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KEAM Mathematics: Algebra and Trigonometry Syllabus

The Mathematics: Algebra and Trigonometry syllabus for KEAM in full: 4 chapters, 12 topics and 28 sub-topics, every one listed below, plus 50 Mathematics: Algebra and Trigonometry flashcards with answers.

4Chapters
12Topics
28Sub-topics
50Flashcards

Mathematics: Algebra and Trigonometry syllabus: all 4 chapters

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

  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. 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. 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. 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

KEAM Mathematics: Algebra and Trigonometry flashcards — 50 questions with 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.

  19. State the relations nCr = nC(n−r) and Pascal's rule.

    nCr = nC(n−r) (symmetry); Pascal's rule: nCr + nC(r−1) = (n+1)Cr.

  20. How many arrangements of n objects are there when p of one kind, q of another, and r of another are identical?

    n! / (p! · q! · r!).

Show the other 30 Mathematics: Algebra and Trigonometry flashcards

  1. Number of circular permutations of n distinct objects, and when clockwise/anticlockwise are not distinguished?

    (n − 1)! for distinct circular arrangements; (n − 1)!/2 when reflections (e.g., a necklace) are not distinguished.

  2. Write the general (binomial) theorem expansion of (a + b)^n.

    (a + b)^n = Σ_{r=0}^{n} nCr · a^(n−r) · b^r.

  3. Give the general term T_{r+1} in the expansion of (a + b)^n.

    T_{r+1} = nCr · a^(n−r) · b^r.

  4. How do you find the term independent of x in a binomial expansion?

    Write the general term, simplify the power of x to a single exponent in r, set that exponent to 0, solve for r, then substitute back.

  5. Where does the middle term occur in the expansion of (a+b)^n for n even and n odd?

    If n is even: one middle term, the (n/2 + 1)th. If n is odd: two middle terms, the ((n+1)/2)th and ((n+3)/2)th.

  6. Give the nth term and the sum of the first n terms of an arithmetic progression (AP).

    a_n = a + (n−1)d; S_n = (n/2)[2a + (n−1)d] = (n/2)(a + l) where l is the last term.

  7. Give the nth term and the sum of the first n terms of a geometric progression (GP).

    a_n = a·r^(n−1); S_n = a(r^n − 1)/(r − 1) for r ≠ 1.

  8. Give the sum to infinity of a GP and its convergence condition.

    S_∞ = a/(1 − r), valid only when |r| < 1.

  9. State the relation between AM, GM, and HM of two positive numbers.

    AM ≥ GM ≥ HM, and GM² = AM × HM (so GM is the geometric mean of AM and HM).

  10. Give the standard summation formulas for Σn, Σn², and Σn³.

    Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².

  11. Define a symmetric matrix and a skew-symmetric matrix.

    Symmetric: A' = A (a_ij = a_ji). Skew-symmetric: A' = −A (a_ij = −a_ji), so all diagonal entries are 0.

  12. State the conditions for matrix addition and matrix multiplication to be defined.

    Addition: matrices must be of the same order. Multiplication AB: number of columns of A must equal number of rows of B.

  13. Is matrix multiplication commutative? State key properties.

    No, AB ≠ BA in general. Multiplication is associative (AB)C = A(BC) and distributive A(B+C) = AB + AC.

  14. How can any square matrix be expressed in terms of symmetric and skew-symmetric matrices?

    A = (1/2)(A + A') + (1/2)(A − A'), where (A+A')/2 is symmetric and (A−A')/2 is skew-symmetric.

  15. Give the formula for the determinant of a 2×2 matrix [[a, b],[c, d]].

    Determinant = ad − bc.

  16. State the properties det(AB) and det(kA) for an n×n matrix.

    det(AB) = det(A)·det(B); det(kA) = k^n · det(A) for an n×n matrix.

  17. How does the determinant change when two rows (or columns) are interchanged, or when a row is a multiple of another?

    Interchanging two rows/columns multiplies the determinant by −1; if two rows/columns are identical or proportional, the determinant is 0.

  18. Give the formula for the area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) using a determinant.

    Area = (1/2) |det[[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]]|.

  19. Define the adjoint of a square matrix and give A·(adj A).

    adj A = transpose of the cofactor matrix. A·(adj A) = (adj A)·A = |A|·I.

  20. State the formula for the inverse of a matrix and the condition for its existence.

    A⁻¹ = (1/|A|)·adj A, which exists iff |A| ≠ 0 (A is non-singular).

  21. For the system AX = B, how do you classify solutions using |A|?

    If |A| ≠ 0: unique solution X = A⁻¹B (consistent). If |A| = 0 and (adj A)B = 0: infinitely many solutions. If |A| = 0 and (adj A)B ≠ 0: no solution (inconsistent).

  22. State Cramer's rule for a system of equations.

    For AX = B with D = |A| ≠ 0, each variable x_i = D_i / D, where D_i is D with the i-th column replaced by B.

  23. Give the values of sin, cos, and tan at 0°, 30°, 45°, 60°, and 90°.

    sin: 0, 1/2, 1/√2, √3/2, 1. cos: 1, √3/2, 1/√2, 1/2, 0. tan: 0, 1/√3, 1, √3, undefined.

  24. State the three Pythagorean trigonometric identities.

    sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

  25. Write the compound angle formulas for sin(A±B) and cos(A±B).

    sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB.

  26. Give the double angle formulas for sin 2θ, cos 2θ, and tan 2θ.

    sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1; tan 2θ = 2tanθ/(1 − tan²θ).

  27. Give the general solutions of sin θ = 0, cos θ = 0, and tan θ = 0.

    sin θ = 0 ⇒ θ = nπ; cos θ = 0 ⇒ θ = (2n+1)π/2; tan θ = 0 ⇒ θ = nπ, for n ∈ ℤ.

  28. State the principal value ranges (domains/ranges) of sin⁻¹x, cos⁻¹x, and tan⁻¹x.

    sin⁻¹x: domain [−1,1], range [−π/2, π/2]; cos⁻¹x: domain [−1,1], range [0, π]; tan⁻¹x: domain ℝ, range (−π/2, π/2).

  29. State the complementary identities sin⁻¹x + cos⁻¹x and tan⁻¹x + cot⁻¹x.

    sin⁻¹x + cos⁻¹x = π/2 (for x ∈ [−1,1]); tan⁻¹x + cot⁻¹x = π/2 (for all real x).

  30. State the Law of Sines and the Law of Cosines for a triangle.

    Law of Sines: a/sinA = b/sinB = c/sinC = 2R. Law of Cosines: a² = b² + c² − 2bc·cosA (and cyclic forms).

KEAM Mathematics: Algebra and Trigonometry FAQ

What is in the KEAM Mathematics: Algebra and Trigonometry syllabus?

KEAM Mathematics: Algebra and Trigonometry is split into 4 chapters: Sets, Functions and Number Systems, Combinatorics and Series, Matrices and Determinants and Trigonometry. Together they contain 12 topics and 28 sub-topics.

How many chapters are in KEAM Mathematics: Algebra and Trigonometry?

4 chapters, containing 12 topics and 28 sub-topics. The largest chapter is Sets, Functions and Number Systems, with 3 topics.

Are there flashcards for KEAM Mathematics: Algebra and Trigonometry?

Yes — a 50-card Mathematics: Algebra and Trigonometry 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.