Further Mathematics Syllabus For WAEC 2024

Further Mathematic syllabus is only for those that are doing further mathematics and this is because further maths is one of the elective subjects of WAEC, so you can decide to do it or register it or leave it, it is not compulsory.

This content will show you the full syllabus which contains the topics and subtopics you are expected to read or cover up before the day of your WAEC examination.

Further Mathematics Syllabus For WAEC

Below are all the topics WAEC have approved for you to study;

AREAS COMMON TO BOTH ALTERNATIVES:

1. Circular Measure

Topic Content Notes:

• Lengths of Arcs of Circles and Radians:
  • Lengths of arcs, perimeters of sectors, and segments measured in radians.

2. Trigonometry

Topic Content Notes:

• Sine, Cosine, and Tangent of Angles:
  • For $0°\le \theta \le 360°$.
• Trigonometric Ratios of Angles 30°, 45°, and 60°:
  • Identify without the use of tables.
• Heights and Distances
• Angles of Elevation and Depression
• Bearings, Positive and Negative Angles:
  • Simple cases only.
• Compound and Multiple Angles:
  • Identities and solution of trigonometric ratios.
• Graphical Solution of Simple Trigonometric Equations:
  • $a\cos x+b\sin x=c$.
• Solution of Triangles:
  • Include the notion of radians and trigonometric ratios of negative angles.

3. Indices, Logarithms, and Surds

Topic Content Notes:

• Indices:
  • Elementary theory of indices. Meaning of a0a^0a0, a−na^{-n}a−n, ana^nan.
• Logarithms:
  • Elementary theory of logarithms.
  • Calculations involving multiplication, division, power, and nth roots.
  • Applications such as reducing a relation y=axby = a x^by=axb (where $a,b$ are constants) to a linear form log⁡10y=blog⁡10x+log⁡10a \log_{10} y = b \log_{10} x + \log_{10} alog10​y=blog10​x+log10​a.
• Surds:
  • Surds of the form a\sqrt{a}a​, aaa \sqrt{a}aa​, and abn\frac{a}{b \sqrt{n}}bn​a​.
  • Rationalization of the denominator.
• Sequences:
  • Finite and infinite sequences.
  • Linear and exponential sequences.
  • Un=U1+(n−1)dU_n = U_1 + (n – 1)dUn​=U1​+(n−1)d where $d$ is the common difference.
  • Sn=n(U1+Un)2S_n = \frac{n (U_1 + U_n)}{2}Sn​=2n(U1​+Un​)​.
  • Un=U1rn−1U_n = U_1 r^{n-1}Un​=U1​rn−1 where $r$ is the common ratio.
  • Sn=U11−rn1−rS_n = U_1 \frac{1 – r^n}{1 – r}Sn​=U1​1−r1−rn​ for $r<1$ or Sn=U1rn−1r−1S_n = U_1 \frac{r^n – 1}{r – 1}Sn​=U1​r−1rn−1​ for $r>1$.
• Use of the Binomial Theorem:
  • Proof not required.
  • Expansion of (a+b)n(a + b)^n(a+b)n for a positive integral index.
  • Use of (1+x)n≈1+nx(1 + x)^n \approx 1 + nx(1+x)n≈1+nx for any rational $n$, where $x$ is sufficiently small.

4. Algebraic Equations

Topic Content Notes:

• Factors and Factorization:
  • Solution of quadratic equations using:
    • Completing the square.
    • Quadratic formula.
  • Symmetric properties of the equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.
  • Sum and product of roots.
  • Solution of two simultaneous equations where one is linear and the other is quadratic.
  • Graphical and analytical methods permissible.

5. Polynomials

Topic Content Notes:

• Addition, Subtraction, and Multiplication of Polynomials:
  • Factor and remainder theorems.
  • Zeros of a polynomial function.
  • Graphs of polynomial functions of degree $n\le 3$.
  • Division of a polynomial of degree not greater than 4 by a polynomial of lower degree.

6. Rational Functions and Partial Fractions

Topic Content Notes:

• Basic Operations:
  • Zeros, domain, and range.
  • Resolution of rational functions into partial fractions.
  • Rational functions of the form F(x)G(x)\frac{F(x)}{G(x)}G(x)F(x)​, where $G(x)$ and $F(x)$ are polynomials, $G(x)$ must be factorable into linear and quadratic factors. Degree of numerator less than that of denominator which is less than or equal to 4.

7. Linear Inequalities

Topic Content Notes:

• Graphical and Analytical Solutions:
  • Solution of simultaneous linear inequalities in two variables.
  • Quadratic inequalities.

