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Chapter 1: Quadratic Equations
Introduction
An equation of the form ax² + bx + c = 0, where a ≠ 0, is called a quadratic equation.
Methods of Solving
1. Factorization
x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3
2. Completing the Square
x² + bx + c = 0
(x + b/2)² = (b/2)² - c
x = -b/2 ± √(D)/2a
3. Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Discriminant (D)
D = b² - 4ac
- D > 0: Two real and unequal roots
- D = 0: Two real and equal roots
- D < 0: No real roots
Sum and Product of Roots
α + β = -b/a
αβ = c/a
Chapter 2: Theory of Quadratic Equations
Nature of Roots
- Positive D: Real roots
- Perfect Square D: Rational roots
- Zero D: Equal roots
Formation of Quadratic Equation
x² - (sum of roots)x + (product of roots) = 0
Maximum and Minimum Values
For ax² + bx + c:
Maximum = c - b²/4a (when a < 0)
Minimum = c - b²/4a (when a > 0)
Chapter 3: Variations
Direct Variation
x ∝ y means x = ky (k = constant)
Inverse Variation
x ∝ 1/y means xy = k
Joint Variation
x ∝ yz means x = kyz
K-Method
If x ∝ y and x = 4 when y = 2
x = ky → 4 = k(2) → k = 2
When y = 5: x = 2(5) = 10
Chapter 4: Partial Fractions
Types of Proper Fractions
1. Linear Factors (Distinct)
(x+3)/(x-1)(x-2) = A/(x-1) + B/(x-2)
2. Linear Factors (Repeated)
2x/(x-1)² = A/(x-1) + B/(x-1)²
3. Non-Repeated Irreducible Quadratic
5x/(x²+1)(x-1) = (Ax+B)/(x²+1) + C/(x-1)
Chapter 5: Sets and Functions
Sets
A collection of well-defined objects.
Types of Sets
- Empty Set (∅): No elements
- Subset (⊆): All elements in A are in B
- Universal Set (U): Contains all sets under consideration
- Complement (A'): Elements not in A
Operations on Sets
- Union (∪): All elements in A or B
- Intersection (∩): Common elements
- Difference (A-B): Elements in A but not in B
De Morgan's Laws
(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'
Functions
A relation where each input has exactly one output.
- One-to-One: f(x₁) = f(x₂) → x₁ = x₂
- Onto: Every element in codomain has pre-image
- Bijective: Both one-to-one and onto
Chapter 6: Basic Statistics
Measures of Central Tendency
Mean
x̄ = Σxᵢ/n
Median
Middle value when data arranged in order.
- Odd n: (n+1)/2 th term
- Even n: Average of n/2 and (n/2 + 1)th terms
Mode
Most frequent value.
Measures of Dispersion
Variance
σ² = Σ(xᵢ - x̄)² / n
Standard Deviation
σ = √(Variance)
Chapter 7: Trigonometry
Trigonometric Ratios
sin θ = Opposite/Hypotenuse
cos θ = Adjacent/Hypotenuse
tan θ = Opposite/Adjacent
Reciprocal Relations
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Angle Formulas
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
Double Angle Formulas
sin 2A = 2 sinA cosA
cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
Chapter 8: Circles
Theorems
1. Equal chords subtend equal angles at the center
2. Perpendicular from center to chord bisects it
3. Angle subtended by diameter is a right angle
If AB is diameter, then ∠APB = 90°
4. Tangent is perpendicular to radius
Circumcircle and Incircle
- Circumcircle: Circle passing through all vertices
- Incircle: Circle touching all sides
Arc Length
L = (θ/360°) × 2πr
Sector Area
A = (θ/360°) × πr²
Chapter 9: Practical Geometry
Tangent from External Point
Two tangents can be drawn from an external point to a circle. They are equal in length.
Circle Through Three Points
A unique circle passes through three non-collinear points.
Construction Problems
- Construct triangle given SAS, ASA, SSS
- Draw tangent to circle from external point
- Construct circumcircle and incircle
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