Advertisement Space
Chapter 1: Matrices and Determinants
Matrix
A rectangular array of numbers arranged in rows and columns.
A = [aᵢⱼ]ₘₓₙ
m = rows, n = columns
Types of Matrices
- Row Matrix: 1 × n (single row)
- Column Matrix: m × 1 (single column)
- Square Matrix: m = n
- Zero Matrix: All elements = 0
- Identity Matrix: Diagonal = 1, others = 0 (denoted I)
Matrix Operations
Addition: A + B = [aᵢⱼ + bᵢⱼ]
Scalar Multiplication: kA = [kaᵢⱼ]
Multiplication: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ
Determinant
Only square matrices have determinants.
|A| for 2×2: |a b; c d| = ad - bc
|A| for 3×3: Use expansion method
Find determinant of A = [3, 2; 1, 4]
|A| = 3×4 - 2×1 = 12 - 2 = 10
Chapter 2: Real and Complex Numbers
Types of Numbers
- Natural Numbers (N): 1, 2, 3, ...
- Whole Numbers (W): 0, 1, 2, ...
- Integers (Z): ..., -2, -1, 0, 1, 2, ...
- Rational Numbers (Q): p/q where q ≠ 0
- Irrational Numbers: Cannot be expressed as p/q (π, √2)
- Real Numbers (R): Rational + Irrational
Complex Numbers
z = a + bi
a = real part, b = imaginary part
i = √(-1), i² = -1
Operations on Complex Numbers
(a+bi) + (c+di) = (a+c) + (b+d)i
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
Chapter 3: Logarithms
Definition
If aˣ = N, then logₐN = x
Logarithm Laws
log(MN) = log M + log N
log(M/N) = log M - log N
log Mⁿ = n log M
logₐ 1 = 0
logₐ a = 1
Change of Base
logₐN = log N / log a
log₁₀100 = 2 because 10² = 100
Chapter 4: Algebraic Expressions
Algebraic Sentences
- Equation: Contains = sign (e.g., 2x + 3 = 7)
- Identity: True for all values (e.g., (a+b)² = a²+2ab+b²)
- Formula: General rule expressed mathematically
Factorization
- Common Factor: a²b + ab² = ab(a + b)
- Difference of Squares: a² - b² = (a+b)(a-b)
- Trinomial: x² + (a+b)x + ab = (x+a)(x+b)
Chapter 5: Fundamental Concepts of Geometry
Terms
- Point: Has no dimension, only position
- Line: Extends infinitely in both directions
- Ray: Has one endpoint, extends infinitely
- Line Segment: Has two endpoints
- Plane: Flat surface extending infinitely
Angle Types
- Acute: 0° < θ < 90°
- Right: θ = 90°
- Obtuse: 90° < θ < 180°
- Straight: θ = 180°
- Reflex: 180° < θ < 360°
Chapter 6: Lines and Angles
Parallel Lines
Two lines in a plane that never meet.
Transversal
A line crossing two or more parallel lines.
Angle Pairs
- Corresponding: Equal (F-position)
- Alternate Interior: Equal (Z-position)
- Alternate Exterior: Equal
- Co-interior: Supplementary (C-position)
If lines are parallel:
Corresponding angles = Equal
Alternate interior = Equal
Co-interior = 180°
Chapter 7: Triangles
Types of Triangles
- By Sides: Scalene, Isosceles, Equilateral
- By Angles: Acute, Right, Obtuse
Triangle Properties
- Sum of angles = 180°
- Exterior angle = Sum of two opposite interior angles
- Side opposite largest angle is longest
Congruency (SSS, SAS, ASA, AAS, RHS)
Chapter 8: Quadrilaterals
Types
- Parallelogram: Opposite sides parallel
- Rectangle: Parallelogram + right angles
- Rhombus: Parallelogram + equal sides
- Square: Rectangle + Rhombus
- Trapezoid: One pair parallel
Properties
Parallelogram: Opposite sides =, Opposite angles =, Diagonals bisect each other
Rectangle: Diagonals =
Rhombus: Diagonals ⟂
Square: All properties
Chapter 9: Areas of Triangles and Quadrilaterals
Area Formulas
Triangle: A = ½ × base × height
Parallelogram: A = base × height
Rectangle: A = length × width
Square: A = side²
Rhombus: A = ½ × d₁ × d₂
Heron's Formula
s = (a+b+c)/2 (semi-perimeter)
A = √[s(s-a)(s-b)(s-c)]
Advertisement Space