MATHEMATICAL SUBJECTS Mathematics should be visualised as the vehicle for aiding a student to think, reason, analyse and articulate logically.

Transcription

1 MATHEMATICAL SUBJECTS Mathematics should be visualised as the vehicle for aiding a student to think, reason, analyse and articulate logically. Apart from being treated as a subject of its own, Mathematics should be related to any subject involving analysis and reasoning. Aims To help the student. (i) develop manipulative and computational skills; (ii) understand basic fundamental concepts and their applications; (iii) acquire skills of clear mathematical expression and logical mathematical reasoning. Objectives To enable the student to: (a) develop interest in Mathematics and appreciate both its beauty and as an aid to other disciplines. (b) develop an understanding of the relationship between mathematical variables. (c) acquire knowledge and understanding of the terms, symbols, concepts, principles, processes and proofs. (d) develop the necessary skills to work with: (i) mathematical tables; (ii) calculators; (iii) computers; where available. (e) develop the ability to interpret algorithms for problem-solving. (f) apply mathematical knowledge and skills in solving everyday life problems.

2 456 MATHEMATICS O-LEVEL Examination Format: There will be two papers, each of which will contain questions on any part of the syllabus and the solution of any question may request knowledge of more than one branch of the syllabus. Paper 1: (2 ½ hours) It will consist of two sections, A and B. section A will contain ten short compulsory questions, carrying 40 marks. Section B will contain seven questions. Candidates will be required to attempt five questions, carrying 60 marks. (100 Marks) Paper 2: (2 ½ hours) It will consist of two sections, A and B. Section A will contain ten short compulsory questions carrying 40 marks. Section B will contain seven questions. Candidates will be required to attempt five questions, carrying 60 marks. (100 marks) Note: 1. S.I Units will be used in questions involving mass and measure of length; the use of the centimetres, litre and hectare will continue. 2. The 24-hour clock will be used for quoting time of a day, for example 3:15a.m.. will be denoted by hours, 3:15 pm by hours. 3. Candidates will be expected to be familiar with the solidus notation for the expression of compound units. E.g. 5 m/s for 5 metres per second or 5 ms -1 ; 1.36x 10 4 kg/m 3 or 1.36x10 4 kg m 3 for 1.36x10 4 kilograms per cubic metre 4. Mathematical tables may be used (UNEB four figure tables are recommended). 5. Calculators may also be used. They could be simple or scientific but must be silent and non programmable.

3 Detailed syllabus: Topic Notes 1. Numerical concepts: (a) Ordinary operations of arithmetic Factors, multiple and divisors. Prime and composite numbers. Sequences and number patterns. (b) Fractions, decimals, expressions of recurring decimals as equivalent fractions. Scale and representative fractions. Ratio, percentage. Direct and inverse proportions. Knowledge of addition, subtraction, multiplication and division of real numbers is expected. Knowledge of L.C.M and H.C.F and their application is expected. (c) Estimates and approximations. This section is to include the use of Significant figures; decimal notations A x 10 n,where n is an places. Integer and 1 < A < 10. (d) Rules of operations for indices and logarithms. Examples of integral and fractional indices will be set. Tables of squares, square roots, logarithms and reciprocals to be used. (e) Simple manipulation of surds. Simple identities involving square roots. Rationalism of surds included. 2. Algebraic symbols, Expressions and Equations: (a) Use of algebraic symbols to This also includes evaluation of

4 represent statements. Generalizations of arithmetical relations in symbols, the interpretation of statements given in symbolic form. algebraic expression and change of subject in formula. (b) Equations, inequalities and identities. Solution of linear equations and inequalities in one real variable, solution in two real variables. Solution in two real variables by any algebraic method. (c) Factors of expressions, including trinomials. e.g. factors of ( a 2 -b 2 ), ax +bx, x 2 + 2xy + y 2,ax 2 + bx + c and a 3 + b 3. (d) Solutions of quadratic equations and inequalities in one real variable 3. Extension of set theory, relations: (a) Knowledge of union, intersection and complement. (b) Relations, mappings and functions. Graphical representation: arrow Only questions involving factorization will be set. Derivation and use of formula. λ= -b + ( b 2 _ 4ac) 2a is not expected. Application of set theory in solving variety of problems. Venn diagrams may be used. Knowledge of Domain and range is required.

