Complete Mathematics Notes for Class 9
These comprehensive Mathematics notes cover all 13 chapters of the Class 9 Punjab Textbook Board syllabus. Each chapter includes clear theory, solved examples, step-by-step exercise solutions, important definitions, and practice MCQs with answer keys.
The notes are ideal for exam preparation and concept building, offering structured content that helps students understand and apply key mathematical principles with confidence.
Key Features:
- Chapter-wise PDF downloads available for offline study
- Solved exercises with step-by-step solutions
- Practice MCQs with detailed answer keys
- Important definitions, formulas, and theorems highlighted
- Concept maps and summary tables for quick revision
Topics Covered:
- Real Numbers & Number Systems
- Logarithms & Their Applications
- Algebraic Expressions & Factorization
- Linear Equations & Inequalities
- Trigonometry & Geometry
- Coordinate Geometry & Graphs
- Probability & Statistics
Chapter-wise Notes
π Chapter 1 β Real Numbers
Key Concepts: This chapter explains the number system, dividing numbers into rational and irrational. It covers properties of real numbers, representation on the number line, and operations with radicals.
β’ \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
β’ \(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\)
β’ \((a + b)(a - b) = a^2 - b^2\)
β’ Rationalizing denominator: \(\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}\)
β’ Commutative: \(a + b = b + a\); \(a \times b = b \times a\)
β’ Associative: \((a + b) + c = a + (b + c)\)
β’ Distributive: \(a \times (b + c) = a \times b + a \times c\)
β’ Additive Identity: \(a + 0 = a\)
β’ Multiplicative Identity: \(a \times 1 = a\)
π Exercise 1.1 β Real Numbers (Class 9 Math)
View PDFπ Exercise 1.2 β Real Numbers (Class 9 Math)
View PDFπ Exercise 1.3 β Real Numbers (Class 9 Math)
View PDFπ Review Exercise β Chapter 1 Real Numbers
View PDFπ Definitions β Chapter 1 Real Numbers
View PDFπ MCQs β Chapter 1 Real Numbers
View PDFπ Chapter 2 β Logarithms
Key Concepts: This chapter introduces logarithms as the inverse of exponents. It covers the fundamental laws of logarithms, common logarithms (base 10), and their applications in simplifying calculations. The relationship between exponents and logarithms is explored through various examples and problems.
β’ \(\log_a(mn) = \log_a m + \log_a n\)
β’ \(\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n\)
β’ \(\log_a(m^n) = n \log_a m\)
β’ \(\log_a a = 1\)
β’ \(\log_a 1 = 0\)
β’ Change of Base: \(\log_a b = \frac{\log_n b}{\log_n a}\)
π Exercise 2.1 β Logarithms (Class 9 Math)
View PDFπ Exercise 2.2 β Logarithms (Class 9 Math)
View PDFπ Exercise 2.3 β Logarithms (Class 9 Math)
View PDFπ Exercise 2.4 β Logarithms (Class 9 Math)
View PDFπ Review Exercise β Logarithms (Class 9 Math)
View PDFπ Definitions β Logarithms (Class 9 Math)
View PDFπ MCQs β Logarithms (Class 9 Math)
View PDFπ Chapter 3 β Sets and Functions
Key Concepts: This chapter introduces fundamental concepts of sets and functions. It covers set operations (union, intersection, difference), types of sets (finite, infinite, equal, equivalent), and various types of functions (linear, quadratic, constant). Venn diagrams are used extensively to visualize set relationships.
β’ \(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\)
β’ \(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\)
β’ \(A - B = \{x \mid x \in A \text{ and } x \notin B\}\)
β’ \(A' = \{x \mid x \notin A\}\)
β’ \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)
π Exercise 3.1 β Sets and Functions (Class 9 Math)
View PDFπ Exercise 3.2 β Sets and Functions (Class 9 Math)
View PDFπ Exercise 3.3 β Sets and Functions (Class 9 Math)
View PDFπ Review Exercise β Sets and Functions (Class 9 Math)
View PDFπ Definitions β Sets and Functions (Class 9 Math)
View PDFπ MCQs β Sets and Functions (Class 9 Math)
View PDFπ Chapter 4 β Factorization and Algebraic Manipulation
Key Concepts: This chapter covers various techniques for factorizing algebraic expressions including common factors, grouping, difference of squares, perfect squares, and sum/difference of cubes. It also includes algebraic manipulation techniques such as simplification of complex fractions and rational expressions.
