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FAST-NU Entry Test Advanced Mathematics Syllabus

The Advanced Mathematics syllabus for FAST-NU Entry Test in full: 11 chapters and 42 topics, every one listed below, plus 51 Advanced Mathematics flashcards with answers.

11Chapters
42Topics
51Flashcards

Advanced Mathematics syllabus: all 11 chapters

Expand any chapter to see its topics and sub-topics.

  1. Number Systems and Complex Numbers

     2 topics
    • Real and Rational Numbers
    • Complex Numbers

  2. Sets, Functions and Groups

     4 topics
    • Set Operations and Venn Diagrams
    • Relations and Functions
    • Inverse and Composite Functions
    • Binary Operations and Groups

  3. Matrices and Determinants

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

  4. Quadratic Equations

     3 topics
    • Roots and Nature of Roots
    • Sum and Product of Roots
    • Equations Reducible to Quadratic Form

  5. Partial Fractions and Sequences

     4 topics
    • Resolution into Partial Fractions
    • Arithmetic Progressions
    • Geometric Progressions
    • Harmonic and Infinite Series

  6. Binomial Theorem and Induction

     4 topics
    • Mathematical Induction
    • Binomial Theorem for Positive Index
    • Binomial Series for Rational Index
    • General and Middle Terms

  7. Permutations, Combinations and Probability

     4 topics
    • Factorial and Counting Principles
    • Permutations
    • Combinations
    • Probability

  8. Trigonometry

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

  9. Analytic Geometry

     4 topics
    • Straight Lines
    • Circle
    • Conic Sections
    • Vectors in Plane and Space

  10. Differential Calculus

     4 topics
    • Limits and Continuity
    • Derivatives from First Principles
    • Rules of Differentiation
    • Application of Derivatives

  11. Integral Calculus

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

FAST-NU Entry Test Advanced Mathematics flashcards — 51 questions with 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.

Show the other 31 Advanced Mathematics flashcards

  1. 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.

  2. 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).

  3. Define the composite function (f∘g)(x).

    (f∘g)(x) = f(g(x)): apply g first, then f. Its domain is the set of x in dom(g) for which g(x) lies in dom(f).

  4. State the condition for a function to have an inverse and how to find f⁻¹.

    A function has an inverse iff it is bijective (one-to-one). To find f⁻¹: set y = f(x), swap x and y, then solve for y.

  5. What is the key relationship between f and f⁻¹?

    f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Their graphs are reflections of each other across the line y = x.

  6. What is (f∘g)⁻¹ in terms of f⁻¹ and g⁻¹?

    (f∘g)⁻¹ = g⁻¹∘f⁻¹ — the inverse of a composition reverses the order of the inverses.

  7. Define a binary operation on a set S.

    A binary operation * on S is a rule assigning to each ordered pair (a, b) ∈ S×S a unique element a*b ∈ S (closure under *).

  8. What four axioms must (G, *) satisfy to be a group?

    Closure, Associativity, existence of an Identity element e (a*e = a), and existence of an Inverse for each element (a*a⁻¹ = e).

  9. What additional property makes a group abelian (commutative)?

    A group is abelian if a*b = b*a for all a, b ∈ G — the operation is commutative.

  10. Distinguish a semigroup, a monoid, and a group.

    Semigroup: closed + associative. Monoid: semigroup + identity. Group: monoid + every element has an inverse.

  11. Define a square matrix, a diagonal matrix, and a scalar matrix.

    Square: equal rows and columns (n×n). Diagonal: square with all off-diagonal entries 0. Scalar: a diagonal matrix with all diagonal entries equal.

  12. Define an identity matrix and a null (zero) matrix.

    Identity I: diagonal matrix with 1s on the diagonal (AI = IA = A). Null matrix: all entries are 0 (A + 0 = A).

  13. What is a symmetric matrix versus a skew-symmetric matrix?

    Symmetric: Aᵀ = A (aᵢⱼ = aⱼᵢ). Skew-symmetric: Aᵀ = −A (aᵢⱼ = −aⱼᵢ), forcing zeros on the main diagonal.

  14. State the rule for matrix multiplication conformability and the order of the product.

    To compute A·B, columns of A must equal rows of B. If A is m×n and B is n×p, AB is m×p. Matrix multiplication is generally non-commutative (AB ≠ BA).

