ETEA Engineering Test 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 ETEA Engineering Test, not a summary of it.

1. 1

   Number Systems

   2 topics

   • Real and Complex Numbers
   • Operations on Complex Numbers
2. 2

   Sets, Functions and Groups

   3 topics

   • Sets and Venn Diagrams
   • Relations and Functions
   • Binary Operations and Groups
3. 3

   Matrices and Determinants

   3 topics

   • Algebra of Matrices
   • Determinants and Properties
   • Inverse of a Matrix and Cramer's Rule
4. 4

   Quadratic Equations

   3 topics

   • Solution of Quadratic Equations
   • Nature of Roots and Relations
   • Partial Fractions
5. 5

   Sequences and Series

   3 topics

   • Arithmetic Progression
   • Geometric Progression
   • Harmonic Progression and Means
6. 6

   Permutation, Combination and Probability

   3 topics

   • Permutations
   • Combinations
   • Probability
7. 7

   Mathematical Induction and Binomial Theorem

   2 topics

   • Principle of Mathematical Induction
   • Binomial Theorem and Series
8. 8

   Trigonometry

   4 topics

   • Trigonometric Identities
   • Trigonometric Functions and Graphs
   • Solution of Triangles
   • Inverse Trigonometric Functions
9. 9

   Functions and Limits

   2 topics

   • Types of Functions
   • Limits and Continuity
10. 10

    Differentiation

    3 topics

    • Derivative and Rules of Differentiation
    • Differentiation of Trigonometric and Exponential Functions
    • Applications of Derivatives
11. 11

    Integration

    3 topics

    • Indefinite Integrals and Methods
    • Definite Integrals
    • Area Under the Curve and Differential Equations
12. 12

    Analytic Geometry and Vectors

    4 topics

    • Straight Line and Pair of Lines
    • Conic Sections
    • Linear Inequalities and Programming
    • Vectors in Space

Mathematics flashcards for ETEA Engineering Test

21 of 52 cards from the Mathematics deck — real questions with worked answers.

1. What is the standard form of a complex number, and what do its parts represent?

   A complex number is written as $z = a + bi$, where $a$ is the real part $\operatorname{Re}(z)$, $b$ is the imaginary part $\operatorname{Im}(z)$, and $i = \sqrt{-1}$ so that $i^{2} = -1$.

2. How do the natural, integer, rational, irrational, and real number sets relate to one another?

   They nest: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. The irrationals $\mathbb{Q}'$ together with $\mathbb{Q}$ make up $\mathbb{R}$, i.e. $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$, and $\mathbb{Q} \cap \mathbb{Q}' = \varnothing$. All of these sit inside the complex numbers $\mathbb{C}$.

3. What distinguishes a rational number from an irrational number?

   A rational number can be written as $\frac{p}{q}$ with $p,q \in \mathbb{Z}$ and $q \neq 0$ (its decimal terminates or repeats). An irrational number cannot be expressed as such a ratio; its decimal is non-terminating and non-repeating, e.g. $\sqrt{2}$, $\pi$, $e$.

4. Define the modulus and conjugate of a complex number $z = a + bi$.

   The modulus is $|z| = \sqrt{a^{2} + b^{2}}$, and the conjugate is $\bar{z} = a - bi$. They satisfy $z\bar{z} = a^{2} + b^{2} = |z|^{2}$.

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

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

6. What is the pattern of the powers of $i$?

   They cycle with period 4: $i^{1}=i$, $i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$. In general $i^{n}$ depends only on $n \bmod 4$, so e.g. $i^{n}$ for $n \equiv 0$ gives $1$.

7. State De Moivre's theorem for a complex number in polar form.

   If $z = r(\cos\theta + i\sin\theta)$, then for integer $n$: $$z^{n} = r^{n}\big(\cos(n\theta) + i\sin(n\theta)\big).$$

8. What are the moduli/argument rules for the product and quotient of complex numbers?

   For products $|z_1 z_2| = |z_1||z_2|$ and $\arg(z_1 z_2) = \arg z_1 + \arg z_2$; for quotients $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ and $\arg\!\left(\frac{z_1}{z_2}\right) = \arg z_1 - \arg z_2$.

9. State the formulas for the union, intersection, and difference of two sets.

   $A \cup B = \{x : x \in A \text{ or } x \in B\}$; $A \cap B = \{x : x \in A \text{ and } x \in B\}$; $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$.

10. State De Morgan's laws for sets.

    $$(A \cup B)' = A' \cap B', \qquad (A \cap B)' = A' \cup B'.$$ The complement of a union is the intersection of complements, and vice versa.

11. What is the formula for the number of elements in the union of two finite sets (inclusion–exclusion)?

    $$n(A \cup B) = n(A) + n(B) - n(A \cap B).$$ For three sets: $n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)$.

12. How many subsets and how many proper subsets does a set with $n$ elements have?

    It has $2^{n}$ subsets in total (the power set) and $2^{n} - 1$ proper subsets (all subsets except the set itself).

13. Define a relation and a function from set $A$ to set $B$.

    A relation from $A$ to $B$ is any subset of the Cartesian product $A \times B$. A function $f : A \to B$ is a relation in which every element of $A$ is paired with exactly one element of $B$.

14. Distinguish injective, surjective, and bijective functions.

    Injective (one-one): distinct inputs give distinct outputs, $f(x_1)=f(x_2)\Rightarrow x_1=x_2$. Surjective (onto): every element of the codomain is an image. Bijective: both injective and surjective, so an inverse $f^{-1}$ exists.

15. What is the domain and range of a relation, and how do they differ for a function?

    The domain is the set of all first components and the range is the set of all second components of the ordered pairs. For a function the domain must include every element of the starting set, each appearing exactly once as a first component.

16. What four properties define a group $(G, *)$?

    1) Closure: $a*b \in G$. 2) Associativity: $(a*b)*c = a*(b*c)$. 3) Identity: there is $e$ with $a*e = e*a = a$. 4) Inverse: each $a$ has $a^{-1}$ with $a*a^{-1} = e$. If $*$ is also commutative, the group is abelian.

17. What is a binary operation on a set $S$, and what does closure mean?

    A binary operation $*$ on $S$ is a rule assigning to each ordered pair $(a,b)$ with $a,b \in S$ a unique element $a*b$. The operation is closed on $S$ if $a*b \in S$ for all $a,b \in S$, i.e. $*: S \times S \to S$.

18. What is a semigroup, a monoid, and how do they build up to a group?

    A semigroup has closure and associativity. A monoid is a semigroup that also has an identity element. A group is a monoid in which every element additionally has an inverse.

19. How are two matrices added, and what condition must hold?

    They must have the same order ($m \times n$). Addition is entrywise: if $A=[a_{ij}]$ and $B=[b_{ij}]$ then $A+B = [a_{ij}+b_{ij}]$.

20. State the rule for matrix multiplication, including the order requirement.

    For $A_{m\times n}$ and $B_{n\times p}$ (inner dimensions equal), the product $C = AB$ is $m\times p$ with $$c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}.$$ Matrix multiplication is generally not commutative: $AB \neq BA$.

21. Define the transpose of a matrix and give the transpose of a product.

    The transpose $A^{t}$ swaps rows and columns: $(A^{t})_{ij} = a_{ji}$. For products $(AB)^{t} = B^{t} A^{t}$ (order reverses).

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