5 diagrams (including papygrams), cartesian graphs. Composite functions, e.g. fg(x). 4. Graphs: (a) Rectangular Cartesian co- Graphs will be limited to the ordinates in two dimensions equivalent of the graph of the Locus regarded as a set of function. Points satisfying a given condition Ax 3 + Bx 2 + Cx + D + E/x + F/x 2 The Cartesian equation of locus where at least three of the coefficients in simple cases. Gradient of a line. A, B, C, D, E, F are zero e.g a line or The gradient of curve estimated curve as a set of options whose from a tangent. Use of gradient and co-ordinates form the solution set Y intercept to determine the equation of an equation in two variables. of a line. Estimation of best line through a set of points. (b) Simple application of intersection of lines and curves to the solution of simultaneous and quadratic equations. To include intersection with the axes. In the case of two simultaneous equations at least one will be linear. ( c ) Graphs of inequalities in one or two Questions may be set which involve variables. The intersection of regions. Questions involving the maximum or minimum value of a function of two variables ( e.g. x + 2y) with in a given region will be confined to integral values such as can readily be found by inspection. 5. Vectors and Matrices: (a) notion of vectors, basic e.g. a ~ + b = c ~ b =ka, operation on vectors, conditions k a = b.

6 of vectors to be parallel and collinear. Magnitude of a vector. (b) Addition and multiplication of Some simple applications of matrices. Determinant and inverse matrices, e.g. as stores of of a 2x2 matrix. information and as a tool to solve systems of linear equations. 6. Geometrical Concepts: (a) The transformation of reflection Knowledge of the use of rotation, translation and enlargement transformation matrices is in two dimensions. Invariant properties expected. the knowledge of under each of these transformations congruence and similarity and the resulting ideas of symmetry of triangles is expected. congruence, and similarity. Properties the notation MT (p) for the of similar and congruent figures. image of p under the transformation Combination of transformations. T followed by the transformation M. (b) Simple geometric constructions. This includes use of rule and compasses, protractor or any other appropriate aids to bisect angles and line segments and draw perpendiculars. Construction of angles of 30 0,45 0 and Construction of a triangle, quadrilateral or other simple polygon, circumscribed and inscribed circles, of a triangle from sufficient data. Knowledge of the sum of angles of a triangle or quadrilateral and sum of the exterior angles of a polygon is assumed. (c) Angle properties of parallelograms

7 Right-angled triangles and Pythagoras Theorem and its converse. Since, cosine and tangent of acute, obtuse and reflex angles. This includes angles in the four Quadrants. Extension of this to sine and cosine waves is expected. Knowledge of the use of trigonometrical tables are expected. (d) Nets of the surface of solids, sketching solids. Angle between a line and a plane and between two planes. This to include cuboid, triangular prism, tetrahedron, square-based pyramid and octahedron. Skew lines are not required. (e) The circle: i. Symmetry properties ( formal These to include angles subtended proofs not to be asked). by equal arcs (chords); angle at the centre is twice that at the circumference. Angle in the semicircle is a right angle, angles in opposite segments are supplementary. ii. Tangent property (formal proofs Tangents are perpendicular to radius not to be asked) tangents are of equal length if drawn from an external point If a straight line touches a circle and from the point of contract a chord be drawn, the angles which this a chord makes with the tangent are equal to the angles in the alternate segments.

8 (f) Bearings. Knowledge of angles of elevation and depression and scale drawing will be expected. 7. Miscellaneous Application (a) Conversation from one currency to another given the relevant exchange rate. Interest, Discount. Compound interest will be limited to only as far as 2 years. (b) Principles of taxation and insurance. Knowledge of such terms as hire purchase, income per head, taxable income, control premium, insured value will be expected. Note that this topic should only deal with the mathematical manipulation and knowledge of the mentioned terms, but not to the depth of commerce or Principles of Accounts. (c) Mensuration of the triangles, parallelogram, circle and figures formed by combination of these. Volumes and surface areas of prism (including right cylinder),pyramid Properties of similar solids including (including right cone),and sphere. scale factors. (d) The application of Pythagoras theorem. Trigonometrical methods involving simple two and three-dimensional figures. Knowledge of the use of sine, cosine ratios and trigonometrical tables in solving simple problems will be expected.

9 (e) Estimation of the area under a graph and its interpretation. Application of gradient and area under a graph to easy linear kinematics involving distance-time and speed-time graphs. (f) Representation and interpretation of data in tabular or graphical form. e.g. bar chart, pie chart, frequency polygon and line graph. (g) Mode mean and median. Modal class, estimation of mean and Median from grouped data. (h) Basic concepts of probability. Simple combination of probabilities. Only problems which can be solved by the numeration of an event within a finite possibility space will be set. This may be extended to include union and intersection of events.