β’ \(a^2 - b^2 = (a - b)(a + b)\)
β’ \(a^2 \pm 2ab + b^2 = (a \pm b)^2\)
β’ \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
β’ \(x^2 + (a + b)x + ab = (x + a)(x + b)\)
β’ \(acx^2 + (ad + bc)x + bd = (ax + b)(cx + d)\)
π Exercise 4.1 β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ Exercise 4.2 β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ Exercise 4.3 β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ Exercise 4.4 β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ Review Exercise β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ Definitions β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ MCQs β Factorization & Algebraic Manipulation (Class 9 Math)
View PDFπ Chapter 5 β Linear Equations and Inequalities
βKey Concepts: This chapter covers fundamental concepts of linear equations and inequalities, including their definitions, properties, and applications in real-world problem solving. It introduces key terms like feasible region, objective function, and optimal solutions for systems of inequalities.
β’ Linear Equation: \(ax + b = 0\), where \(a \neq 0\)
β’ Inequality Symbols: \(>\), \(<\), \(\geq\), \(\leq\)
β’ Linear Inequality: \(ax + b < 0\) (or \(>\), \(\leq\), \(\geq\))
π Exercise 5.1 β Linear Equations (Class 9 Math)
View PDFπ Exercise 5.2 β Linear Equations (Class 9 Math)
View PDFπ Exercise 5.3 β Linear Equations (Class 9 Math)
View PDFπ Review Exercise β Linear Equations (Class 9 Math)
View PDFπ Definitions β Linear Equations (Class 9 Math)
View PDFπ MCQs β Linear Equations(Class 9 Math)
View PDFπ Chapter 6 β Trigonometry
βKey Concepts: This chapter introduces trigonometric ratios (sine, cosine, tangent) for acute angles in right-angled triangles. It covers fundamental trigonometric identities, applications to solve right triangles, and angle measurement in degrees. The chapter also includes practical applications of trigonometry in real-world problems.
β’ \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
β’ \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
β’ \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\)
β’ \(\csc\theta = \frac{1}{\sin\theta}\)
β’ \(\sec\theta = \frac{1}{\cos\theta}\)
β’ \(\cot\theta = \frac{1}{\tan\theta}\)
β’ \(\sin^2\theta + \cos^2\theta = 1\)
β’ \(1 + \tan^2\theta = \sec^2\theta\)
β’ \(1 + \cot^2\theta = \csc^2\theta\)
β’ \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
β’ \(\sin(90^\circ - \theta) = \cos\theta\)
β’ \(\cos(90^\circ - \theta) = \sin\theta\)
π Exercise 6.1 β Trigonometric Ratios (Class 9 Math)
View PDFπ Exercise 6.2 β Trigonometric Identities (Class 9 Math)
View PDFπ Exercise 6.3 β Trigonometry (Class 9 Math)
View PDFπ Review Exercise β Trigonometry (Class 9 Math)
View PDFπ Definitions β Trigonometry (Class 9 Math)
View PDFπ MCQs β Trigonometry (Class 9 Math)
View PDFπ Chapter 7 β Coordinate Geometry
βKey Concepts: This chapter introduces the Cartesian coordinate system and covers fundamental concepts including distance between points, midpoint formula, slope of a line, and graphing linear equations. It includes applications of coordinate geometry in solving geometric problems.
β’ Distance Formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
β’ Midpoint Formula: \(M(x,y)=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
β’ Slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
β’ Point-Slope Form: \(y - y_1 = m(x - x_1)\)
β’ Slope-Intercept Form: \(y = mx + c\)
β’ General Form: \(ax + by + c = 0\)
π Exercise 7.1 β Coordinate Geometry (Class 9 Math)
View PDFπ Exercise 7.2 β Coordinate Geometry (Class 9 Math)
View PDFπ Exercise 7.3 β Coordinate Geometry (Class 9 Math)
View PDFπ Review Exercise β Coordinate Geometry (Class 9 Math)
View PDFπ Definitions β Coordinate Geometry (Class 9 Math)
View PDFπ MCQs β Coordinate Geometry (Class 9 Math)
View PDFπ Chapter 8 β Logic
βKey Concepts: This chapter introduces the basics of mathematical logic including statements, truth tables, logical connectives (and, or, not, if-then, if and only if), and logical equivalence. It also covers methods of proof such as direct proof and proof by contradiction.
β’ Conjunction (AND): \( p \land q \)
β’ Disjunction (OR): \( p \lor q \)
β’ Negation (NOT): \( \lnot p \)
β’ Conditional: \( p \to q \)
β’ Biconditional: \( p \leftrightarrow q \)
β’ Tautology: Always true (e.g., \( p \lor \lnot p \))
β’ Contradiction: Always false (e.g., \( p \land \lnot p \))
π Chapter 9 β Similar Figures
βKey Concepts: This chapter explains the concept of similarity in geometric figures, including the criteria for similarity of triangles (AA, SAS, SSS), properties of similar figures, and applications of similarity in problem solving. It also discusses the relationship between ratios of corresponding sides, areas, volumes, and perimeters.
