Further Mathematics

Topics

Notes

I. Pure Mathematics

(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

All the four operations on surds Rationalising the denominator of surds

(3) Binary Operations

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, the inverse relation 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;

. 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 → ax 2 +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 3x2 + 5x + 2 = 0 are  and β, form the equation with roots . 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

(i) Rational functions of the form

(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

*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.

(10) Binomial Theorem

Use of the binomial theorem for positive integral index only. Proof of the theorem not required.

(11) Sequences and Series

(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.)

*(v) Recurrence Series

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 + …

Generating the terms of a recurrence series and finding an explicit formula for the sequence.

(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.

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, 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

(i) Trigonometric Ratios and Rules

(ii) Compound and Multiple Angles.

(iii) Trigonometric Functions and Equations

Sine, Cosine and Tangent of general angles (0o≤θ≤360o ). Identify trigonometric ratios of angles 30O , 45O , 60o without use of tables. Use basic trigonometric ratios and reciprocals to prove given trigonometric identities. Evaluate sine, cosine and tangent of negative angles. Convert degrees into radians and vice versa. Application to real life situations such as heights and distances, perimeters, solution of triangles, angles of elevation and depression, bearing(negative and positive angles) including use of sine and cosine rules, etc, Simple cases only. sin (A± B),cos (A ± B), tan(A ± B).

Use of compound angles in simple identities and solution of trigonometric ratios e.g. finding sin 75o , cos 150o etc, finding tan 45o without using mathematical tables or calculators and leaving your answer as a surd, etc.

Use of simple trigonometric identities to find trigonometric ratios of compound and multiple angles (up to 3A).

Relate trigonometric ratios to Cartesian Coordinates of points (x, y) on the circle x2 + y2 = r2 . f:x →sin x, g: x → a cos x + b sin x = c.

Graphs of sine, cosine, tangent and functions of the form asinx + bcos x. Identifying maximum and minimum point, increasing and decreasing portions.

Graphical solutions of simple trigonometric equations e.g. asin x + bcos x = k. Solve trigonometric equations up to quadratic equations e.g. 2sin2 x – sin x – 3 =0, for 0o ≤ x ≤ 360o . *Express f(x) = asin x + bcos x in the form Rcos (x ±α) or Rsin (x ± α) for 0o ≤  ≤ 90o and use the result to calculate the minimum and maximum points of a given functions.

(14)Co-ordinate Geometry

(i) Straight Lines

(ii) Conic Sections

Mid-point of a line segment Coordinates of points which divides a given line in a given ratio. Distance between two points; Gradient of a line;

Equation of a line:

(i) Intercept form;

(ii) Gradient form; Conditions for parallel and perpendicular lines. Calculate the acute angle between two intersecting lines e.g. if m 1 and m 2 are the gradients of two intersecting lines.

If m 1 m 2 = -1, then the lines are perpendicular.

*The distance from an external point P(x 1, y 1) to a given line ax + by + c.

Loci of variable points which
move under given conditions
Equation of a circle:
 (i) Equation in terms of
 centre, (a, b), and
 radius, r,

 (ii) The general form: x2+y2+2gx+2fy+c = 0.
Tangents and normals to circles
Equations of parabola in
rectangular Cartesian
coordinates.
Finding the equation of a
tangent and normal to a
parabola at a given point.
*Sketch graphs of given
parabola and find the equation
of the axis of symmetry. 

(15)Differentiation

(i) The idea of a limit

(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

(i) Intuitive treatment of limit

Relate to the gradient of a curve

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

(16)Integration

(i) Indefinite Integral

(ii) Definite Integral

(iii) Applications of the Definite Integral

(i) Integration of polynomials of the form axn ; n ≠ -1.

(ii) Integration of sum and difference of polynomials

(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.

Simple problems on integration
by substitution.
Integration of simple
trigonometric functions .
(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.

. 

II. Statistics and Probability

(17)Statistics

(i) Tabulation and Graphical representation of data

(ii) Measures of location (iii) Measures of Dispersion

(iv)Correlation

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-quartile range from an Ogive.

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

(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.

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 distribution P(x=r)=n Crp r q n-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 

III. Vectors and Mechanics

(19)Vectors

(i) Definitions of scalar and vector Quantities.

(ii) Representation of Vectors.

(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 application.

**(ix) Vector (cross) product and its application.

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.

Position vector of A relative to O. 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.

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 toestablish such trigonometric formulae

(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.

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.

(21)Dynamics

(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:

Conservation of Linear Momentum(exclude coefficient of restitution).

Distinguish between momentum and impulse. Objects projected at an angle to the horizontal.