Waec Syllabus for Further Mathematics and Recommended TextBooks

See the original Waec Syllabus for Further Mathematics and Recommended TextBooks  for this year. Note that this syllabus is for both Internal May/June and external Nov/Dec GCE candidates.

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WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

233

AIMS OF THE SYLLABUS
The aims of the syllabus are to test candidates on:
(i) further conceptual and manipulative skills in Mathematics;
(ii) an intermediate course of study which bridges the gap between Elementary
Mathematics and Higher Mathematics;
(iii) aspects of mathematics that can meet the needs of potential Mathematicians,
Engineers, Scientists and other professionals.

EXAMINATION FORMAT
There will be two papers both of which must be taken.
PAPER 1: (Objective) – 1½ hours (50 marks)
PAPER 2: (Essay) – 2½ hours (100 marks)
PAPER 1 (50 marks) – This will contain forty multiple-choice questions, testing the
areas common to the two alternatives of the syllabus, made up
of twenty-four from Pure Mathematics, eight from Statistics and
Probability and eight from Vectors and Mechanics. Candidates
are expected to attempt all the questions.

PAPER 2 – This will contain two sections – A and B.
SECTION A (48 marks) – This will consist of eight compulsory questions that are
elementary in type, drawn from the areas common to both
alternatives as for Paper 1 with four questions drawn from Pure
Mathematics, two from Statistics and Probability and two from
Vectors and Mechanics.

SECTION B (52 marks) – This will consist of ten questions of greater length and difficulty
consisting of three parts as follows:

PART I (PURE MATHEMATICS) – There will be four questions with two drawn from the
common areas of the syllabus and one from each
alternatives X and Y.

PART II (STATISTICS AND PROBABILITY) – There will be three questions with two drawn
from common areas of the syllabus and one
from alternative X.

PART III (VECTORS AND MECHANICS) – There will be three questions with two drawn
from common areas of the syllabus and one from
alternative X.
Candidates will be expected to answer any four questions with at least one from each part.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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234

Electronic calculators of the silent, cordless and non-programmable type may be used in these papers.
Only the calculator should be used; supplementary material such as instruction leaflets, notes on
programming must in no circumstances be taken into the examination hall. Calculators with paper
type output must not be used. No allowance will be made for the failure of a calculator in the
examination.

A silent, cordless and non-programmable calculator is defined as follows:

(a) It must not have audio or noisy keys or be operated in such a way as to disturb other
candidates;

(b) It must have its own self-contained batteries (rechargeable or dry) and not always be
dependent on a mains supply;

(c) It must not have the facility for magnetic card input or plug-in modules of programme
instructions.

DETAILED SYLLABUS

In addition to the following topics, harder questions may be set on 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.

NOTE: Alternative X shall be for Further Mathematics candidates since the topics therein are
peculiar to Further Mathematics.

Alternative Y shall be for Mathematics (Elective) candidates since the topics therein
are peculiar to Mathematics (Elective).

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

235

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

1. Circular Measure
and Radians

Lengths of Arcs of circles
Perimeters of Sectors and
Segments measure in radians

2. Trigonometry

(i) Sine, Cosine and
Tangent of angles

(ii) Trigonometric ratios of
the angles 300, 450, 600

(iii) Heights and distances

(iv) Angles of elevation and
depression

(v) Bearings, Positive and
negative angles.

(vi) Compound and multiple
angles.

(vii) Graphical solution of
simple trig. equation.

(viii) Solution of triangles.

For O0 ≤ θ ≤ 3600

Identify without use of
tables.

Simple cases only.

Their use in simple
Identities and solution
of trig. ratios.

a cos x + b sin x = c

Include the notion of
radian and trigonometric
ratios of negative angles.

3. Indices, Logarithms
and Surds.
(a) Indices

(b) Logarithms

(i) Elementary theory of
Indices.

(ii) Elementary theory of
Logarithm

log a xy = logax + logay,

logaxn = nlogax

(iii) Applications

1
Meaning of a0, a-n, a n

Calculations involving
multiplication,
division, power and nth
roots:
1
log an, log √a, log a n

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 y = abx .

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

(c) Surds

(d) Sequences:
Linear and
Exponential
sequences

(e) Use of the
Binomial
Theorem for a
positive integral
index.

Surds of the form

a , a√ a and a + b√ n
√b

where a is rational.
b is a positive integer and n is
not a perfect square.

(i) Finite and infinite
sequences

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

(iii) Sn = n (U1 + Un)
2
(iv) Un = U1 rn-1
where r is the common ratio.

(v) Sn = U1(1 – rn ) ; r < 1
1 – r
or
Sn = U1 (rn – 1) ; r > 1
r – 1
Proof of Binomial Theorem not
required.
Expansion of (a + b)n
Use of (1 + x)n  1 + nx for any
rational n, where x is sufficiently
small e.g. 0(0.998)⅓

Rationalisation of the
Denominator:

a + √b
√c – √d

4. Algebraic Equations

(a) Factors and Factorisation.
Solution of Quadratic
equations using:-

(i) completing the square,
(ii) formula.