  15. Define the transpose of a matrix and state (AB)ᵀ.

    Transpose Aᵀ swaps rows and columns. (AB)ᵀ = BᵀAᵀ (order reverses); also (Aᵀ)ᵀ = A and (A+B)ᵀ = Aᵀ+Bᵀ.

  16. How is the determinant of a 2×2 matrix [[a,b],[c,d]] computed?

    det = ad − bc. For 2×2 matrices the determinant is the cross-product difference of the diagonals.

  17. How do you expand the determinant of a 3×3 matrix by cofactors along the first row?

    For [[a,b,c],[d,e,f],[g,h,i]]: det = a(ei−fh) − b(di−fg) + c(dh−eg), using alternating + − + signs with 2×2 minors.

  18. State two key properties of determinants under row operations.

    Swapping two rows multiplies det by −1; multiplying a row by k multiplies det by k. If two rows are identical or a row is all zeros, det = 0.

  19. What is det(AB) and det(Aᵀ) in terms of det(A) and det(B)?

    det(AB) = det(A)·det(B), and det(Aᵀ) = det(A). Also det(kA) = kⁿ·det(A) for an n×n matrix.

  20. Define a singular matrix and a non-singular matrix.

    Singular: det = 0 (no inverse exists). Non-singular: det ≠ 0 (invertible). Only non-singular square matrices have inverses.

  21. State the adjoint formula for the inverse of a matrix.

    A⁻¹ = (1/det A)·adj(A), where adj(A) is the transpose of the cofactor matrix. Requires det A ≠ 0.

  22. Give the quick formula for the inverse of a 2×2 matrix [[a,b],[c,d]].

    A⁻¹ = 1/(ad−bc) · [[d, −b],[−c, a]], provided ad − bc ≠ 0.

  23. State the property (AB)⁻¹ and the relation between A⁻¹ and A.

    (AB)⁻¹ = B⁻¹A⁻¹ (order reverses). Also A·A⁻¹ = A⁻¹·A = I and det(A⁻¹) = 1/det(A).

  24. State Cramer's rule for solving a linear system AX = B.

    For a non-singular system, each variable xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with its i-th column replaced by the constants B.

  25. Describe the matrix-inverse method for solving AX = B.

    If A is non-singular, the unique solution is X = A⁻¹B. Compute A⁻¹ and multiply by the constant vector B.

  26. How does the determinant of the coefficient matrix indicate the nature of a linear system's solutions?

    If det A ≠ 0: unique solution. If det A = 0: the system has either no solution (inconsistent) or infinitely many solutions (dependent).

  27. What is the discriminant of a quadratic ax² + bx + c = 0, and what does it determine?

    Discriminant D = b² − 4ac. It determines the nature (real/complex, distinct/equal) of the roots without solving the equation.

  28. Give the nature of roots for each case of the discriminant D = b² − 4ac.

    D > 0: two real distinct roots. D = 0: two real equal (repeated) roots. D < 0: two complex conjugate roots. D a perfect square (D>0): rational distinct roots.

  29. State the quadratic formula for the roots of ax² + bx + c = 0.

    x = [−b ± √(b²−4ac)] / (2a), giving the two roots of the quadratic equation.

  30. For ax² + bx + c = 0 with roots α and β, what are the sum and product of the roots?

    Sum α + β = −b/a; Product αβ = c/a. (Vieta's formulas for a quadratic.)

  31. How do you form a quadratic equation given the sum S and product P of its roots?

    The equation is x² − Sx + P = 0, i.e. x² − (sum of roots)x + (product of roots) = 0.

FAST-NU Entry Test Advanced Mathematics FAQ

What is in the FAST-NU Entry Test Advanced Mathematics syllabus?

FAST-NU Entry Test Advanced Mathematics is split into 11 chapters: Number Systems and Complex Numbers, Sets, Functions and Groups, Matrices and Determinants, Quadratic Equations, Partial Fractions and Sequences, Binomial Theorem and Induction, Permutations, Combinations and Probability, Trigonometry, Analytic Geometry, Differential Calculus and Integral Calculus. Together they contain 42 topics.

How many chapters are in FAST-NU Entry Test Advanced Mathematics?

11 chapters, containing 42 topics. The largest chapter is Trigonometry, with 5 topics.

Are there flashcards for FAST-NU Entry Test Advanced Mathematics?

Yes — a 51-card Advanced Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.