Class-X CBSE Mathematics Study Notes

Class-X CBSE Board Mathematics Study Notes

Linear Equations

• Definition: An equation of the form Ax + B = 0, where A and B are constants, and A ≠ 0.
• Types:
  • One Variable: e.g., 2x + 3 = 0
  • Two Variables: e.g., 3x + 4y = 12
• Solution Methods:
  • Graphical method
  • Substitution method
  • Elimination method
• Graph: Represents a straight line on a coordinate plane.

Geometry

• Basic Concepts:
  • Points, lines, angles, polygons, circles.
• Triangles:
  • Types: Equilateral, Isosceles, Scalene.
  • Properties: Angle sum property, congruence criteria (SSS, SAS, ASA, AAS, RHS).
• Quadrilaterals:
  • Types: Parallelogram, rectangle, square, rhombus, trapezium.
  • Properties: Sum of interior angles = 360°.
• Circles:
  • Key terms: Radius, diameter, chord, tangent, sector.
  • Properties: Angle subtended by a diameter is a right angle.

Quadratic Equations

• Standard Form: ax² + bx + c = 0, where a ≠ 0.
• Solutions:
  • Factoring method
  • Completing the square
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
• Nature of Roots:
  • Real and distinct: b² - 4ac > 0
  • Real and equal: b² - 4ac = 0
  • Complex roots: b² - 4ac < 0

Polynomials

• Definition: An expression of the form P(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.
• Types:
  • Monomial: One term (e.g., 5x²)
  • Binomial: Two terms (e.g., x² - 3)
  • Trinomial: Three terms (e.g., x² + 4x + 4)
• Operations:
  • Addition, subtraction, multiplication, and division.
• Factorization: Expressing a polynomial as a product of its factors.

Statistics

• Data Types: Qualitative and quantitative.
• Measures of Central Tendency:
  • Mean: Average of data set.
  • Median: Middle value when data is arranged in order.
  • Mode: Most frequently occurring value.
• Graphical Representation:
  • Bar graphs, histograms, pie charts.

Coordinate Geometry

• Coordinate System: Cartesian plane with x-axis and y-axis.
• Distance Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).
• Section Formula: For dividing a line segment in ratio m:n:
  • (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n).
• Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Slope of a Line: m = (y₂ - y₁)/(x₂ - x₁).

Linear Equations

• An equation format: Ax + B = 0, with constants A and B, where A ≠ 0.
• Types of linear equations:
  • One variable (e.g., 2x + 3 = 0) involves a single variable.
  • Two variables (e.g., 3x + 4y = 12) involves an interaction between two variables.
• Methods for solving linear equations include:
  • Graphical method, visual representation of equations.
  • Substitution method, replacing variables to simplify.
  • Elimination method, removing one variable to solve for another.
• The graph of a linear equation forms a straight line in a coordinate plane.

Geometry

• Fundamental elements include points, lines, angles, polygons, and circles.
• Triangles:
  • Types include Equilateral (all sides equal), Isosceles (two sides equal), and Scalene (all sides different).
  • Properties: The angle sum property states the sum of internal angles equals 180°, with congruence criteria like SSS, SAS, ASA, AAS, and RHS.
• Quadrilaterals:
  • Types encompass Parallelogram, rectangle, square, rhombus, and trapezium.
  • The sum of interior angles in any quadrilateral is 360°.
• Circles:
  • Key terms consist of radius, diameter, chord, tangent, and sector.
  • Special property: An angle subtended by a diameter creates a right angle.

Quadratic Equations

• Standard form is represented as ax² + bx + c = 0, ensuring a ≠ 0.
• Solution methods include:
  • Factoring, breaking down into simpler expressions.
  • Completing the square, rearranging to form a perfect square.
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a facilitates finding roots.
• Nature of roots determined by the discriminant (b² - 4ac):
  • Real and distinct roots occur when b² - 4ac > 0.
  • Real and equal roots result when b² - 4ac = 0.
  • Complex roots arise when b² - 4ac < 0.

Polynomials

• Defined as P(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, consisting of terms with coefficients and variables.
• Types of polynomials include:
  • Monomial: One term (e.g., 5x²).
  • Binomial: Two terms (e.g., x² - 3).
  • Trinomial: Three terms (e.g., x² + 4x + 4).
• Operations on polynomials allow for addition, subtraction, multiplication, and division.
• Factorization involves expressing a polynomial as a product of its factors.

Statistics

• Types of data categorized as qualitative or quantitative.
• Measures of central tendency consist of:
  • Mean, calculated as the average value of a dataset.
  • Median, the middle value when data is sorted.
  • Mode, the value that appears most frequently.
• Graphical representation methods include bar graphs, histograms, and pie charts for visual data analysis.

Coordinate Geometry

• Built upon the Cartesian coordinate system, featuring x-axis and y-axis.
• Distance formula calculates the distance between two points: d = √((x₂ - x₁)² + (y₂ - y₁)²).
• Section formula assists in dividing a line segment in the ratio m:n:
  • Formula: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
• Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) computes the exact middle point between two coordinates.
• The slope of a line determined by m = (y₂ - y₁)/(x₂ - x₁) illustrates the line's steepness or incline.

Explore essential concepts of Class-X CBSE Mathematics including linear equations, geometry basics, and quadratic equations. This study guide provides definitions, types, solution methods, and key properties, ensuring a comprehensive understanding for students. Perfect for exam preparation!