Chapter 1: Real Numbers

  • Euclid’s Division Lemma: a = bq + r, where 0 ≤ r < b.

  • HCF × LCM = Product of Two Numbers

Chapter 2: Polynomials

  • Quadratic Polynomial: ax² + bx + c, where a ≠ 0.

  • Sum of Roots: α + β = -b/a.

  • Product of Roots: αβ = c/a.

Chapter 3: Pair of Linear Equations in Two Variables

  • Standard Form: ax + by + c = 0.

  • Solution using Substitution/Elimination Method.

  • Condition for Consistency:

    • Unique Solution: a1/a2 ≠ b1/b2.

    • Infinite Solutions: a1/a2 = b1/b2 = c1/c2.

    • No Solution: a1/a2 = b1/b2 ≠ c1/c2.

Chapter 4: Quadratic Equations

  • Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a.

  • Nature of Roots:

    • b² - 4ac > 0: Two distinct real roots.

    • b² - 4ac = 0: Two equal real roots.

    • b² - 4ac < 0: No real roots.

Chapter 5: Arithmetic Progression

  • nth Term: a_n = a + (n-1)d.

  • Sum of n Terms: S_n = n/2 [2a + (n-1)d].

  • Sum of n Terms (Alternate): S_n = n/2 (a + l), where l is the last term.

Chapter 6: Triangles

  • Similarity Criterion: AA, SAS, SSS.

  • Pythagoras Theorem: In a right triangle, a² + b² = c².

  • Area Ratio of Similar Triangles: Ratio of areas = (Ratio of corresponding sides)².

Chapter 7: Coordinate Geometry

  • Distance Formula: d = √[(x2 - x1)² + (y2 - y1)²].

  • Midpoint Formula: M = [(x1 + x2)/2, (y1 + y2)/2].

  • Area of Triangle: A = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.

Chapter 8: Introduction to Trigonometry

  • Trigonometric Ratios:

    • sinθ = Opposite/Hypotenuse.

    • cosθ = Adjacent/Hypotenuse.

    • tanθ = Opposite/Adjacent.

    • cosecθ = 1/sinθ.

    • secθ = 1/cosθ.

    • cotθ = 1/tanθ.

  • Trigonometric Identities:

    • sin²θ + cos²θ = 1.

    • 1 + tan²θ = sec²θ.

    • 1 + cot²θ = cosec²θ.

Chapter 9: Some Applications of Trigonometry

  • Height and Distance Formulas:

    • Angle of Elevation and Depression.

    • tanθ = Opposite/Adjacent.

Chapter 10: Circles

  • Tangent Properties:

    • A tangent to a circle is perpendicular to the radius at the point of tangency.

    • Length of Tangents from an External Point: Equal.

Chapter 11: Constructions

  • Division of a Line Segment: Using basic proportionality theorem.

  • Construction of Tangents to a Circle: From a point outside it.

Chapter 12: Areas Related to Circles

  • Circumference of Circle: 2πr.

  • Area of Circle: πr².

  • Area of Sector: (θ/360) × πr².

  • Length of Arc: (θ/360) × 2πr.

Chapter 13: Surface Areas and Volumes

  • Surface Area:

    • Sphere: 4πr².

    • Hemisphere: 3πr².

    • Cylinder: 2πr(h + r).

    • Cone: πr(l + r).

  • Volume:

    • Sphere: (4/3)πr³.

    • Hemisphere: (2/3)πr³.

    • Cylinder: πr²h.

    • Cone: (1/3)πr²h.

Chapter 14: Statistics

  • Mean: Mean = ∑fx / ∑f.

  • Median: Median = L + [(N/2 - CF)/f] × h.

  • Mode: Mode = L + [(f1 - f0)/(2f1 - f0 - f2)] × h.

Chapter 15: Probability

  • Probability Formula: P(E) = Number of favorable outcomes / Total number of outcomes.

  • Range of Probability: 0 ≤ P(E) ≤ 1.