8. Logic

Topic Content Notes:

• Truth Tables:
  • Using $P\text{ or }Q$, $P\text{ and }Q$, $P⟹Q$, $Q⟹P$.
• Rules of Syntax:
  • True or false statements.
  • Rule of logic applied to arguments, implications, and deductions.
  • Use of truth tables.

9. Co-ordinate Geometry: Straight Line

Topic Content Notes:

• Distance Between Two Points:
• Midpoint of a Line Segment:
• Gradient of a Line:
  • Gradient as the ratio of vertical change to horizontal change.
• Conditions for Parallel and Perpendicular Lines.

10. Differentiation

Topic Content Notes:

• Idea of a Limit:
  • Intuitive treatment of the limit, relating to the gradient of a curve.
• Derivative of a Function:
  • Meaning and determination from first principles in simple cases only.
  • Differentiation of polynomials.
  • Product and quotient rules.
  • Differentiation of implicit functions.
• Second Derivatives and Rates of Change:
  • Concept of maxima and minima.
  • Curve sketching (up to cubic functions) and linear kinematics.

11. Integration

Topic Content Notes:

• Indefinite Integral:
  • Exclude $n=-1$.
  • Integration of sums and differences of polynomials.
• Definite Integral:
  • Applications to plane areas and rates of change.
  • Approximation restricted to the trapezium rule.
• Volume of Solid of Revolution.

12. Sets

Topic Content Notes:

• Idea of a Set Defined by a Property:
  • Set notations and their meanings.
  • Disjoint sets, universal set, and complement of a set.
  • Venn diagrams and their use in solving problems.
  • Commutative, associative, and distributive properties over union and intersection.

13. Mappings and Functions

Topic Content Notes:

• Domain and Co-domain of a Function:
  • One-to-one, onto, identity, and constant mapping.
  • Inverse of a function.
  • Composition of functions.

14. Matrices and Linear Transformation

Topic Content Notes:

• Matrix Representation:
  • Equal matrices.
  • Addition, subtraction, and multiplication of matrices.
  • Inverse of a 2×2 matrix.
  • Special linear transformations: reflection, rotation.

15. Determinants and Binary Operations

Topic Content Notes:

• Evaluation of Determinants:
  • Application to areas of triangles and quadrilaterals.
  • Solution of simultaneous linear equations.
• Binary Operations:
  • Closure, commutativity, associativity, and distributivity.

STATISTICS AND PROBABILITY

1. Graphical Representation of Data

Topic Content Notes:

• Frequency Tables:
• Cumulative Frequency Tables:
• Histogram:
  • Including unequal class intervals.
• Frequency Curves and Ogives:
  • For grouped data of equal and unequal class intervals.

2. Measures of Central Tendency and Location

Topic Content Notes:

• Mean, Median, Mode, Quartiles, and Percentiles:
  • Mode and modal group for grouped data from a histogram.
  • Median from grouped data and from ogives.
  • Mean for grouped data, use of an assumed mean required.

3. Measures of Dispersion

Topic Content Notes:

• Range, Interquartile Range, Variance, and Standard Deviation:
  • For grouped and ungrouped data using an assumed mean or true mean.

4. Correlation

Topic Content Notes:

• Scatter Diagrams:
  • Meaning of correlation: positive, negative, and zero correlations from scatter diagrams.
  • Use data without ties.
• Line of Fit:
  • Use of line of best fit to predict one variable from another.

5. Probability

Topic Content Notes:

• Meaning and Calculation of Probability:
  • Relative frequency, use of simple sample spaces.
  • Equally likely events, mutually exclusive events.
  • Addition and multiplication of probabilities.
  • Binomial probability distribution.

6. Permutations and Combinations

Topic Content Notes:

• Simple Cases of Arrangements:
  • Simple problems on permutations and combinations.

VECTORS AND MECHANICS

1. Vectors

Topic Content Notes:

• Definitions and Representations:
  • Scalar and vector quantities.
  • Addition, subtraction, and multiplication of vectors.
  • Position and free vectors.
• Parallelogram Law and Triangle Law:
  • Use of coordinates.
  • Scalar product.

2. Forces

Topic Content Notes:

• Resultant of Forces Acting at a Point:
  • Equilibrium.
• Moments and Couples:
  • Simple examples.
  • Applications of moments and couples.
• Equilibrium of a Rigid Body:
  • Force systems.
  • Friction.
  • Simple cases of frictional forces acting on a particle.

