AMUEEE 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 AMUEEE, not a summary of it.

1. 1

   Sets and Functions

   2 topics

   • Sets
   • Relations & Functions
2. 2

   Algebra

   2 topics

   • Principle of Mathematical Induction
   • Complex Numbers and Quadratic Equations
3. 3

   Coordinate Geometry

   2 topics

   • Straight Lines
   • Conic Sections
4. 4

   Calculus

   2 topics

   • Limits and Derivatives
   • Continuity and Differentiability
5. 5

   Mathematical Reasoning

   2 topics

   • Mathematically inclined statements
   • Validating the statements

Mathematics flashcards for AMUEEE

21 of 54 cards from the Mathematics deck — real questions with worked answers.

1. What is the formula for the number of subsets of a finite set with n elements?

   2^n subsets (including the empty set and the set itself); the number of proper subsets is 2^n - 1.

2. State De Morgan's laws for two sets A and B.

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

3. For finite sets, what is the inclusion-exclusion formula for n(A ∪ B ∪ C)?

   n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C).

4. Define the power set of a set A.

   The power set P(A) is the set of all subsets of A, including the empty set and A itself.

5. What is the symmetric difference A Δ B of two sets?

   A Δ B = (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B); elements in exactly one of the sets.

6. If set A has m elements and set B has n elements, how many relations exist from A to B?

   2^(mn), since a relation is any subset of the Cartesian product A × B which has mn elements.

7. Define a one-one (injective) function.

   A function f is injective if distinct elements have distinct images: f(x₁)=f(x₂) implies x₁=x₂.

8. Define an onto (surjective) function.

   A function f: A→B is surjective if every element of B is the image of at least one element of A, i.e. range = codomain.

9. What condition makes a function bijective (invertible)?

   A function is bijective if it is both one-one (injective) and onto (surjective); only bijective functions are invertible.

10. What is the domain and range of the function f(x) = 1/x?

    Domain: all real numbers except 0 (ℝ − {0}); Range: all real numbers except 0 (ℝ − {0}).

11. How many functions can be defined from a set with m elements to a set with n elements?

    n^m functions.

12. State the Principle of Mathematical Induction.

    If P(1) is true, and P(k) true implies P(k+1) true for all natural k, then P(n) is true for all natural numbers n.

13. Using induction conventions, prove the formula for 1+2+...+n.

    1+2+...+n = n(n+1)/2; verified by base case n=1 and the inductive step P(k)⇒P(k+1).

14. What is the sum of the first n squares, 1²+2²+...+n²?

    n(n+1)(2n+1)/6, a standard result provable by mathematical induction.

15. Define the imaginary unit i and give the values of i², i³, i⁴.

    i = √(−1); i² = −1, i³ = −i, i⁴ = 1 (powers of i cycle every 4).

16. What is the modulus of a complex number z = a + bi?

    |z| = √(a² + b²).

17. What is the multiplicative inverse of z = a + bi?

    z⁻¹ = (a − bi)/(a² + b²) = conjugate of z divided by |z|².

18. State the polar (modulus-argument) form of a complex number.

    z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).

19. Give the quadratic formula for ax² + bx + c = 0.

    x = [−b ± √(b² − 4ac)] / (2a).

20. What does the discriminant b²−4ac tell you about the roots of a quadratic?

    If >0: two distinct real roots; if =0: two equal real roots; if <0: two complex conjugate roots.

21. For ax²+bx+c=0, give the sum and product of the roots.

    Sum of roots = −b/a; product of roots = c/a.

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