Class 9 Mathematics Complete Guide - Algebra, Geometry, Arithmetic
Complete Class 9 Mathematics guide covering all chapters: algebra, geometry, arithmetic, statistics with formulas, examples, and practice problems for CDC curriculum.
Introduction to Class 9 Mathematics#
Class 9 Mathematics builds the foundation for SEE-level mathematics. The CDC curriculum covers Algebra, Geometry, Arithmetic, and Statistics. Strong basics here make Class 10 much easier.
Algebra#
Unit 1: Sets#
Set— A well-defined collection of distinct objects.
| Notation | Meaning | Example |
|---|---|---|
| A = {1, 2, 3} | Roster form | - |
| B = {x : x is even, x < 10} | Set-builder | B = {2, 4, 6, 8} |
| A ⊂ B | A is subset of B | {1,2} ⊂ {1,2,3} |
| A ∪ B | Union | {1,2} ∪ {2,3} = {1,2,3} |
| A ∩ B | Intersection | {1,2} ∩ {2,3} = {2} |
| A' | Complement | U = {1..5}, A={1,2} → A'={3,4,5} |
| n(A) | Cardinal number | n({a,b,c}) = 3 |
Important Formulas
- n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C)
- De Morgan's Laws: (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B'
Unit 2: Polynomials#
Algebraic Expressions
| Type | Form | Degree |
|---|---|---|
| Monomial | 3x², 5 | 2, 0 |
| Binomial | x² + 2x, 3x - 5 | max of terms |
| Trinomial | x² + 2x + 1 | max of terms |
| Polynomial | aₙxⁿ + ... + a₁x + a₀ | n |
Operations
Addition: (2x² + 3x + 1) + (x² - 2x + 4) = 3x² + x + 5 Subtraction: (3x² + 2x - 1) - (x² - x + 2) = 2x² + 3x - 3 Multiplication: (x + 2)(x - 3) = x² - x - 6 Division: (x² + 5x + 6) ÷ (x + 2) = x + 3 (remainder 0)
Factorization
| Method | Form | Example |
|---|---|---|
| Common factor | ax + ay = a(x + y) | 3x² + 6x = 3x(x + 2) |
| Grouping | ax + ay + bx + by = (a+b)(x+y) | x² + 5x + 6 = (x+2)(x+3) |
| Difference of squares | a² - b² = (a+b)(a-b) | x² - 9 = (x+3)(x-3) |
| Perfect square | a² ± 2ab + b² = (a ± b)² | x² + 6x + 9 = (x+3)² |
| Quadratic trinomial | x² + (p+q)x + pq = (x+p)(x+q) | x² + 7x + 12 = (x+3)(x+4) |
Remainder and Factor Theorems
- Remainder Theorem: If f(x) is divided by (x - a), remainder = f(a)
- Factor Theorem: (x - a) is factor of f(x) iff f(a) = 0
f(x) = x³ - 6x² + 11x - 6 f(1) = 1 - 6 + 11 - 6 = 0 → (x-1) is factor Divide: (x-1)(x² - 5x + 6) = (x-1)(x-2)(x-3)
Unit 3: Sequences and Series#
Arithmetic Progression (AP)
| Formula | Expression |
|---|---|
| nth term | aₙ = a + (n-1)d |
| Sum of n terms | Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l) |
| Arithmetic Mean | A = (a + b)/2 |
Find 10th term of AP: 3, 7, 11, ... a = 3, d = 4 a₁₀ = 3 + 9×4 = 39
Sum of first 20 terms of 2, 5, 8, ... S₂₀ = 20/2 [2×2 + 19×3] = 10 × 61 = 610
Geometric Progression (GP)
| Formula | Expression |
|---|---|
| nth term | aₙ = arⁿ⁻¹ |
| Sum of n terms | Sₙ = a(rⁿ - 1)/(r - 1) for r ≠ 1 |
| Sum to infinity | S∞ = a/(1-r) for |
| Geometric Mean | G = √(ab) |
3, 6, 12, 24, ... Find 8th term. a = 3, r = 2 a₈ = 3 × 2⁷ = 384
Sum of infinite GP: 1 + 1/2 + 1/4 + 1/8 + ... a = 1, r = 1/2 S∞ = 1/(1 - 1/2) = 2
Unit 4: Quadratic Equations#
| Method | When to Use |
