ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y    TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   1. Circular Measure Lengths of Arcs of circles    and Radians Perimeters of Sectors and    Segments measure in radians      2. Trigonometry (i) Sine, Cosine and For O0  ≤ θ ≤ 3600    Tangent of angles    (ii) Trigonometric ratios of Identify without use of    the angles 300, 450, 600 tables.    (iii) Heights and distances    (iv) Angles of elevation and    depression    (v) Bearings,  Positive and Simple cases only.    negative angles.    (vi) Compound and multiple Their use in simple    angles. Identities and solution    of trig. ratios.    (vii) Graphical solution of a cos x + b sin x = c    simple trig. equation.    (viii)  Solution of triangles. Include the notion of    radian and trigonometric    ratios of negative angles.      3. Indices, Logarithms    and Surds. 1    (a) Indices (i) Elementary theory of Meaning of a0, a-n, a n    Indices.    (b) Logarithms (ii) Elementary theory of Calculations involving       Logarithm multiplication,    division,  power and nth       log a xy = logax + logay, roots:    1       logax n  = nlogax log an, log √a, log a n    (iii) Applications 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 .

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y    TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   (c) Surds Surds of the form Rationalisation of the    Denominator:       a , a√ a  and a + b√ n    a + √b       √b          √c - √d    where a is rational.    b is a positive integer and  n is    not a perfect square.   (d) Sequences: (i) Finite and infinite    Linear and    sequences    Exponential       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   (e) Use of the Proof of Binomial Theorem not    Binomial required.    Theorem for a Expansion of (a + b) n    positive integral    index. Use of (1 + x)n » 1 + nx for any    rational n, where x is sufficiently    small e.g.  0(0.998)   4.  Algebraic Equations (a) Factors and Factorisation.    Solution of Quadratic    equations using:-    (i) completing the square, The condition       (ii) formula. b² - 4ac ≥ 0 for the    equation to have real roots.    (a) Symmetric properties of    the equation Sum and product of roots.    ax² + bx + c = 0    (b)  Solution of two simulta-    neous equations where one Graphical and analytical        
is linear and the other methods permissible.quadratic.

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES   AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   5.  Polynomials (i) Addition, subtraction    and multiplication of    polynomials.    (ii) Factor and remainder Not exceeding degree 4    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.


  ax + b   6.  Rational Functions e.g. f:  x   →       and Partial Fractions px²   +  qx + r  
(i) The four basic operations. 
 (ii)Zeros, domain and range; (iii)  Resolution of     rational func-   Sketching not required.        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)

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES    ALTERNATIVE X ALTERNATIVE Y    TOPIC CONTENT NOTES (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 Validity of compound    P or Q, P and Q. statements involving    P implies Q, Q implies P. implications and    connectives.    (ii) Rule of syntax:  true or Include the use of symbols:    false statements, rule of ~ P    logic applied to arguments,    implications and p v q, p ^q, p Þ q    deductions    Use of Truth tables.         9. Co-ordinate (a) (i)  Distance between two    Geometry: points;    Straight line  
(ii) Mid-point of a line segment; 
 (iii)   Gradient of a line; Gradient of a line as ratio    of vertical change and    horizontal change.  
(iv) Conditions for parallel and perpendicular lines. 
(b) Equation of a line: 
(i) Intercept form; 
(ii) Gradient form; 
(iii) The general form. 
 Conic Sections (c) (i) Equation of a circle; (i) Equation in terms of (iii)  Equations of     parabola in    centre and radius e.g.    (ii) Tangents and normals rectangular    (x-a)² + (y-b)² = r²;    are required for circle. Cartesian       (ii) The general form: coordinates.       x² + y² + 2gx + 2fy +    c = 0;   10.   Differentiation (a) (i) The idea of a limit (i) Intuitive treatment    of limit.    Relate to the     
gradient of a curve.


WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES   AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   (ii) The derivative of (ii)  Its meaning and its    a function. determination from    first principles in    simple cases only.  
e.g. axn  + b,
n ≤ 3, (n Î I)
(iii) Differentiation of polynomials e.g. 2x4 – 4x3 + 3x²  - x + 7and (a + bxn ) m(iv) Product andApplication of differentiation Quotient rules.
Differentiation of implicit functions such as ax² + by² = c

  (b) (i) Second derivatives (i) The equation of a    and Rates of change; tangent to a curve at    a point.    (ii) Concept of maxima (ii) Restrict turning    and minima. points to maxima    and minima.    (iii) Include curve    sketching (up to    cubic functions) and    linear kinematics.      11.  Integration (i) Indefinite 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) Definite Integral (ii)  Simple problemson integrationby substitution.