β’ AA (Angle-Angle): Two corresponding angles equal \( (\angle A = \angle P, \angle B = \angle Q) \)
β’ SAS (Side-Angle-Side): Proportional sides and included angle equal \( \left(\frac{AB}{PQ} = \frac{AC}{PR}, \angle A = \angle P\right) \)
β’ SSS (Side-Side-Side): All corresponding sides proportional \( \left(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\right) \)
β’ Corresponding angles are equal
β’ Corresponding sides are proportional
β’ Ratio of areas = (Ratio of sides)Β²
β’ Ratio of volumes = (Ratio of sides)Β³ (for 3D figures)
β’ Perimeters are proportional to corresponding sides
π Chapter 10 β Graphs of Functions
βKey Concepts: This chapter introduces the graphs of different types of functions including linear, quadratic, and piecewise functions. It covers determining domain and range, interpreting important features (intercepts, vertices, asymptotes), and solving equations graphically. The chapter highlights the relationship between algebraic equations and their graphical representations.
β’ Linear: \( f(x) = mx + b \) β Straight line
- Slope (\( m \)): Steepness and direction
- y-intercept (\( b \)): Point \((0, b)\)
β’ Quadratic: \( f(x) = ax^2 + bx + c \) β Parabola
- Vertex form: \( f(x) = a(x - h)^2 + k \)
- Vertex: \((h, k)\), Axis of symmetry: \( x = h \)
β’ Piecewise: Defined by different expressions on different intervals
π Exercise 10.1 β Graphs of Functions
View PDFπ Exercise 10.2 β Graphs of Functions
View PDFπ Exercise 10.3 β Graphs of Functions
View PDFπ Review Exercise β Graphs of Functions
View PDFπ Definitions β Graphs of Functions
View PDFπ MCQs β Graphs of Functions
View PDFπ Chapter 11 β Loci and Construction
βKey Concepts: This chapter explains the concept of loci (the set of points satisfying a given condition) and basic geometric constructions using compass and straightedge. It includes constructions of perpendicular bisectors, angle bisectors, parallel lines, circles, and specific angles. The connection between loci and constructions is highlighted throughout.
β’ Perpendicular bisector β Points equidistant from two given points
β’ Angle bisector β Points equidistant from two lines
β’ Circle β Points at a fixed distance from a given point (center)
β’ Parallel lines β Points at a constant distance from a line
β’ Compass & straightedge techniques for accurate constructions
π Chapter 12 β Information Handling
βKey Concepts: This chapter covers methods of organizing and representing data including frequency distributions, histograms, frequency polygons, and pie charts. It also explains measures of central tendency (mean, median, mode) and measures of dispersion (range). The focus is on calculation and interpretation of statistical graphs.
β’ Mean: \( \bar{X} = \frac{\sum x}{n} \)
β’ Median: Middle value of ordered data
β’ Mode: Most frequent value
β’ Range: \( \text{Maximum} - \text{Minimum} \)
β’ Class Mark: \( \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \)
β’ Relative Frequency: \( \frac{\text{Class Frequency}}{\text{Total Frequency}} \)
π Exercise 12.1 β Information Handling
View PDFπ Exercise 12.2 β Information Handling
View PDFπ Exercise 12.3 β Information Handling
View PDFπ Review Exercise β Information Handling
View PDFπ Definitions β Information Handling
View PDFπ MCQs β Information Handling
View PDFπ Chapter 13 β Probability
Key Concepts: This chapter introduces the fundamentals of probability, including sample space, events, and calculation of probabilities for simple cases. It covers theoretical probability, experimental probability, complementary events, and the relationship between theory and real-world applications.
β’ Theoretical Probability: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} \)
β’ Experimental Probability: \( \frac{\text{Number of times event occurs}}{\text{Total trials}} \)
β’ Probability Range: \( 0 \leq P(E) \leq 1 \)
β’ Complementary Events: \( P(\text{not } E) = 1 - P(E) \)
β’ Equally Likely Outcomes: \( P(E) = \frac{n(E)}{n(S)} \)
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Key Math Formulas
Distance Formula:
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Quadratic Equation:
\( ax^2 + bx + c = 0 \)
Solutions: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Pythagorean Identity:
\( \sin^2\theta + \cos^2\theta = 1 \)
Logarithm Product Rule:
\( \log_a(mn) = \log_a m + \log_a n \)
Probability of Event:
\( P(E) = \frac{n(E)}{n(S)} \)
Where \( n(E) \) = favorable outcomes, \( n(S) \) = total outcomes