ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry Syllabus

Mathematics (Part I) - 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 (Part I) - Algebra and Trigonometry in ICS (Intermediate in Computer Science), not a summary of it.

1. 1

   Number Systems

   3 topics

   • Real Numbers and Their Properties
   • Complex Numbers
   • Properties of Real and Complex Numbers
2. 2

   Sets, Functions and Groups

   4 topics

   • Sets and Operations on Sets
   • Venn Diagrams and Laws
   • Relations and Functions
   • Binary Operations and Groups
3. 3

   Matrices and Determinants

   5 topics

   • Types of Matrices
   • Matrix Operations
   • Determinants and Properties
   • Inverse of a Matrix
   • Solving Systems of Linear Equations
4. 4

   Quadratic Equations

   4 topics

   • Solution of Quadratic Equations
   • Nature of the Roots
   • Sum and Product of Roots
   • Equations Reducible to Quadratic Form
5. 5

   Partial Fractions

   3 topics

   • Proper and Improper Fractions
   • Resolution into Partial Fractions
   • Linear and Quadratic Factors
6. 6

   Sequences and Series

   4 topics

   • Arithmetic Progression
   • Geometric Progression
   • Harmonic Progression
   • Infinite Geometric Series
7. 7

   Permutation, Combination and Probability

   4 topics

   • Factorial Notation
   • Permutations
   • Combinations
   • Basic Probability
8. 8

   Mathematical Induction and Binomial Theorem

   4 topics

   • Principle of Mathematical Induction
   • Binomial Theorem for Positive Integral Index
   • General Term and Middle Term
   • Binomial Series
9. 9

   Fundamentals of Trigonometry

   3 topics

   • Units of Angular Measurement
   • Trigonometric Ratios
   • Trigonometric Identities
10. 10

    Trigonometric Identities of Sum and Difference of Angles

    3 topics

    • Sum and Difference Formulae
    • Double and Half Angle Identities
    • Product to Sum Formulae
11. 11

    Application of Trigonometry

    4 topics

    • Solution of Triangles
    • Law of Sines and Cosines
    • Area of a Triangle
    • Circles Connected with Triangles
12. 12

    Inverse Trigonometric Functions and Trigonometric Equations

    3 topics

    • Graphs of Trigonometric Functions
    • Inverse Trigonometric Functions
    • Solving Trigonometric Equations

Mathematics (Part I) - Algebra and Trigonometry flashcards for ICS (Intermediate in Computer Science)

20 of 52 cards from the Mathematics (Part I) - Algebra and Trigonometry deck — real questions with worked answers.

1. Define a rational number and give its standard form.

   A number expressible as p/q where p and q are integers and q ≠ 0. In standard form q > 0 and HCF(p, q) = 1 (the fraction is in lowest terms).

2. What distinguishes a rational number from an irrational number in terms of decimal expansion?

   A rational number has a terminating or recurring (repeating) decimal expansion; an irrational number has a non-terminating, non-recurring decimal expansion.

3. State the closure, commutative, associative, identity and inverse properties for addition of real numbers.

   For all real a, b, c: closure (a+b is real), commutative (a+b=b+a), associative ((a+b)+c=a+(b+c)), additive identity 0 (a+0=a), additive inverse −a (a+(−a)=0).

4. State the distributive property of multiplication over addition for real numbers.

   For all real a, b, c: a(b+c) = ab + ac (left distributive) and (b+c)a = ba + ca (right distributive).

5. State the trichotomy property of real numbers.

   For any two real numbers a and b, exactly one of the following holds: a < b, a = b, or a > b.

6. State the transitive property of inequality for real numbers.

   For real a, b, c: if a < b and b < c, then a < c (similarly for >, ≤, ≥).

7. What is the additive identity and the multiplicative identity in the real numbers?

   The additive identity is 0 (a + 0 = a); the multiplicative identity is 1 (a · 1 = a).

8. Which real number has no multiplicative inverse, and why?

   0 has no multiplicative inverse because there is no real number x such that 0 · x = 1.

9. Define a complex number and name its real and imaginary parts.

   A complex number is z = a + bi where a, b are real and i = √(−1). Here a is the real part Re(z) and b is the imaginary part Im(z).

10. What is the value of i, i², i³ and i⁴?

    i = √(−1), i² = −1, i³ = −i, i⁴ = 1. Powers of i cycle with period 4.

11. Define the conjugate of a complex number z = a + bi and state z·z̄.

    The conjugate is z̄ = a − bi. Their product z·z̄ = a² + b² (a non-negative real number).

12. Define the modulus (absolute value) of a complex number z = a + bi.

    |z| = √(a² + b²), the distance of the point (a, b) from the origin in the complex plane.

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

    Multiply numerator and denominator by the conjugate of the denominator: (a+bi)(c−di) / (c²+d²), then separate into real and imaginary parts.

14. Give the formula for multiplying two complex numbers (a+bi)(c+di).

    (a+bi)(c+di) = (ac − bd) + (ad + bc)i.

15. When are two complex numbers a+bi and c+di equal?

    They are equal if and only if their real parts are equal and their imaginary parts are equal: a = c and b = d.

16. How is a complex number z = a + bi represented geometrically?

    As the point (a, b) in the Argand (complex) plane, with the x-axis as the real axis and the y-axis as the imaginary axis.

17. State the properties of the conjugate: conjugate of a sum and of a product.

    Conjugate of (z₁ + z₂) = z̄₁ + z̄₂, and conjugate of (z₁ · z₂) = z̄₁ · z̄₂. Also (z̄)¯ = z.

18. Define a set and what is meant by the empty set.

    A set is a well-defined collection of distinct objects (elements). The empty (null) set, denoted ∅ or { }, contains no elements.

19. Distinguish between a subset and a proper subset.

    A ⊆ B means every element of A is in B. A is a proper subset (A ⊂ B) if A ⊆ B and A ≠ B (B has at least one element not in A).

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

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

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