|---|---|
| Factorization | When factors are integers |
| Quadratic Formula | Always works |
| Completing Square | For deriving formula |
| Graphical | For approximate roots |
Quadratic Formula— For ax² + bx + c = 0 (a ≠ 0):
x = [-b ± √(b² - 4ac)] / 2a
Discriminant: D = b² - 4ac
Solve: 2x² - 5x + 2 = 0 D = 25 - 16 = 9 x = (5 ± 3)/4 → x = 2, x = 1/2
If roots of x² - 5x + k = 0 are equal, find k. D = 25 - 4k = 0 → k = 25/4
Unit 5: Inequalities#
| Property | Rule |
|---|---|
| Addition | a > b → a + c > b + c |
| Multiplication by positive | a > b, c > 0 → ac > bc |
| Multiplication by negative | a > b, c < 0 → ac < bc |
| Reciprocal | a > b > 0 → 1/a < 1/b |
| Squaring | a > b > 0 → a² > b² |
Solve: 3x - 7 > 2x + 5 x > 12
Solve: (x-2)/(x+3) > 0 Critical points: x = 2, x = -3 Intervals: (-∞, -3): +/(-) = - ✗ (-3, 2): +/(+) = + ✓ (2, ∞): +/(+) = + ✓ Solution: (-3, 2) ∪ (2, ∞)
Geometry#
Unit 6: Lines and Angles#
| Angle Pair | Property |
|---|---|
| Adjacent | Share vertex and arm |
| Linear pair | Sum = 180° |
| Vertically opposite | Equal |
| Corresponding | Equal (if lines parallel) |
| Alternate interior | Equal (if lines parallel) |
| Alternate exterior | Equal (if lines parallel) |
| Interior on same side | Supplementary (if parallel) |
Unit 7: Triangles#
Congruence Criteria
| Criterion | Condition |
|---|---|
| SSS | Three sides equal |
| SAS | Two sides and included angle |
| ASA | Two angles and included side |
| AAS | Two angles and non-included side |
| RHS | Right angle, hypotenuse, side |
Similarity Criteria
| Criterion | Condition |
|---|---|
| AA | Two angles equal |
| SSS | Sides proportional |
| SAS | Two sides proportional, included angle equal |
- Basic Proportionality (Thales): Line parallel to one side divides other two sides proportionally
- Angle Bisector Theorem: Bisector divides opposite side in ratio of adjacent sides
- Pythagoras: a² + b² = c² (right triangle)
- Converse Pythagoras: If a² + b² = c², angle is 90°
- Apollonius: Sum of squares of two sides = 2(median² + half base²)
Unit 8: Quadrilaterals#
| Quadrilateral | Properties |
|---|---|
| Parallelogram | Opposite sides parallel & equal, diagonals bisect |
| Rectangle | All angles 90°, diagonals equal |
| Rhombus | All sides equal, diagonals perpendicular bisect |
| Square | Rectangle + Rhombus |
| Trapezium | One pair of parallel sides |
| Kite | Two pairs of adjacent equal sides |
- Parallelogram: base × height
- Triangle: ½ × base × height = ½ ab sin C
- Trapezium: ½ (sum of parallel sides) × height
- Rhombus: ½ × product of diagonals
Unit 9: Circles#
| Theorem | Statement |
|---|---|
| 1 | Perpendicular from center bisects chord |
| 2 | Equal chords equidistant from center |
| 3 | Angle at center = 2 × angle at circumference |
| 4 | Angles in same segment are equal |
| 5 | Angle in semicircle = 90° |
| 6 | Opposite angles of cyclic quadrilateral = 180° |
| 7 | Tangent ⟂ radius at point of contact |
| 8 | Tangents from external point are equal |
| 9 | Alternate segment theorem |
