# International islamic university malaysia centre for foundation studies

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## CENTRE FOR FOUNDATION STUDIES

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

1. FUNDAMENTALS

1. Equations

2. Inequalities and their solutions

3. Complex numbers and their algebraic properties

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,

fourth-degree 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.

1. FUNCTIONS AND THEIR GRAPHS

2.1 Functions, Types, Domain, Range and Transformation

2.2 Properties of Functions

2.3 Graphs of functions

2.5 Operations on functions ; Composite functions

2.6 One-to-one functions and their inverses

Learning outcomes

Students should be able to

1. define functions and write a function in its notation. Use arrow diagram.

1. determine whether an equation is a function algebraically and graphically ( by using vertical line test).

1. define the domain and range of a function

1. find domain of a function by using algebraic method for polynomial, rational and radical form of a function.

1. evaluate a function

1. identify whether a function is even or odd graphically or algebraically

1. identify the basic graph of a function. Graph functions of linear, constant, power, root, reciprocal, absolute value and piecewise-defined function.

2. find domain and range of functions by graphical approach.

3. graph a function by using transformation techniques : Vertical and horizontal shifting, Reflection about x-axis and y-axis, Vertical Stretching and Shrinking, and Horizontal Stretching and Shrinking

4. determine and graph the function obtained from a series of transformations – Combine shifting, reflection, stretching and shrinking.

5. graph a quadratic function using the standard form, , identify vertex, ( h,k ) , axis of symmetry and x,y- intercepts. Discuss shape of parabolas, open up or open down

6. find the maximum and minimum values of quadratic function.

1. find the quadratic function given its vertex and one other point.

1. form the sums, differences, products, and quotients of functions and their domains 1. define composite functions – represent a composite function by an arrow diagram. Existence of composite functions may not be discussed.

1. find composite function and its domain.

1. find the components of a composite function

1. evaluate a composite function.

1. determine whether a function is one-to-one function algebraically and graphically (by using horizontal line test).

1. define the inverse of a function with the help of an arrow diagram. Emphasize that the inverse exists only for a one-to-one functions.

1. find the inverse of a function.

2. verify that two functions are inverses by using Theorem on inverse function: 1. discuss the relationship between a function and its inverse with the help of a graph.

Sketch the graph of the functions and its inverse on the same axes.

1. find the domain and range of an inverse function.

1. POLYNOMIAL AND RATIONAL FUNCTIONS

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

1. define polynomials and state the degree of a polynomial and the leading coefficient.

Identify monomial, binomial, trinomial and polynomial

1. perform division of polynomials and write the answer in the form P(x) = Q(x)D(x) +R(x) or Use long division method and synthetic division method.

1. use remainder and factor theorems

2. list potential rational zeros (by applying ‘Rational Zeros Theorem’) and number of real

zeros (by applying ‘Descarte’s Rule of Signs’).

1. find the roots and real zeros of a polynomial (highest degree of polynomial is five).

2. identify the value of a such that ( x + a ) is a factor of P(x) and factorize completely. State Fundamental Theorem of Algebra and Complete Factorization Theorem.

3. form polynomial with given complex zeros. Use Conjugate Zeros Theorem to find complex zeros of a polynomial.

4. find complex zeros of a polynomial up to degree four.

5. form polynomial from its zero.

6. analyze the graph of a polynomial by considering its intercepts, zeros, even/odd multiplicity, end behavior; and location of graph in each interval above/below the x-axis.

7. determine the vertical, horizontal or oblique asymptotes of a rational function.

8. analyze and graph rational functions by considering its intercepts, asymptotes, intersection of graph with horizontal or oblique asymptotes and location of graph in each interval above/below the x-axis

1. EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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.

1. recall the properties of exponential functions: Change exponential expressions to logarithmic expressions and vice versa. Include ln x and log x.

1. graph exponential functions using transformations

1. define logarithmic function with base a

1. identify graph of logarithmic functions and the characteristics of the graph: the domain, range, intercept and asymptote.

1. recall the properties of logarithmic functions. Highlight the fact that exponential and logarithmic functions are inverses.

1. determine the domain of a logarithmic function.

1. graph logarithmic functions using transformations

1. find exact value of a logarithmic expression

1. evaluate exponential functions.

1. state the properties of logarithms: (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 1. SYSTEMS OF EQUATIONS

5.1 Matrices

5.2 Determinant

5.3 Systems of Linear Equations: Substitution and Elimination

5.4 Systems of linear equations: Gaussian Elimination and Gauss-Jordan 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 Gauss-Jordan

Elimination. Use elementary row operations to arrive row-echelon form or reduced row-echelon 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-1A = 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, 10th Edition.

Thomson Brooks/Cole

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