FAST-NU Entry Test Advanced Mathematics Syllabus

Advanced Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Advanced Mathematics in FAST-NU Entry Test, not a summary of it.

1. 1

   Number Systems and Complex Numbers

   2 topics

   • Real and Rational Numbers
   • Complex Numbers
2. 2

   Sets, Functions and Groups

   4 topics

   • Set Operations and Venn Diagrams
   • Relations and Functions
   • Inverse and Composite Functions
   • Binary Operations and Groups
3. 3

   Matrices and Determinants

   4 topics

   • Types of Matrices and Operations
   • Determinants and Properties
   • Inverse of a Matrix
   • Solving Linear Systems
4. 4

   Quadratic Equations

   3 topics

   • Roots and Nature of Roots
   • Sum and Product of Roots
   • Equations Reducible to Quadratic Form
5. 5

   Partial Fractions and Sequences

   4 topics

   • Resolution into Partial Fractions
   • Arithmetic Progressions
   • Geometric Progressions
   • Harmonic and Infinite Series
6. 6

   Binomial Theorem and Induction

   4 topics

   • Mathematical Induction
   • Binomial Theorem for Positive Index
   • Binomial Series for Rational Index
   • General and Middle Terms
7. 7

   Permutations, Combinations and Probability

   4 topics

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

   Trigonometry

   5 topics

   • Trigonometric Identities and Ratios
   • Sum, Difference and Multiple Angles
   • Solution of Triangles
   • Inverse Trigonometric Functions
   • Trigonometric Equations
9. 9

   Analytic Geometry

   4 topics

   • Straight Lines
   • Circle
   • Conic Sections
   • Vectors in Plane and Space
10. 10

    Differential Calculus

    4 topics

    • Limits and Continuity
    • Derivatives from First Principles
    • Rules of Differentiation
    • Application of Derivatives
11. 11

    Integral Calculus

    4 topics

    • Indefinite Integrals
    • Methods of Integration
    • Definite Integrals and Area
    • Differential Equations

Advanced Mathematics flashcards for FAST-NU Entry Test

22 of 51 cards from the Advanced Mathematics deck — real questions with worked answers.

1. How are the real numbers classified into two main disjoint subsets?

   Real numbers (R) split into rational numbers (Q) and irrational numbers (Q'). Q can be written as p/q with integers p, q and q ≠ 0; Q' (e.g. √2, π, e) cannot.

2. What is the defining property of a rational number's decimal expansion?

   A rational number has a decimal expansion that is either terminating or non-terminating but recurring (periodic). Irrational numbers have non-terminating, non-recurring decimals.

3. State the closure, commutative, associative, and distributive properties for real numbers under addition and multiplication.

   Closure: a+b, a·b ∈ R. Commutative: a+b=b+a, ab=ba. Associative: (a+b)+c=a+(b+c), (ab)c=a(bc). Distributive: a(b+c)=ab+ac.

4. What are the additive and multiplicative identities and inverses in R?

   Additive identity: 0 (a+0=a); additive inverse: −a. Multiplicative identity: 1 (a·1=a); multiplicative inverse of a≠0 is 1/a.

5. Prove-style fact: is the sum of a rational and an irrational number rational or irrational?

   It is always irrational. If rational + irrational were rational, then irrational = rational − rational = rational, a contradiction.

6. Define a complex number and its standard (rectangular) form.

   A complex number is z = a + bi where a, b ∈ R and i = √(−1) (i² = −1). 'a' is the real part Re(z), 'b' is the imaginary part Im(z).

7. What are the powers of i: i, i², i³, i⁴?

   i¹ = i, i² = −1, i³ = −i, i⁴ = 1. The pattern repeats every 4 powers, so iⁿ depends on n mod 4.

8. Define the conjugate of z = a + bi and give the value of z·z̄.

   Conjugate z̄ = a − bi. Product z·z̄ = a² + b² = |z|², a non-negative real number.

9. What is the modulus of z = a + bi?

   |z| = √(a² + b²), the distance from the origin to the point (a, b) in the Argand (complex) plane.

10. How do you divide complex numbers, e.g. (a+bi)/(c+di)?

    Multiply numerator and denominator by the conjugate of the denominator: [(a+bi)(c−di)] / (c²+d²), then separate into real and imaginary parts.

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

    If z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos nθ + i sin nθ) for any integer n.

12. Write the polar form of a complex number and the formula for its argument.

    z = r(cos θ + i sin θ) where r = |z| = √(a²+b²) and θ = arg(z) = tan⁻¹(b/a), adjusted for the correct quadrant.

13. List the standard set notations: union, intersection, difference, complement.

    Union A∪B (in A or B); Intersection A∩B (in both); Difference A−B (in A not B); Complement A' (in universal set U but not A).

14. State De Morgan's laws for sets.

    (A∪B)' = A'∩B' and (A∩B)' = A'∪B'. The complement of a union is the intersection of complements, and vice versa.

15. Give the distributive laws for set operations.

    A∩(B∪C) = (A∩B)∪(A∩C) and A∪(B∩C) = (A∪B)∩(A∪C).

16. What is the formula for |A∪B| (inclusion–exclusion for two sets)?

    |A∪B| = |A| + |B| − |A∩B|. For three sets: |A|+|B|+|C| − |A∩B| − |B∩C| − |A∩C| + |A∩B∩C|.

17. What do disjoint sets and a Venn diagram region for A∩B = ∅ represent?

    Disjoint sets share no elements (A∩B = ∅); in a Venn diagram their circles do not overlap.

18. Define a relation from set A to set B.

    A relation R from A to B is any subset of the Cartesian product A×B; it is a set of ordered pairs (a, b) with a∈A, b∈B.

19. What three properties define an equivalence relation?

    Reflexive (aRa), Symmetric (aRb ⇒ bRa), and Transitive (aRb and bRc ⇒ aRc). All three must hold.

20. Define a function and distinguish domain, codomain, and range.

    A function maps each element of the domain to exactly one element of the codomain. Domain = input set; Codomain = declared output set; Range = actual set of outputs.

21. Distinguish injective (one-to-one), surjective (onto), and bijective functions.

    Injective: distinct inputs give distinct outputs. Surjective: every codomain element is hit (range = codomain). Bijective: both injective and surjective.

22. What is the vertical line test for a function?

    A curve represents a function if and only if every vertical line meets the graph in at most one point (each x has one y).

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