Class-X CBSE Mathematics Study Notes

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.

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.

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

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.

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 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).

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.

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.

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.

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.

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.

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.