KEAM Mathematics: Algebra and Trigonometry Syllabus

Mathematics: Algebra and Trigonometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics: Algebra and Trigonometry in KEAM, not a summary of it.

1. 1

   Sets, Functions and Number Systems

   3 topics

   • Sets, Relations and Functions
     • Set operations and Venn diagrams
     • Types of relations and functions
     • Composition and inverse of functions
   • Complex Numbers and Quadratic Equations
     • Algebra of complex numbers and modulus-argument
     • Roots of quadratic equations and nature of roots
   • Mathematical Reasoning and Inequalities
     • Statements, connectives and quantifiers
     • Linear inequalities and graphical solutions
2. 2

   Combinatorics and Series

   3 topics

   • Permutations and Combinations
     • Fundamental principle of counting
     • Permutations with restrictions
     • Combinations and applications
   • Binomial Theorem
     • Binomial theorem for positive integral index
     • General and middle terms
   • Sequences and Series
     • Arithmetic and geometric progressions
     • Arithmetic-geometric and special series
3. 3

   Matrices and Determinants

   3 topics

   • Matrices
     • Matrix operations and types
     • Transpose, symmetric and skew-symmetric matrices
   • Determinants
     • Properties of determinants
     • Minors, cofactors and expansion
   • Inverse and System of Equations
     • Adjoint and inverse of a matrix
     • Solution of linear equations by Cramer's rule
     • Consistency of systems
4. 4

   Trigonometry

   3 topics

   • Trigonometric Functions
     • Trigonometric ratios and identities
     • Compound, multiple and submultiple angles
     • Trigonometric equations
   • Inverse Trigonometric Functions
     • Domain, range and principal values
     • Properties and identities
   • Properties of Triangles
     • Sine and cosine rules and area
     • Heights and distances

Mathematics: Algebra and Trigonometry flashcards for KEAM

18 of 50 cards from the Mathematics: Algebra and Trigonometry deck — real questions with worked answers.

1. For a set with n elements, how many subsets and how many proper subsets does it have?

   Number of subsets = 2^n; number of proper subsets = 2^n - 1 (excluding the set itself).

2. State the distributive laws for sets relating union and intersection.

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

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

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

4. Give the inclusion-exclusion formula for n(A ∪ B ∪ C).

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

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

   Relations from A to B = 2^(mn); functions from A to B = n^m.

6. Define one-one (injective), onto (surjective), and bijective functions.

   Injective: distinct elements have distinct images. Surjective: every element of codomain is an image. Bijective: both injective and surjective (one-one and onto).

7. What are the domain and range of f(x) = 1/√(a² − x²)?

   Domain: (−a, a); Range: [1/a, ∞).

8. Define i and give the values of i², i³, i⁴ and the general powers of i.

   i = √(−1), i² = −1, i³ = −i, i⁴ = 1. Powers cycle with period 4: i^(4k)=1, i^(4k+1)=i, i^(4k+2)=−1, i^(4k+3)=−i.

9. For complex number z = a + bi, define its modulus and conjugate, and give z·z̄.

   |z| = √(a² + b²); conjugate z̄ = a − bi; z·z̄ = |z|² = a² + b².

10. State the polar (modulus-argument) form of a complex number and De Moivre's theorem.

    z = r(cos θ + i sin θ), where r = |z|, θ = arg z. De Moivre: (cos θ + i sin θ)^n = cos nθ + i sin nθ.

11. What are the cube roots of unity and their key properties?

    1, ω, ω² where ω = (−1 + i√3)/2. Properties: 1 + ω + ω² = 0 and ω³ = 1.

12. Give the roots of the quadratic ax² + bx + c = 0 and the relations between roots and coefficients.

    Roots = [−b ± √(b²−4ac)]/(2a). Sum of roots α+β = −b/a; product αβ = c/a.

13. How does the discriminant D = b² − 4ac classify the roots of a real quadratic?

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

14. Write the negation of the compound statements 'p ∧ q' and 'p ∨ q'.

    ~(p ∧ q) = ~p ∨ ~q; ~(p ∨ q) = ~p ∧ ~q (De Morgan's laws for logic).

15. Define the converse, inverse, and contrapositive of 'if p then q', and which is logically equivalent.

    Converse: q→p; Inverse: ~p→~q; Contrapositive: ~q→~p. The contrapositive is logically equivalent to the original implication.

16. State the AM-GM inequality for two positive numbers and the condition for equality.

    For a, b > 0: (a + b)/2 ≥ √(ab), with equality iff a = b.

17. State the triangle inequality for real numbers (modulus form).

    |a + b| ≤ |a| + |b| and ||a| − |b|| ≤ |a − b|.

18. Give the formulas for nPr and nCr.

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

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