** WAEC Syllabus For Further Mathematics 2021/2022 PDF download.** It is obvious that many students are preparing for Waec Further Mathematics Syllabus 2021 without the syllabus. I use this opportunity to inform you that your preparation is not effective Without Waec Further Mathematics Syllabus 2021 syllabus.

This is an outline of the main points of a discourse, the subjects, the contents of WAEC Further mathematics Syllabus 2021/2022 curriculum Given out by The West African Examination Council for all candidates sitting for the 2021 Waec examination.

Table of Contents

## WAEC Further Maths Syllabus 2021

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### WAEC Syllabus For Further Mathematics 2021/2022

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WAEC SYLLABUS 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.

(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), ∪, ∩, { },∉, ∈, ⊂, ⊆,

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

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Surds of the form √

, a√ and a+b√ 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 √ , √ √ , √ √ √

(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 p ν q, p ∧ q, p ⇒ q

Use of Truth tables to deduce conclusions of compound statements. Include negation.

(5) Functions

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.

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 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.

(6) Polynomial Functions

(i) Linear Functions, Equations and Inequality

(ii) Quadratic Functions, Equations and Inequalities

(ii) Cubic Functions and Equations

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 3×2 + 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).

(7) Rational Functions

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(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

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

(8) Indices and Logarithmic Functions

(i) Indices

(ii) Logarithms

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;

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

(ii) Simple cases of selection of objects

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. n pr = n! (n-r)! n Cr= n! r!(n-r)!

(10) Binomial Theorem

(12) Matrices and Linear Transformation

(i) Matrices

(ii) Determinants

(iii) Inverse of 2 x 2 Matrices

(iv) Linear Transformation

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) 1 0 0 −1 reflection in the x – axis (ii) −1 0 0 1 reflection in the y – axis (iii) 0 1 1 0 reflection in the line y = x (iv) cos5 −sin5 sin5 cos5 for anticlockwise rotation through θ about the origin. (v) -8925 sin25 9;25 −-8925 , 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

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