10 457 ADDITIONAL MATHEMATICS Examination format: There will be two papers. Paper 1: (2 ½ hours) This will contain twelve questions. Eight questions will be on pure mathematics, four questions will be on Vectors and Matrices. Candidates will be required to answer eight questions. (100 marks) Paper 2: (2 ½ hours) It will contain twelve questions. Six questions will be on Mechanics and the other six questions will be on Statistics. Candidates will be required to answer eight questions. (100 marks) Detailed syllabus: The topics have been selected in such a way that the interest of students taking Social/Biological Sciences and those taking Physical Sciences are well catered for Topic 1. Pure mathematics (8 questions). Indices, logarithms and surds. Arithmetic and geometric progressions. Factor and Remainder theorems. Use of the Binomial theorem for an integral index, and its use for simple approximations. Elementary permutations and combinations. Notes Including the sum of finite number of items. Questions on the greatest term and on sums and properties of the coefficients will not be asked. Elementary properties of quadratic expressions and equations. Range of values of the function ax 2 + bx + c by graphical and other methods.

11 Gradient of a straight line Relationship between gradients of perpendicular and parallel lines. Representation of a curve by means of a pair of parametric equations. Single parameter only. Equations of tangent and normal. Elementary locus problems. Equations of a circle Including eliminations of parameter. Easy simultaneous equations In two unknowns. Complex numbers At least one linear. Geometrical representation. Modulus and argument. Addition and multiplication. Circular measure, arc-length, area of a sector of a circle, trigonometrical ratios of angles of any magnitude. To be used in graphs of simple trigonometrical functions. Use of a formulae for sin (A + B), cos (A + B), tan (A + B). Application to multiple angles and simple identities. Sine cosine formulae and the formulae (½ bc) sin A. Simple trigonometrical problems in three dimensions. To be used in solving triangles and determination of area of triangles. Proofs of these formulae will not be required.

12 Derivative of kx n where n is a Application to small increments positive integral index. rates of change, velocity and acceleration, maxima and minima (any method of discrimination will be acceptable). Derivatives of simple algebraic, trigonometrical functions including sums, products, quotients, composite functions. Implicit functions and inverse trigonometric functions are excluded. Integration as the inverse of differentiation. Definite integral integration of simple functions: applications to plane areas and volumes of solids and kinematics. Excluding integration by parts and by change variable. 2. Vectors and Matrices (4 questions ) Displacement and position and position vectors ( 2-dimensions). The ratio theorem λop + μoq = (λ + μ) OR. The scalar product a.b of two Vectors is in the form a.b = a b cos θ and also a.b = a 1 b 1 + a 2 b 2 where θ is the angle between a and b, a = ( a 1 ) and b = ( b 1 ). Addition, subtraction and multiplication by scalars. Application of the scalar product to test perpendicularity in a plane. a 2 b 2

13 Matrices and their application. (i) The use of determinant of a 2 x 2 matrix to solve a pair of linear simultaneous equations. Geometrical interpretation of the value of the determinant. 2. Mechanics (6 Questions) Forces, velocities and acceleration as vectors, composition and resolution of velocities, relative velocity. (ii) Matrices applied to probability, e.g. routes through network.. Kinematics of particle moving in a straight line. Include its graphical treatment. Motion with uniform acceleration Discussion of Newton s laws of Motion.. Application to connected bodies. Ideas of mass, force, energy, work and power. Composition and and resolution of forces, moments. An experimental basis is sufficient, proofs of the fundamental Theorems of statistics will not be required.

14 Simple cases of friction. The conservation of momentum in rectilinear motion. Conservation of energy. Equilibrium of a particle and centre of gravity of a rigid body under coplanar forces. The laws of friction between solids in contact. 3. Statistics (6 questions) Scope and limitations of statistics. The tabulation and appropriate representation of numerical data, choice of class intervals. Frequency distributions, histograms, cumulative frequency. Measure of central tendency. Measure of dispersion Moving averages. Index numbers. Addition and multiplication Laws of probability. Mean, median and mode. Interquartile range and standard deviation Discrete variable. Expectation: Expected values. simple probability and frequency distributions. General ideas of correlation. Calculation of a rank correlation coefficient and its interpretation. General ideas of sampling and surveys. Calculated for discrete and mathematically defined continuous distributions. Particularly the Binomial distribution and its mean and standard deviation.

15 Knowledge of normal distributions. Estimation of the limits of a mean of a population from a large sample. Include scatter graphs. Kendall s or Spearman s method of calculation of a rank correlation coefficient.