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MATHEMATICS 1 ( SHE 1114 ) SYLLABUS Aims The Mathematics 1 syllabus aims to develop the understanding of mathematical concepts and their techniques, together with the skills in mathematical reasoning and problem solving, so as to enable students to proceed to programmes related to science and technology at institutions of higher learning. Course Objectives The objectives of this syllabus are to develop the abilities of students to 1. apply mathematical terminology, notations, principles, and methods; 2. analyze, interpret and make mathematical decisions. 3. apply algebra and functions in the field of science, technology and social management. 4. develop understanding of mathematical concepts, applications and skills to analyze and solve problems so that students have a strong foundation to pursue studies in Biological and Physical Sciences, Medical, Pharmacy, Allied Health Sciences, Nursing, Dentistry and ICT programmes. Content
Learning outcomes Students should be able to (a) solve linear equations: – include solving for one variable in terms of others. (b) solve quadratic equations by factorization, completing the square and quadratic formula, recognize the types of roots of a quadratic equation based on the value of the discriminant, _{. Three types of roots : }(i) ,two distinct real roots (ii)_{ } ,two equal real roots (iii)_{ } ,two complex roots (c) solve other types of equations – Equations involving fractional expressions, radicals, fourthdegree equation of quadratic type, fractional powers, and absolute value. (d) use properties of inequalities – (e) solve linear and a pair of simultaneous inequalities. Answer may be given in either set or interval form. (f) solve quadratic inequalities using either graphical or table of signs. (g) solve rational inequalities involving linear and quadratic expression using table of signs. (h) solve absolute value inequalities in linear, quadratic and ratio expressions such as ; by using basic definition or squaring both sides. _{ Define } _{ } _{ Exclude inequalities of the forms:} _{ } _{ ; } (i) define complex numbers._{ } _{Introduce imaginary numbers. Discuss the standard form }_{a +bi}_{ , real and imaginary parts.} _{Real number is a subset of the set of complex number. } (j) discuss the equality of two complex numbers. (k) identify the conjugate of a complex number (l) perform algebraic operations on complex numbers (m) evaluate and write the square root of negative numbers and powers of i in standard form. (n) solve quadratic equations in the complex number.
2.1 Functions, Types, Domain, Range and Transformation 2.2 Properties of Functions 2.3 Graphs of functions 2.4 Quadratic functions 2.5 Operations on functions ; Composite functions 2.6 Onetoone functions and their inverses Learning outcomes Students should be able to
_{ }
Sketch the graph of the functions and its inverse on the same axes.
3.1 Polynomial functions 3.2 Technique of polynomial division 3.3 The real zeros of a polynomial function 3.4 Complex Zeros of a Polynomial Function 3.5 Graph of polynomial functions 3.6 Rational functions and their graphs Learning outcomes Students should be able to
Identify monomial, binomial, trinomial and polynomial
Use long division method and synthetic division method.
zeros (by applying ‘Descarte’s Rule of Signs’).
4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Laws of Logarithms 4.4 Logarithmic & Exponential Equations Learning outcomes Students should be able to (a) identify the graph of exponential functions, and natural exponential function. Highlight the domain, range, intercept and asymptote.
(l) use law of logarithms to evaluate, expand and combine logarithmic expressions _{ } express a logarithmic expression as a sum or difference of logarithms and as a single logarithm. (m) evaluate logarithms with the change of base formula: (n) solve logarithmic and exponential equations algebraically: Exponential equation such as _{ }
5.1 Matrices 5.2 Determinant 5.3 Systems of Linear Equations: Substitution and Elimination 5.4 Systems of linear equations: Gaussian Elimination and GaussJordan Elimination 5.5 Systems of linear equations: Cramer’s Rule 5.6 Systems of linear equations: Inverse Learning outcomes Students should be able to (a) define a matrix and the equality of matrices. Identify the size/dimension of the matrix. (b) identify the different types of matrices such as row, column, zero, diagonal, symmetric, upper triangular, lower triangular, identity and transpose matrices. (c) perform operations on matrices such as addition, subtraction, scalar multiplication and multiplication of two matrices. Able to recognize whether the dimensions of matrices are compatible. (d) discuss the properties of addition, scalar multiplication and matrix multiplication. Stress that the matrix multiplication is not commutative. (e) evaluate 2 by 2 determinant (f) evaluate 3 by 3 determinant using minor and cofactors (g) discuss the properties of determinant (h) solve a system of linear equations in 2 and 3 variables using substitution and elimination method (i) discuss types of solutions namely: unique solution, no solution ( inconsistent system) and infinitely many (j) solve a system of linear equations in 2 and 3 variables using substitution and elimination method (k) discuss types of solutions namely: unique solution, no solution ( inconsistent system) and infinitely many solutions (dependent system) (l) write augmented matrix of a linear system (m) perform row operations on a matrix. (n) solve systems of linear equations using Gaussian Elimination and GaussJordan Elimination. Use elementary row operations to arrive rowechelon form or reduced rowechelon form of a matrix. Include inconsistent and dependent systems. (o) use Cramer’s Rule to solve a system of equations in 2 and 3 variables. For a system of linear equations in the form of AX=B , a unique solution exist when (p) find the inverse of 2 x 2 and 3x 3 matrix using row operation and adjoint matrix. Matrix A is singular or not invertible ( that is matrix A has no inverse matrix) when _{ } _{ Verify that a matrix }_{A}^{1}_{ }_{ is an inverse of matrix }_{A}_{ by using }_{AA}^{1}_{ = }_{A}^{1}_{A = I} (q) solve systems of equations using inverse matrix. Form of examination The examination consists of one paper; the duration for the paper is 3 hours. Candidates are required to take the paper. Reference Books 1. Parveen Kausar, Mathematics for Matriculation: Algebra (2009). Cengage Learning Asia Pte. Ltd. 2. Steward, J., Redlin, L., & Watson, S. (2006). Precalculus. USA: Thomson. 3. Sullivan, M. Precalculus. (2005), USA: Pearson Prentice Hall. 4. Earl Swokowski. Algebra and Trigonometry with Analytic Geometry, 10^{th} Edition. Thomson Brooks/Cole 