PAF Aeronautical 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 PAF Aeronautical Engineering, not a summary of it.

1. 1

   Number Systems

   3 topics

   • Real and Complex Numbers
   • Sets, Functions and Groups
   • Number Theory Basics
2. 2

   Algebra

   6 topics

   • Quadratic Equations
   • Matrices and Determinants
   • Partial Fractions
   • Sequences and Series
   • Permutations, Combinations and Probability
   • Mathematical Induction and Binomial Theorem
3. 3

   Trigonometry

   5 topics

   • Trigonometric Functions and Identities
   • Trigonometric Equations
   • Inverse Trigonometric Functions
   • Solution of Triangles
   • Heights and Distances
4. 4

   Calculus

   4 topics

   • Limits and Continuity
   • Differentiation
   • Integration
   • Differential Equations
5. 5

   Geometry and Vectors

   3 topics

   • Coordinate Geometry
   • Analytical Geometry
   • Vectors
6. 6

   Statistics

   3 topics

   • Measures of Central Tendency
   • Measures of Dispersion
   • Data Representation

Mathematics flashcards for PAF Aeronautical Engineering

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

1. Classify the real number system: what are the two disjoint subsets that make up $\mathbb{R}$, and give an example of each.

   $\mathbb{R}$ splits into rationals $\mathbb{Q}$ (numbers expressible as $\frac{p}{q}$, $q \neq 0$, e.g. $\frac{3}{4}$) and irrationals (non-terminating, non-repeating decimals, e.g. $\sqrt{2}$ or $\pi$).

2. What is the standard (rectangular) form of a complex number, and what do its parts represent?

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

3. Define the complex conjugate of $z = a + bi$ and state one key product identity.

   The conjugate is $\bar{z} = a - bi$. Their product is real: $z\bar{z} = a^{2} + b^{2} = |z|^{2}$.

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

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

5. Evaluate the powers of $i$: what are $i^{1}, i^{2}, i^{3}, i^{4}$, and what is the general pattern?

   $i^{1}=i$, $i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$. Powers cycle with period 4, so $i^{n} = i^{n \bmod 4}$.

6. On the Argand diagram, how is a complex number $z = a + bi$ represented?

   As the point $(a, b)$ (or the vector from the origin to that point) in the plane, with the horizontal axis the real axis and the vertical axis the imaginary axis.

7. Define the modulus and argument of $z = a + bi$.

   Modulus: $|z| = \sqrt{a^{2} + b^{2}}$ (distance from origin). Argument: $\theta = \arg(z) = \tan^{-1}\!\left(\frac{b}{a}\right)$ (angle from positive real axis).

8. Write the polar form of a complex number and state De Moivre's theorem.

   Polar form: $z = r(\cos\theta + i\sin\theta)$ with $r = |z|$. De Moivre: $z^{n} = r^{n}(\cos n\theta + i\sin n\theta)$.

9. What does the modulus identity $|z_1 z_2| = |z_1||z_2|$ tell you, and what is the corresponding rule for arguments?

   The modulus of a product equals the product of moduli; the arguments add: $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$.

10. Define a set, and distinguish a subset from a proper subset.

    A set is a well-defined collection of distinct objects. $A \subseteq B$ (subset) means every element of $A$ is in $B$; $A \subset B$ (proper subset) additionally requires $A \neq B$.

11. State De Morgan's laws for sets.

    $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

12. Define a function, and distinguish injective, surjective and bijective functions.

    A function $f: A \to B$ assigns each element of $A$ exactly one element of $B$. Injective (one-to-one): distinct inputs give distinct outputs. Surjective (onto): every element of $B$ is an image. Bijective: both injective and surjective.

13. What four properties must a set $G$ with binary operation $*$ satisfy to be a group?

    Closure, associativity ($a*(b*c)=(a*b)*c$), existence of an identity $e$ ($a*e=e*a=a$), and existence of an inverse for each element ($a*a^{-1}=e$). An abelian group additionally requires commutativity.

14. State the Fundamental Theorem of Arithmetic.

    Every integer greater than 1 can be expressed uniquely (up to order) as a product of prime numbers.

15. Define the GCD and LCM of two integers and state their relationship.

    GCD is the greatest common divisor; LCM is the least common multiple. For positive integers $a, b$: $\gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b$.

16. State the divisibility rule for 3 and for 9.

    An integer is divisible by 3 if the sum of its digits is divisible by 3; it is divisible by 9 if the digit sum is divisible by 9.

17. What is the general form of a quadratic equation, and what is the quadratic formula for its roots?

    General form: $ax^{2} + bx + c = 0$, $a \neq 0$. Roots: $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

18. Define the discriminant of a quadratic and give its formula.

    The discriminant is $\Delta = b^{2} - 4ac$; it determines the nature of the roots of $ax^{2}+bx+c=0$.

19. Classify the nature of the roots of $ax^{2}+bx+c=0$ based on the discriminant $\Delta = b^{2}-4ac$.

    If $\Delta > 0$: two distinct real roots. If $\Delta = 0$: one repeated (equal) real root. If $\Delta < 0$: two complex conjugate (non-real) roots.

20. When are the roots of a quadratic with rational coefficients rational, and when are they irrational conjugates?

    If $\Delta = b^{2}-4ac$ is a perfect square (and $\geq 0$), the roots are rational; if $\Delta > 0$ but not a perfect square, the roots are irrational and occur as conjugate pairs $p \pm \sqrt{q}$.

21. For $ax^{2}+bx+c=0$ with roots $\alpha$ and $\beta$, state the sum and product of the roots.

    Sum: $\alpha + \beta = -\dfrac{b}{a}$. Product: $\alpha\beta = \dfrac{c}{a}$.

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