WASSCE / WAEC Further Maths Syllabus

 √.√√

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 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

(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;

(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!

(n-r)!

nCr= 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,…, un.(ii) u1, u2,….

Recognizing the pattern of a sequence. e.g.

(i)Un = U1 + (n-1)d, where d is the common difference.

(ii)Un= U1 rn-1 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]

*(v) Recurrence Series

(ii) Determinants

(iii)Inverse of 2 x 2 Matrices

(iv)Linear Transformation

(iii)Sn= U1(1-rn) , r<1

l – r

(iv)Sn=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.

Finding the images of points under given linear transformation

(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.

(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.

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 inter-quartilerange

from an Ogive.

(ii)Mean deviation, variance and standard deviation for grouped and ungrouped

(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)=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

n is large and p is small.

(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 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

Ois PPPPPQNO.

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, 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.

(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.