(a) Symmetric properties of
the equation
ax² + bx + c = 0

(b) Solution of two simulta-
neous equations where one
is linear and the other
quadratic.

The condition
b² – 4ac ≥ 0 for the
equation to have real roots.

Sum and product of roots.

Graphical and analytical
methods permissible.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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237

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

5. Polynomials

(i) Addition, subtraction
and multiplication of
polynomials.

(ii) Factor and remainder
theorems

(iii) Zeros of a polynomial
function.

(iv) Graphs of Polynomial
functions of degree
n ≤ 3.

(v) Division of a polynomial
of degree not greater
than 4 by a Polynomial
of lower degree.

Not exceeding degree 4

6. Rational Functions
and Partial Fractions

ax + b
e.g. f: x →
px² + qx + r

(i) The four basic operations.

(ii) Zeros, domain and range;
Sketching not required.

(iii) Resolution of
rational func-
tions into
partial frac-
tions.

Rational func-
tions of the
form
F(x)
Q(x) =
G(x)

G(x) ≠ 0

where G(x) and
F(x) are polyno-
mials, G(x) must
be factorisable
into linear and
quadratic factors
(Degree of Nume-
rator less than that
of denominator
which is less than
or equal to 4)

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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238

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

7. Linear Inequalities

Graphical and Analytical
Solution of simultaneous linear
Inequalities in 2 variables and
Quadratic inequalities.

8. Logic

(i) The truth table, using not
P or Q, P and Q.
P implies Q, Q implies P.

(ii) Rule of syntax: true or
false statements, rule of
logic applied to arguments,
implications and
deductions

Validity of compound
statements involving
implications and
connectives.

Include the use of symbols:
~ P

p v q, p ^q, p  q
Use of Truth tables.

9. Co-ordinate
Geometry:
Straight line

Conic Sections

(a) (i) Distance between two
points;

(ii) Mid-point of a line
segment;

(iii) Gradient of a line;

(iv) Conditions for parallel
and perpendicular
lines.

(b) Equation of a line:

(i) Intercept form;
(ii) Gradient form;
(iii) The general form.

(c) (i) Equation of a circle;
(ii) Tangents and normals
are required for circle.

Gradient of a line as ratio
of vertical change and
horizontal change.

(i) Equation in terms of
centre and radius e.g.
(x-a)² + (y-b)² = r²;

(ii) The general form:

x² + y² + 2gx + 2fy +
c = 0;

(iii) Equations of
parabola in
rectangular
Cartesian
coordinates.

10. Differentiation

(a) (i) The idea of a limit

(i) Intuitive treatment
of limit.
Relate to the
gradient of a curve.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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239

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

(ii) The derivative of
a function.

Application of differentiation

(b) (i) Second derivatives
and Rates of change;

(ii) Concept of maxima
and minima.

(ii) Its meaning and its
determination from
first principles in
simple cases only.

e.g. axn + b,

n ≤ 3, (n  I)

(iii) Differentiation of
polynomials e.g.

2×4 – 4×3 + 3x² – x + 7
and (a + bxn ) m

(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.

(iv) Product and
Quotient rules.
Differentiation
of implicit
functions such
as ax² + by² = c

11. Integration

(i) Indefinite Integral

(ii) Definite Integral

(i) Exclude n = -1 in

∫ xndx.

(ii) Integration of sum
and difference of
polynomials e.g.
4 x³ + 3x² – 6x + 5
include linear
kinematics.
Relate to the area
under a curve.

(ii) Simple problems
on integration
by substitution.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

240

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

(iii) Applications of the
definite integral

(iii) Plane areas and
Rate of change.

(iii) Volume of solid
of revolution.

(iv) Approximation
restricted to
trapezium rule.

12. 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}, , 
empty set { }, , , ,C,
U (universal set) or
A1 (Complement of set
A).

13. Mappings and
Functions

(i) Domain and co-domain
of a function.

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

(iii) Inverse of a function;

(iv) Composition of
functions.

The notation: e.g.
: x  3x + 4
g: x  x²
where x  R.

Graphical representation
of a function.

Image and the range.

Notation: fog (x) = f(g(x))
Restrict to simple
algebraic functions only.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

241

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

14. Matrices:
(a) Algebra of
Matrices.

(b) Linear
Transformation

(i) Matrix representation

(ii) Equal matrices

(iii) Addition of matrices

(iv) Multiplication of a
Matrix by a scalar.

(v) Multiplication of
matrices.

Restrict to 2 x 2 matrices
Introduce the notation A,
B, C for a matrix.

(i) The notation I
for the unit
identity matrix.

(ii) Zero or null
matrix.

Some special
matrices:
(i) Reflection in
the x-axis;

Reflection in the
y-axis.

The clockwise and
anti-clockwise
rotation about the
origin.

(ii) Inverse of a
2 x 2 matrix;

(i) Restrict to the
Cartesian plane;

(ii) Composition of
linear
transformation;

(iii) Inverse of a
linear trans-
formation;

(iv) Some special
linear trans-
formations:
Identity
Transforma-
tion,

Reflection in
the x-axis

Reflection in
the y-axis;

Reflection in
the line y = x

Clockwise and anti-
clockwise rotation
about the origin.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

(c) Determinants

Evaluation of deter-
minants of 2 x 2
and 3 x 3 matrices.

