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ECAT Mathematics Syllabus
The Mathematics syllabus for ECAT in full: 11 chapters and 41 topics, every one listed below, plus 51 Mathematics flashcards with answers.
Mathematics syllabus: all 11 chapters
Expand any chapter to see its topics and sub-topics.
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Number Systems, Sets, Functions and Groups
3 topics- Complex numbers and their operations
- Sets, relations and functions
- Binary operations and groups
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Matrices and Determinants
4 topics- Types of matrices and operations
- Determinants and properties
- Inverse of a matrix and adjoint
- Solution of linear systems
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Quadratic Equations
4 topics- Solution of quadratic equations
- Relation between roots and coefficients
- Nature of roots
- Equations reducible to quadratic form
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Partial Fractions, Sequences and Series
4 topics- Resolution into partial fractions
- Arithmetic sequences and series
- Geometric and harmonic sequences
- Sum to infinity and means
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Permutation, Combination and Probability
3 topics- Permutations
- Combinations
- Basic probability
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Mathematical Induction and Binomial Theorem
3 topics- Principle of mathematical induction
- Binomial theorem for positive index
- Binomial series for any index
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Trigonometry
5 topics- Measurement of angles and circular functions
- Trigonometric identities
- Trigonometric equations
- Inverse trigonometric functions
- Solution of triangles
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Functions, Limits and Differentiation
4 topics- Limits and continuity
- Differentiation from first principles
- Rules of differentiation
- Applications of derivatives
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Integration
4 topics- Indefinite integrals and methods
- Definite integrals
- Area under the curve
- Differential equations
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Analytic Geometry
3 topics- Straight lines and pair of lines
- Circle
- Conic sections
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Vectors
4 topics- Vectors in plane and space
- Scalar (dot) product
- Vector (cross) product
- Scalar triple product
ECAT Mathematics flashcards — 51 questions with answers
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.
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.
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).
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.
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).
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).
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|.
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
Show the other 31 Mathematics flashcards
How do you compute the determinant of a 2×2 matrix [[a, b], [c, d]]?
det = ad - bc.
How does swapping two rows of a matrix affect its determinant?
Swapping any two rows (or two columns) multiplies the determinant by -1 (changes its sign).
What happens to a determinant if two rows (or columns) are identical or proportional?
The determinant is 0.
How is the determinant of a product related to individual determinants? det(AB) = ?
det(AB) = det(A) × det(B).
If a single row of a matrix is multiplied by a scalar k, what happens to the determinant?
The determinant is multiplied by k. (For an n×n matrix, det(kA) = k^n det(A).)
What is the adjoint (adjugate) of a matrix?
The adjoint is the transpose of the cofactor matrix (matrix of signed minors).
State the formula for the inverse of a matrix A in terms of its adjoint and determinant.
A^(-1) = (1 / det(A)) × adj(A), valid only when det(A) ≠ 0.
When does a square matrix A have an inverse (is non-singular)?
A is invertible if and only if det(A) ≠ 0. If det(A) = 0, the matrix is singular and has no inverse.
What is the inverse of a 2×2 matrix [[a, b], [c, d]]?
(1/(ad-bc)) × [[d, -b], [-c, a]], provided ad - bc ≠ 0.
State Cramer's Rule for solving a system of linear equations.
For AX = B, each unknown x_i = det(A_i) / det(A), where A_i is A with its i-th column replaced by B. Requires det(A) ≠ 0.
What does it mean if the determinant of the coefficient matrix of a square linear system is zero?
The system has either no solution (inconsistent) or infinitely many solutions; it does not have a unique solution.
What is the quadratic formula for solving ax^2 + bx + c = 0?
x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a ≠ 0.
What is the discriminant of a quadratic equation and its symbol?
The discriminant is D = b^2 - 4ac. It determines the nature of the roots of ax^2 + bx + c = 0.
For ax^2 + bx + c = 0, what is the sum and product of the roots in terms of a, b, c?
Sum of roots = -b/a; Product of roots = c/a.
If the discriminant b^2 - 4ac > 0, what is the nature of the roots?
The roots are real, distinct (unequal). If D is also a perfect square (with rational coefficients), they are rational; otherwise irrational.
If the discriminant b^2 - 4ac = 0, what is the nature of the roots?
The roots are real and equal (a repeated/double root), each equal to -b/(2a).
If the discriminant b^2 - 4ac < 0, what is the nature of the roots?
The roots are complex (imaginary) and occur as a conjugate pair.
How do you form a quadratic equation given that its roots are α and β?
x^2 - (α + β)x + αβ = 0, i.e. x^2 - (sum of roots)x + (product of roots) = 0.
What substitution reduces the equation x^4 - 5x^2 + 4 = 0 to a quadratic?
Let y = x^2. The equation becomes y^2 - 5y + 4 = 0, solve for y, then find x = ±sqrt(y).
When is a rational fraction called 'proper' for partial fraction decomposition?
When the degree of the numerator is less than the degree of the denominator. Otherwise divide first to get a polynomial plus a proper fraction.
What is the partial fraction form for a repeated linear factor (x-a)^2 in the denominator?
A/(x-a) + B/(x-a)^2 — one term for each power up to the highest.
What partial fraction form is used for an irreducible quadratic factor (x^2 + bx + c) in the denominator?
A linear numerator over the quadratic: (Ax + B)/(x^2 + bx + c).
What is the nth term (general term) of an arithmetic sequence with first term a and common difference d?
a_n = a + (n - 1)d.
What is the sum of the first n terms of an arithmetic series?
S_n = (n/2)[2a + (n-1)d] = (n/2)(a + l), where l is the last term.
What is the arithmetic mean (A.M.) between two numbers a and b?
A.M. = (a + b)/2.
What is the nth term of a geometric sequence with first term a and common ratio r?
a_n = a × r^(n-1).
What is the sum to infinity of a geometric series, and when does it converge?
S_infinity = a/(1 - r), valid only when |r| < 1 (the series converges).
What is the relationship between the Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) of two positive numbers?
G^2 = A × H, and A ≥ G ≥ H (equality when the numbers are equal).
What is the formula for the number of permutations of n distinct objects taken r at a time?
P(n, r) = n! / (n - r)!
What is the formula for combinations of n distinct objects taken r at a time, and how does it relate to permutations?
C(n, r) = n! / [r!(n - r)!] = P(n, r) / r!. Combinations ignore order; permutations count order.
How is the probability of an event E defined for equally likely outcomes, and what is its range?
P(E) = (number of favorable outcomes) / (total number of possible outcomes). Its value lies between 0 and 1 inclusive, and P(E) + P(not E) = 1.
ECAT Mathematics FAQ
What is in the ECAT Mathematics syllabus?
ECAT Mathematics is split into 11 chapters: Number Systems, Sets, Functions and Groups, Matrices and Determinants, Quadratic Equations, Partial Fractions, Sequences and Series, Permutation, Combination and Probability, Mathematical Induction and Binomial Theorem, Trigonometry, Functions, Limits and Differentiation, Integration, Analytic Geometry and Vectors. Together they contain 41 topics.
How many chapters are in ECAT Mathematics?
11 chapters, containing 41 topics. The largest chapter is Trigonometry, with 5 topics.
Are there flashcards for ECAT Mathematics?
Yes — a 51-card Mathematics 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.