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y    TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)    (iii) Volume of solid    of revolution.    (iii) Applications of the (iii) Plane areas and    definite integral Rate of change. (iv) Approximation    restricted  to    trapezium rule.      12. Sets (i) Idea of a set defined by a {x : x is real}, È, Ç    property. empty set { }, Æ, Î, Ï,C,    Set notations and their U (universal set) or    meanings. 1    A (Complement of set    (ii) Disjoint sets, Universal A).       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      13. Mappings and The notation:  e.g.    Functions ¦:  x ® 3x + 4    g:  x ® x²    where x Î R.    (i)  Domain and co-domain Graphical representation       of a function. of a function.    (ii)  One-to-one, onto, Image and the range.    identity and constant       mapping;    (iii)  Inverse of a function;    (iv) Composition of Notation:  fog (x) = f(g(x))    functions. Restrict to simple    algebraic functions only.  

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y    TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   14. Matrices: (i) Matrix representation Restrict to 2 x 2 matrices Some special   (a) Algebra of     Introduce the notation A, matrices:    Matrices. (ii) Equal matrices B, C for a matrix. (i) Reflection in       the x-axis;    (iii) Addition of matrices (i) The notation I    for the unit Reflection in the       identity matrix. y-axis.    (iv) Multiplication of a    Matrix by a scalar. (ii) Zero or null The clockwise and    anti-clockwise    (v) Multiplication of matrix. rotation about the    origin.       matrices.    (ii) Inverse of a    2 x 2 matrix;    (i) Restrict to the    Cartesian plane;   (b)  Linear (ii) Composition of    Transformation    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.     


WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y    TOPIC CONTENT NOTES (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       Binary Operations:    15. Operations Closure, Commutativity,    Associativity and Distributivity,    Identity elements and inverses.        

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
PART IISTATISTICS AND PROBABILITY
  ADDITIONAL TOPICS /NOTES   AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   1.   Graphical (i) Frequency tables.   representation of   data (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 Central tendency; Include:   location Mean, median, mode, quartiles    and percentiles (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 (a) Determination of:    Dispersion    (i) Range, Inter-Quartile Simple applications.    range from an ogive.    (ii) Variance and For grouped and ungrouped    standard deviation. data using an assumed    mean or true mean.   4. Correlation (i) Scatter diagrams Meaning of correlation: Rank correlation    positive, negative and Spearman’s Rank    zero correlations from Correlation    scatter diagrams. Coefficient.    Use data without ties  
Meaning and applications.
 (ii)Line of fit Use of line of best fit to    predict one variable from    another.   

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES   AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths    Maths) Elective)   5.  Probability Meaning of probability E.g. tossing 2 dice once,    Relative frequency drawing balls from a box    Calculation of Probability. without replacement.    Use of simple sample spaces. Equally likely events and    Addition and multiplication of mutually exclusive events Probability Distribu-    probabilities. only to be used. 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 Simple cases of number of e.g.  (i)  arrangement of   Combinations. arrangements on a line. students in a row.  
(ii) drawing balls from a box. Simple problems only.
  n n!    P r =    (n – r) !       nCr n!   Simple cases of combination =       of objects. r!(n – r)!     

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
PART III
VECTORS AND MECHANICS
  ADDITIONAL TOPICS /NOTES    AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (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. (iii) Addition and    subtraction of    vectors,    Multiplication of    vector by vectors and    by scalars.    Equation of vectors.    (iv) Commutative, (iv) Illustrate through    Associative and diagram,    Distributive properties. diagrammatic    representation.    Illustrate by solving    problems in    elementary plane    geometry e.g.    concurrency of    medians and    diagonals.    (v) The parallelogram Law. The notation    i for the unit vector    (vi) Unit Vectors.    1    0and  
j for the unit vector
01
along the x and y axis respectively.
(vii) Position and free Vectors. 
WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
  ADDITIONAL TOPICS /NOTES   AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES    ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (For Candidates (For Candidates    offering Further offering Maths       Maths) Elective)   (viii) Resolution and (viii)   Not more than    Composition of three vectors need    Vectors. be composed.  

Using the dot product to establish such trigonome-
(ix) Scalar (dot) product and tric formulae as its application.
(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. (iv) Apply to simple    problems e.g.    suspension of    particles by strings.   (v) Lami’s theorem (v) Apply to simple    problems on    equivalent system of    forces.    (vi) Composition   (vi) Determination of and resolution     of general    Resultant.    coplanar       forces on rigid    bodies.    (viii) Moments of    forces.  



WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

  ADDITIONAL TOPICS /NOTES   AREAS COMMON TO THE TWO ALTERNATIVES FOR ALTERNATIVES       ALTERNATIVE X ALTERNATIVE Y   TOPIC CONTENT NOTES (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 Application of the    (i) Rectilinear motion; equations of motions:       V =  u + at;    (ii) Newton’s Law of    motion. S = ut + ½ at²;       (iii) Consequences of    Newton’s Laws: V² =   u² + 2as.    The impulse and    momentum    equations:    Conservation of Linear    Momentum.    (iv) Motion under Motion along    gravity. inclined planes.