IAT 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 IAT, not a summary of it.

1. 1

   Sets and Functions

   3 topics

   • Sets
   • Relations and Functions
   • Trigonometric Functions
2. 2

   Algebra

   6 topics

   • Principle of Mathematical Induction
   • Complex Numbers and Quadratic Equations
   • Linear Inequalities
   • Permutations and Combinations
   • Binomial Theorem
   • Sequence and Series
3. 3

   Coordinate Geometry

   3 topics

   • Straight Lines
   • Conic Sections
   • Introduction to Three-dimensional Geometry
4. 4

   Calculus

   6 topics

   • Limits and Derivatives
   • Continuity and Differentiability
   • Applications of Derivatives
   • Integrals
   • Applications of Integrals
   • Differential Equations
5. 5

   Vectors and Three-Dimensional Geometry

   2 topics

   • Vectors
   • Three-Dimensional Geometry
6. 6

   Linear Programming

   1 topic

   • Linear Programming
7. 7

   Mathematical Reasoning

   1 topic

   • Mathematical Reasoning

Mathematics flashcards for IAT

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is the formula for the number of elements in the union of two finite sets A and B?

   n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

2. If a set has n elements, how many subsets and how many proper subsets does it have?

   It has 2^n subsets and 2^n − 1 proper subsets.

3. State De Morgan's laws for sets.

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

4. For a relation from set A to set B where n(A)=p and n(B)=q, how many relations are possible?

   2^(pq), since a relation is any subset of A × B which has pq elements.

5. Define a one-one (injective) function and an onto (surjective) function.

   One-one: distinct elements of the domain have distinct images (f(a)=f(b) ⇒ a=b). Onto: every element of the codomain is the image of at least one element of the domain (range = codomain).

6. What is the domain and range of the function f(x) = sin⁻¹(x)?

   Domain: [−1, 1]; Range (principal value): [−π/2, π/2].

7. Write the three reciprocal Pythagorean trigonometric identities.

   sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = cosec²x.

8. State the formulas for sin(A+B) and cos(A+B).

   sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB.

9. What is the general solution of sin x = sin y?

   x = nπ + (−1)^n y, where n ∈ ℤ.

10. What is the general solution of cos x = cos y?

    x = 2nπ ± y, where n ∈ ℤ.

11. State the principle of mathematical induction (its two steps).

    To prove P(n) for all natural numbers n: (1) Base step — show P(1) is true; (2) Inductive step — assume P(k) is true and prove P(k+1) is true. Then P(n) holds for all n ∈ ℕ.

12. What is the modulus and conjugate of the complex number z = a + ib?

    Modulus |z| = √(a² + b²); conjugate z̄ = a − ib.

13. State the values of the powers of i (i = √−1): i², i³, i⁴.

    i² = −1, i³ = −i, i⁴ = 1 (the pattern repeats with period 4).

14. What is the nature of roots of a quadratic ax² + bx + c = 0 based on the discriminant D = b² − 4ac?

    D > 0: real and distinct; D = 0: real and equal; D < 0: complex conjugate (no real roots).

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

    Sum α + β = −b/a; Product αβ = c/a.

16. When you multiply or divide both sides of an inequality by a negative number, what happens?

    The direction of the inequality sign reverses.

17. State the formulas for nPr and nCr.

    nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!].

18. How many ways can n distinct objects be arranged in a circle (circular permutations)?

    (n − 1)! ways.

19. State the general (r+1)th term in the binomial expansion of (a + b)^n.

    T(r+1) = nCr · a^(n−r) · b^r.

20. In the expansion of (a + b)^n, how many terms are there and what is the sum of the binomial coefficients?

    There are n + 1 terms, and the sum of all binomial coefficients is 2^n.

21. What is the nth term and the sum of n terms of an arithmetic progression (AP)?

    nth term: a_n = a + (n−1)d; Sum: S_n = n/2 [2a + (n−1)d] = n/2 (a + l).

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