ECAT 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 ECAT, not a summary of it.

1. 1

   Number Systems, Sets, Functions and Groups

   3 topics

   • Complex numbers and their operations
   • Sets, relations and functions
   • Binary operations and groups
2. 2

   Matrices and Determinants

   4 topics

   • Types of matrices and operations
   • Determinants and properties
   • Inverse of a matrix and adjoint
   • Solution of linear systems
3. 3

   Quadratic Equations

   4 topics

   • Solution of quadratic equations
   • Relation between roots and coefficients
   • Nature of roots
   • Equations reducible to quadratic form
4. 4

   Partial Fractions, Sequences and Series

   4 topics

   • Resolution into partial fractions
   • Arithmetic sequences and series
   • Geometric and harmonic sequences
   • Sum to infinity and means
5. 5

   Permutation, Combination and Probability

   3 topics

   • Permutations
   • Combinations
   • Basic probability
6. 6

   Mathematical Induction and Binomial Theorem

   3 topics

   • Principle of mathematical induction
   • Binomial theorem for positive index
   • Binomial series for any index
7. 7

   Trigonometry

   5 topics

   • Measurement of angles and circular functions
   • Trigonometric identities
   • Trigonometric equations
   • Inverse trigonometric functions
   • Solution of triangles
8. 8

   Functions, Limits and Differentiation

   4 topics

   • Limits and continuity
   • Differentiation from first principles
   • Rules of differentiation
   • Applications of derivatives
9. 9

   Integration

   4 topics

   • Indefinite integrals and methods
   • Definite integrals
   • Area under the curve
   • Differential equations
10. 10

    Analytic Geometry

    3 topics

    • Straight lines and pair of lines
    • Circle
    • Conic sections
11. 11

    Vectors

    4 topics

    • Vectors in plane and space
    • Scalar (dot) product
    • Vector (cross) product
    • Scalar triple product

Mathematics flashcards for ECAT

21 of 51 cards from the Mathematics deck — real questions with worked answers.

1. What is the value of i^2, i^3, and i^4 where i is the imaginary unit?

   i^2 = -1, i^3 = -i, i^4 = 1. The powers of i cycle with period 4.

2. How do you find the modulus of a complex number z = a + bi?

   |z| = sqrt(a^2 + b^2). It is the distance from the origin to the point (a, b) in the Argand plane.

3. What is the conjugate of z = a + bi, and what is the product z times its conjugate?

   The conjugate is z-bar = a - bi. Their product is z * z-bar = a^2 + b^2 = |z|^2 (a real number).

4. How do you divide two complex numbers, e.g. (a+bi)/(c+di)?

   Multiply numerator and denominator by the conjugate of the denominator: (a+bi)(c-di) / (c^2 + d^2), then simplify into real and imaginary parts.

5. State De Moivre's theorem for a complex number in polar form.

   (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), valid for all integers n (and rational n for principal values).

6. What is the polar (trigonometric) form of a complex number z = a + bi?

   z = r(cos θ + i sin θ), where r = |z| = sqrt(a^2+b^2) and θ = arg(z) = arctan(b/a) (adjusted for quadrant).

7. Define the Cartesian product A × B of two sets A and B.

   A × B is the set of all ordered pairs (a, b) where a is in A and b is in B. Its cardinality is |A| × |B|.

8. What are the three defining properties of an equivalence relation?

   Reflexive (a R a), Symmetric (a R b implies b R a), and Transitive (a R b and b R c implies a R c).

9. What is the difference between a relation and a function from set A to set B?

   A relation is any subset of A × B. A function assigns to EACH element of A exactly ONE element of B (every input has exactly one output).

10. Define an injective (one-to-one) function.

    A function f is injective if distinct inputs give distinct outputs: f(x1) = f(x2) implies x1 = x2. No two elements map to the same image.

11. Define a surjective (onto) 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. the range equals the codomain.

12. What four properties must a set with a binary operation satisfy to be a group?

    Closure, Associativity, existence of an Identity element, and existence of an Inverse for every element.

13. What additional property does an abelian (commutative) group have beyond the group axioms?

    Commutativity: a * b = b * a for all elements a and b in the group.

14. Define a binary operation on a set S.

    A binary operation is a function that takes two elements of S and produces a single element of S (i.e. * : S × S → S), satisfying closure.

15. When are two matrices equal?

    Two matrices are equal if and only if they have the same order (same dimensions) and all corresponding entries are equal.

16. What is a symmetric matrix? Give its defining condition.

    A square matrix A is symmetric if A^T = A, i.e. a_ij = a_ji for all i, j.

17. What is the difference between a diagonal matrix and a scalar matrix?

    A diagonal matrix has zeros off the main diagonal. A scalar matrix is a diagonal matrix in which all the diagonal entries are equal.

18. Under what condition can two matrices A and B be multiplied (AB)?

    The number of columns of A must equal the number of rows of B. If A is m×n and B is n×p, then AB is m×p.

19. Is matrix multiplication commutative? State the general rule.

    No, matrix multiplication is generally not commutative: AB ≠ BA in general. It is, however, associative and distributive over addition.

20. What is the transpose of a matrix, and what is (AB)^T equal to?

    The transpose interchanges rows and columns. (AB)^T = B^T A^T (the reverse-order law).

21. How do you compute the determinant of a 2×2 matrix [[a, b], [c, d]]?

    det = ad - bc.

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