Application of deter-
minants to:

(i) Areas of triangles
and quadrila-
terals.

(ii) Solution of 3
simultaneous
linear equations

15. Operations

Binary Operations:
Closure, Commutativity,
Associativity and Distributivity,
Identity elements and inverses.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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243

PART II
STATISTICS AND PROBABILITY

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

1. Graphical
representation of
data

(i) Frequency tables.

(ii) Cumulative frequency
tables.

(iii) Histogram (including
unequal class intervals)

(iv) Frequency curves and
ogives for grouped data
of equal and unequal
class intervals.

2. Measures of
location

Central tendency;
Mean, median, mode, quartiles
and percentiles

Include:

(i) Mode and modal
group for grouped
data from a
histogram;

(ii) Median from
grouped data and
from ogives;

(iii) Mean for grouped
data, use of an
assumed mean
required.

3. Measures of
Dispersion

(a) Determination of:

(i) Range, Inter-Quartile
range from an ogive.

(ii) Variance and
standard deviation.

Simple applications.

For grouped and ungrouped
data using an assumed
mean or true mean.

4. Correlation

(i) Scatter diagrams

(ii) Line of fit

Meaning of correlation:
positive, negative and
zero correlations from
scatter diagrams.

Use of line of best fit to
predict one variable from
another.

Rank correlation
Spearman’s Rank
Correlation
Coefficient.

Use data without ties
Meaning and
applications.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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244

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

5. Probability

Meaning of probability
Relative frequency
Calculation of Probability.
Use of simple sample spaces.
Addition and multiplication of
probabilities.

E.g. tossing 2 dice once,
drawing balls from a box
without replacement.
Equally likely events and
mutually exclusive events
only to be used.

Probability Distribu-
tion.

Binomial Probability

P(x = r) = n Crprqn-r

where Probability of
success = P

Probability of failure
= q, p + q = 1 and n is
the number of trials.
Simple problems
only.

6. Permutations and
Combinations.

Simple cases of number of
arrangements on a line.

Simple cases of combination
of objects.

e.g. (i) arrangement of
students in a row.

(ii) drawing balls
from a box.
Simple problems
only.

n n!
P r =
(n – r) !

n!
nCr =
r!(n – r)!

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

245

PART III
VECTORS AND MECHANICS

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

1. Vectors

(i) Definitions of scalar and
vector quantities.

(ii) Representation of Vectors.

(iii) Algebra of vectors.

(iv) Commutative,
Associative and
Distributive properties.

(v) The parallelogram Law.

(vi) Unit Vectors.

(vii) Position and free
Vectors.

(iii) Addition and
subtraction of
vectors,
Multiplication of
vector by vectors and
by scalars.

Equation of vectors.

(iv) Illustrate through
diagram,
diagrammatic
representation.
Illustrate by solving
problems in
elementary plane
geometry e.g.
concurrency of
medians and
diagonals.

The notation

i for the unit vector

1
0 and

j for the unit vector

0
1

along the x and y axis
respectively.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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246

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

(viii) Resolution and
Composition of
Vectors.

(ix) Scalar (dot) product and
its application.

(viii) Not more than
three vectors need
be composed.

Using the dot product to
establish such trigonome-
tric formulae as

(i) Cos (a ± b) = cos a
cos b ± sin a sin b

(ii) sin (a ± b) = sin a
cos b ± sin b cos a

(iii) c² = a² + b² -2abCos c

Finding angle between two
vectors.

2. Statics

(i) Definition of a force.

(ii) Representation of
Forces.

(iii) Composition and
resolution of coplanar
forces acting at a point.

(iv) Equilibrium of particles.

(v) Lami’s theorem

(vi) Determination of
Resultant.

(iv) Apply to simple
problems e.g.
suspension of
particles by strings.

(v) Apply to simple
problems on
equivalent system of
forces.

(vi) Composition
and resolution
of general
coplanar
forces on rigid
bodies.

(viii) Moments of
forces.

WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
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247

AREAS COMMON TO THE TWO ALTERNATIVES

ADDITIONAL TOPICS /NOTES
FOR ALTERNATIVES

TOPIC CONTENT NOTES

ALTERNATIVE X ALTERNATIVE Y
(For Candidates (For Candidates
offering Further offering Maths
Maths) Elective)

Friction:
Distinction
between smooth
and rough planes.
Determination of
the coefficient of
friction required.

3. Dynamics

(a) (i) The concepts of
Motion, Time and
Space.

(ii) The definitions of
displacement,
velocity,
acceleration and
speed.

(iii) Composition of
velocities and
accelerations.

(b) Equations of motion

(i) Rectilinear motion;

(ii) Newton’s Law of
motion.

(iii) Consequences of
Newton’s Laws:

The impulse and
momentum
equations:
Conservation of Linear
Momentum.

(iv) Motion under
gravity.

Application of the
equations of motions:

V = u + at;

S = ut + ½ at²;

V² = u² + 2as.

Motion along
inclined planes.


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