FSC Part 1 Mathematics Notes - MatricNotes

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Chapter 1: Number Systems

Complex Numbers

Numbers of the form a + bi where i = √(-1)

i² = -1
(a+bi) + (c+di) = (a+c) + (b+d)i
(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Modulus and Argument

|z| = √(a² + b²) (Modulus)
arg(z) = tan⁻¹(b/a) (Argument)

Polar Form

z = r(cosθ + i sinθ)
z = r cis θ

De Moivre's Theorem

(cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)

Chapter 2: Matrices and Determinants

Types of Matrices

Row Matrix: Single row
Column Matrix: Single column
Square Matrix: Rows = Columns
Diagonal Matrix: Non-diagonal elements = 0
Scalar Matrix: Diagonal elements equal
Identity Matrix (I): Diagonal = 1, others = 0
Zero Matrix: All elements = 0

Matrix Operations

(A + B)ᵀ = Aᵀ + Bᵀ
(AB)ᵀ = BᵀAᵀ
(A⁻¹)ᵀ = (Aᵀ)⁻¹

Determinant

|A| = ad - bc (2×2)

Adjoint and Inverse

A⁻¹ = adj(A) / |A|

Cramer's Rule

System of linear equations solved using determinants.

Chapter 3: Quadratic Equations

Nature of Roots

D = b² - 4ac
Sum of roots = -b/a
Product of roots = c/a

Cube Roots of Unity

1, ω, ω²
ω = (-1 + i√3)/2
1 + ω + ω² = 0

Polynomial Equations

Roots satisfy: xⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ = 0

Chapter 4: Sequences and Series

Arithmetic Progression (AP)

aₙ = a + (n-1)d
Sₙ = n/2 [2a + (n-1)d]

Geometric Progression (GP)

aₙ = arⁿ⁻¹
Sₙ = a(rⁿ - 1)/(r-1) (r ≠ 1)
S∞ = a/(1-r) (|r| < 1)

Harmonic Progression (HP)

Reciprocals form AP.

Chapter 5: Permutations and Combinations

Fundamental Principle

If one task can be done in m ways and another in n ways, together: m × n ways.

Permutations

nPₙ = n!
nPᵣ = n!/(n-r)!

Combinations

nCᵣ = n!/(r!(n-r)!)
nCᵣ = nCₙ₋ᵣ

Binomial Theorem

(x + a)ⁿ = Σ nCᵣ xⁿ⁻ᵣ aᵣ
General Term: Tᵣ₊₁ = nCᵣ xⁿ⁻ᵣ aᵣ

Chapter 6: Trigonometry

Fundamental Identities

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

Sum and Difference Formulas

sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)

Product to Sum

2 sinA cosB = sin(A+B) + sin(A-B)
2 cosA cosB = cos(A+B) + cos(A-B)

Half Angle Formulas

sin(θ/2) = ±√((1-cosθ)/2)
cos(θ/2) = ±√((1+cosθ)/2)

Chapter 7: Limits and Continuity

Standard Limits

lim(x→0) sinx/x = 1
lim(x→0) (eˣ - 1)/x = 1
lim(x→0) (1+x)^(1/x) = e
lim(x→∞) (1+1/x)^x = e

Indeterminate Forms

0/0, ∞/∞, 0×∞, ∞-∞, 0⁰, ∞⁰

Chapter 8: Differentiation

Derivative Definition

f'(x) = lim(h→0) [f(x+h) - f(x)]/h

Rules

d/dx (c) = 0
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (uv) = u'v + uv'
d/dx (u/v) = (u'v - uv')/v²

Chain Rule

d/dx [f(g(x))] = f'(g(x)) × g'(x)

Trigonometric Derivatives

d/dx (sinx) = cosx
d/dx (cosx) = -sinx
d/dx (tanx) = sec²x

Exponential and Logarithmic

d/dx (eˣ) = eˣ
d/dx (lnx) = 1/x