Chapter 1
Real Numbers & Quadratic Equations
HCF and LCM Relation
HCF(a, b) × LCM(a, b) = a × b
Fundamental theorem of arithmetic product property.
Variables: Applicable ONLY for two positive integers a and b (does NOT hold for 3 numbers)
Exam Tip: If HCF(a, b) = 1, then a and b are co-prime, and LCM(a, b) = a × b.
Quadratic Formula & Nature of Roots (Discriminant)
x = (-b ± √D) / (2a) where D = b² - 4ac
D > 0: Two distinct real roots; D = 0: Two equal real roots (-b/2a); D < 0: No real roots (complex conjugates).
Variables: Quadratic equation: ax² + bx + c = 0 (a ≠ 0)
Exam Tip: Sum of roots: α + β = -b/a. Product of roots: αβ = c/a. Equation can be written as x² - (α + β)x + αβ = 0.
Chapter 2
Trigonometry & Applications
Fundamental Trigonometric Identities
sin²θ + cos²θ = 1 | 1 + tan²θ = sec²θ | 1 + cot²θ = cosec²θ
Derived directly from Pythagoras Theorem (p² + b² = h²) dividing by h², b², and p².
Variables: 0° ≤ θ ≤ 90° for standard right-angled triangle ABC
Exam Tip: sec²θ - tan²θ = 1 => (secθ - tanθ)(secθ + tanθ) = 1. Therefore, secθ - tanθ is the reciprocal of secθ + tanθ!
Standard Trigonometric Values Table
sin(0, 30, 45, 60, 90) = (0, 1/2, 1/√2, √3/2, 1) | cos(θ) is reversed | tan(θ) = sin(θ)/cos(θ)
Standard angle values essential for Heights & Distances elevation and depression problems.
Variables: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° is undefined
Exam Tip: In heights and distances, Angle of Elevation from observer = Angle of Depression from target (alternate interior angles).