3. Kinematics

Topic Content Notes:

• Rectilinear Motion with Uniform Acceleration:
  • s=ut+12at2s = ut + \frac{1}{2}at^2s=ut+21​at2, $v=u+at$, v2=u2+2asv^2 = u^2 + 2asv2=u2+2as.
• Motion of Projectiles:
  • Simple cases of projectile motion.
  • Time of flight, range, and maximum height.
  • Uniform circular motion.

4. Momentum and Energy

Topic Content Notes:

• Linear Momentum:
  • Law of conservation of linear momentum.
• Kinetic and Potential Energy:
  • Simple problems on conservation of mechanical energy.
  • Work done by a force.
  • Power.

5. Work, Energy, and Power

Topic Content Notes:

• Work Done by a Force:
  • $W=Fd\cos \theta$ where $\theta$ is the angle between the force and displacement vectors.
  • Graphical interpretation of work done from force-displacement graphs.
• Energy:
  • Potential energy (gravitational and elastic).
  • Kinetic energy (12mv2\frac{1}{2}mv^221​mv2).
  • Conservation of energy in mechanical systems.
• Power:
  • P=WtP = \frac{W}{t}P=tW​ and $P=Fv$ for constant force and velocity.
  • Efficiency of machines: Efficiency=Useful power outputTotal power input×100%\text{Efficiency} = \frac{\text{Useful power output}}{\text{Total power input}} \times 100\%Efficiency=Total power inputUseful power output​×100%.

AREAS EXCLUSIVE TO DIFFERENT ALTERNATIVES:

ALTERNATIVE A

1. Complex Numbers

Topic Content Notes:

• Introduction to Complex Numbers:
  • Definition and basic operations (addition, subtraction, multiplication, and division).
  • Complex conjugate and modulus of a complex number.
  • Polar form of a complex number.
• De Moivre’s Theorem:
  • Applications to finding powers and roots of complex numbers.
• Solving Polynomial Equations:
  • Solutions involving complex roots.

2. Group Theory

Topic Content Notes:

• Definition and Examples:
  • Groups, subgroups, cyclic groups.
• Properties of Groups:
  • Closure, associativity, identity, and inverses.
• Order of an Element and a Group:
  • Lagrange’s theorem.
• Permutations and Symmetric Groups:
  • Cycle notation for permutations.

ALTERNATIVE B

1. Calculus (Further Differentiation and Integration)

Topic Content Notes:

• Further Differentiation:
  • Differentiation of trigonometric, exponential, and logarithmic functions.
  • Differentiation of inverse trigonometric functions.
  • Higher-order derivatives.
• Further Integration:
  • Integration by parts.
  • Integration of rational functions using partial fractions.
  • Improper integrals.

2. Numerical Methods

Topic Content Notes:

• Numerical Solutions to Equations:
  • Bisection method, Newton-Raphson method.
• Numerical Integration:
  • Trapezoidal rule, Simpson’s rule.

3. Advanced Statistics

Topic Content Notes:

• Probability Distributions:
  • Binomial, Poisson, and normal distributions.
• Statistical Inference:
  • Confidence intervals, hypothesis testing.
  • Simple linear regression and correlation analysis.
• Sampling Methods:
  • Random, stratified, and systematic sampling.

ADDITIONAL STUDY GUIDANCE

To excel in the WASSCE Mathematics (Core) exam, focus on the following strategies:

1. Understand Fundamental Concepts:
   • Ensure a solid grasp of the basics before moving on to more advanced topics.
2. Practice Problem-Solving:
   • Regularly practice solving past exam papers and sample questions.
3. Use Visual Aids:
   • Employ diagrams and graphs to understand and solve problems, especially in geometry and functions.
4. Master Calculator Skills:
   • Become proficient with your calculator to efficiently perform calculations during the exam.
5. Review and Revise:
   • Regularly review and revise topics to reinforce learning and retention.
6. Seek Clarification:
   • Don’t hesitate to ask for help from teachers or peers when encountering difficult topics.
7. Time Management:
   • Practice completing questions within the allotted time to improve speed and accuracy during the exam.

Frequently Asked Questions

What topics are common to both Mathematics alternatives?

Common topics include trigonometry, indices, logarithms, surds, algebraic equations, polynomials, rational functions, linear inequalities, and calculus.

What additional topics are covered in Alternative A?

Alternative A includes complex numbers and group theory.

What additional topics are covered in Alternative B?

Alternative B includes further differentiation and integration, numerical methods, and advanced statistics.

Conclusion

To excel in WASSCE Mathematics, understand fundamental concepts, practice problem-solving, use visual aids, master calculator skills, review regularly, seek help when needed, and manage time efficiently during exams.