FSc Pre-Engineering Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in FSc Pre-Engineering, not a summary of it.

1. 1

   Number Systems

   3 topics

   • Rational and Irrational Numbers
   • Properties of Real Numbers
   • Complex Numbers
2. 2

   Sets, Functions and Groups

   4 topics

   • Sets and Set Operations
   • Venn Diagrams and Laws of Algebra of Sets
   • Relations and Functions
   • Binary Operations and Groups
3. 3

   Matrices and Determinants

   4 topics

   • Types of Matrices
   • Algebra of Matrices
   • Determinants and Inverse of a Matrix
   • Solution of Linear Systems
4. 4

   Quadratic Equations

   4 topics

   • Solution of Quadratic Equations
   • Nature of Roots
   • Relation Between Roots and Coefficients
   • Systems of Equations Involving Quadratics
5. 5

   Partial Fractions

   2 topics

   • Proper and Improper Fractions
   • Resolution into Partial Fractions
6. 6

   Sequences and Series

   4 topics

   • Arithmetic Progression
   • Geometric Progression
   • Harmonic Progression and Means
   • Sum to Infinity of Geometric Series
7. 7

   Permutation, Combination and Probability

   4 topics

   • Factorial and Counting Principles
   • Permutations
   • Combinations
   • Basic Probability
8. 8

   Mathematical Induction and Binomial Theorem

   3 topics

   • Principle of Mathematical Induction
   • Binomial Theorem for Positive Index
   • Binomial Series for Rational Index
9. 9

   Trigonometry

   5 topics

   • Measurement of Angles and Trigonometric Functions
   • Fundamental Identities and Allied Angles
   • Sum, Difference and Multiple Angle Formulae
   • Graphs and Inverse Trigonometric Functions
   • Solution of Triangles
10. 10

    Functions and Limits

    3 topics

    • Types of Functions
    • Limits of Functions
    • Continuity
11. 11

    Differentiation

    4 topics

    • Derivative and Rules of Differentiation
    • Derivatives of Trigonometric and Inverse Functions
    • Derivatives of Exponential and Logarithmic Functions
    • Applications of Derivatives
12. 12

    Integration

    4 topics

    • Antiderivatives and Standard Integrals
    • Methods of Integration
    • Definite Integrals and Area Under Curve
    • Differential Equations
13. 13

    Analytic Geometry and Vectors

    4 topics

    • Straight Lines
    • Conic Sections
    • Vectors in Plane and Space
    • Scalar and Vector Products

Mathematics flashcards for FSc Pre-Engineering

21 of 60 cards from the Mathematics deck — real questions with worked answers.

1. What is a rational number?

   A number that can be expressed in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$. Its decimal expansion is either terminating or recurring.

2. What is an irrational number? Give two examples.

   A real number that cannot be written as $\frac{p}{q}$ with integers $p,q$ ($q\neq 0$); its decimal expansion is non-terminating and non-recurring. Examples: $\sqrt{2}$ and $\pi$.

3. How can you tell from its decimal expansion whether a number is rational or irrational?

   Rational numbers have terminating or recurring (repeating) decimal expansions; irrational numbers have non-terminating, non-recurring decimal expansions.

4. State the closure, commutative, and associative properties of real numbers under addition.

   Closure: $a+b \in \mathbb{R}$. Commutative: $a+b = b+a$. Associative: $(a+b)+c = a+(b+c)$, for all $a,b,c \in \mathbb{R}$.

5. What are the additive and multiplicative identities and inverses in $\mathbb{R}$?

   Additive identity: $0$ (since $a+0=a$); additive inverse of $a$ is $-a$. Multiplicative identity: $1$ (since $a\cdot 1=a$); multiplicative inverse of $a\neq 0$ is $\frac{1}{a}$.

6. State the distributive law of multiplication over addition for real numbers.

   $a(b+c) = ab + ac$ and $(b+c)a = ba + ca$ for all $a,b,c \in \mathbb{R}$.

7. State the trichotomy and transitive properties of order in $\mathbb{R}$.

   Trichotomy: for any $a,b$, exactly one of $a<b$, $a=b$, $a>b$ holds. Transitive: if $a<b$ and $b<c$ then $a<c$.

8. What is a complex number and what is the imaginary unit $i$?

   A complex number has the form $z = a + bi$ where $a,b \in \mathbb{R}$; $a$ is the real part and $b$ the imaginary part. The imaginary unit satisfies $i = \sqrt{-1}$, so $i^{2} = -1$.

9. Evaluate $i^{2}$, $i^{3}$, and $i^{4}$.

   $i^{2} = -1$, $\quad i^{3} = -i$, $\quad i^{4} = 1$.

10. When are two complex numbers $a+bi$ and $c+di$ equal?

    They are equal if and only if their real and imaginary parts are equal separately: $a = c$ and $b = d$.

11. Define the conjugate of a complex number $z = a + bi$.

    The conjugate is $\bar{z} = a - bi$ (the sign of the imaginary part is reversed).

12. Define the modulus of $z = a + bi$ and give the formula.

    The modulus is the distance from the origin in the Argand plane: $|z| = \sqrt{a^{2} + b^{2}}$.

13. What is the value of $z \cdot \bar{z}$ for $z = a + bi$?

    $z \bar{z} = a^{2} + b^{2} = |z|^{2}$, which is always a non-negative real number.

14. How do you add and subtract complex numbers $(a+bi)$ and $(c+di)$?

    Add/subtract real and imaginary parts separately: $(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i$.

15. How do you multiply two complex numbers $(a+bi)(c+di)$?

    Expand and use $i^{2}=-1$: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$.

16. How do you divide complex numbers, e.g. $\frac{a+bi}{c+di}$?

    Multiply numerator and denominator by the conjugate of the denominator: $\frac{a+bi}{c+di} \cdot \frac{c-di}{c-di} = \frac{(ac+bd) + (bc-ad)i}{c^{2}+d^{2}}$.

17. State two useful properties of the modulus of a product and quotient of complex numbers.

    $|z_{1} z_{2}| = |z_{1}||z_{2}|$ and $\left|\dfrac{z_{1}}{z_{2}}\right| = \dfrac{|z_{1}|}{|z_{2}|}$ (for $z_2 \neq 0$).

18. What is a set and what does the notation $a \in A$ mean?

    A set is a well-defined collection of distinct objects. $a \in A$ means $a$ is an element (member) of the set $A$.

19. Define subset, proper subset, and the empty set.

    $A \subseteq B$ means every element of $A$ is in $B$. $A$ is a proper subset ($A \subset B$) if $A \subseteq B$ and $A \neq B$. The empty set $\varnothing$ contains no elements.

20. Define the union and intersection of two sets $A$ and $B$.

    Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$. Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$.

21. Define the difference $A \setminus B$ and the complement $A'$ of a set.

    Difference: $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$. Complement: $A' = U \setminus A$, the elements of the universal set $U$ not in $A$.

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