

Identify trigonometric ratios of angles 30^O, 45^O, 60^o 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).

Use of compound angles in simple identities and solution of trigonometric ratios e.g. finding sin 75^o, cos 150^oetc, finding tan 45^o 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).

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 0^o ≤ x ≤ 360^o.

*Express f(x) = asin x + bcos x in the form Rcos (x ) or Rsin (x ) for 0^o ≤  ≤ 90^oand use the result to calculate the minimum and maximum points of a given functions.

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 = . 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 = ||.

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 of the axis of symmetry.

(i) Intuitive treatment of limit.

Relate to the gradient of

a curve. e.g. f¹(x) = .

(ii) Its meaning and its

determination from first

principles (simple cases        only).

e.g. axⁿ + b, n ≤ 3, (n ÎI )

e.g. ax^m – bx^{m – 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 = e^ax, 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.

òxⁿ dx =  + c, n ≠ -1.

(ii) Integration of sum and

difference of polynomials.

e.g. ∫(4x³+3x²-6x+5) dx

**(iii)Integration of polynomials

of the form axⁿ; n = -1.

i.e. ò x ^-1 dx = ln x

Simple problems on integration by substitution.

Integration of simple trigonometric functions of the form .

(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 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.

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 distribution

P(x=r)=ⁿC_rp^rq^n-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) =  , where λ = np,

n is large and p is small.

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

con-currency 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 O is .

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).

Applying triangle, parallelogram and polygon laws to composition of forces acting at a point. e.g. find the resultant of two forces (12N, 030^o) and (8N, 100^o) acting at a point.

*Find the resultant of vectors by scale drawing.

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)  =.

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.

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.