Chord length 12 cm, distance from center 5 cm. Find radius. r² = 5² + 6² = 61 → r = √61 cm
Cyclic quadrilateral ABCD, ∠A = 70°. Find ∠C. ∠C = 180° - 70° = 110°
Unit 10: Constructions#
| Construction | Steps Summary |
|---|---|
| Perpendicular bisector | Arcs from endpoints, join intersections |
| Angle bisector | Arc from vertex, arcs from intersections, join |
| Triangle (SSS) | Draw base, arcs from endpoints with other sides |
| Triangle (SAS) | Draw side, angle, other side |
| Triangle (ASA) | Draw side, two angles |
| Circle through 3 points | Perpendicular bisectors of chords intersect at center |
| Tangent at point | Perpendicular to radius |
| Tangents from external | Circle on diameter from external to center |
Arithmetic#
Unit 11: Percentage, Profit and Loss#
| Formula | Expression |
|---|---|
| Profit | SP - CP |
| Loss | CP - SP |
| Profit % | (Profit/CP) × 100 |
| Loss % | (Loss/CP) × 100 |
| SP with profit | CP(1 + P%/100) |
| SP with loss | CP(1 - L%/100) |
| CP from SP & profit | SP × 100/(100 + P%) |
| Discount | MP - SP |
| Discount % | (Discount/MP) × 100 |
Article marked 40% above CP, sold at 15% discount. Profit %? CP = 100, MP = 140 SP = 140 × 0.85 = 119 Profit = 19%
Two successive discounts of 20% and 10% = single discount? Effective = 1 - 0.8×0.9 = 0.28 = 28%
Unit 12: Simple and Compound Interest#
| Type | Formula |
|---|---|
| Simple Interest | SI = PRT/100 |
| Amount (SI) | A = P + SI = P(1 + RT/100) |
| Compound Interest (annual) | A = P(1 + R/100)ᵀ |
| Compound Interest (half-yearly) | A = P(1 + R/200)²ᵀ |
| Compound Interest (quarterly) | A = P(1 + R/400)⁴ᵀ |
| CI | A - P |
Find CI on 5000 at 10% for 2 years, compounded annually. A = 5000(1.1)² = 6050 CI = 1050
Difference between CI and SI on 10000 at 5% for 2 years? SI = 1000, CI = 1025, Difference = 25
Unit 13: Ratio, Proportion and Variation#
| Concept | Formula/Rule |
|---|---|
| Ratio | a = a/b |
| Proportion | a = c → ad = bc |
| Continued proportion | a = b → b² = ac |
| Direct variation | x ∝ y → x = ky |
| Inverse variation | x ∝ 1/y → xy = k |
| Joint variation | x ∝ yz → x = kyz |
Statistics#
Unit 14: Measures of Central Tendency#
| Measure | Formula |
|---|---|
| Mean (ungrouped) | x̄ = Σx/n |
| Mean (grouped) | x̄ = Σfx/Σf |
| Median (odd n) | Middle value |
| Median (even n) | Average of middle two |
| Median (grouped) | L + (N/2 - cf)/f × h |
| Mode (grouped) | L + (f₁-f₀)/(2f₁-f₀-f₂) × h |
Unit 15: Measures of Dispersion#
| Measure | Formula |
|---|---|
| Range | Max - Min |
| Mean Deviation | Σf |
| Variance | σ² = Σf(x - x̄)²/Σf |
| Standard Deviation | σ = √Variance |
| Coefficient of Variation | CV = σ/x̄ × 100% |
Data: 10, 15, 20, 25, 30 Mean = 100/5 = 20 Median = 20 Mode = No mode (all unique)
Variance = [(100+25+0+25+100)/5] = 50 SD = √50 ≈ 7.07 CV = 7.07/20 × 100 = 35.35%
Practice Problems#
Exam Preparation Tips#
| Week | Focus | Activities |
|---|---|---|
| 1-2 | Algebra | Sets, polynomials, factorization |
| 3-4 | Algebra | Quadratics, sequences, inequalities |
| 5-6 | Geometry | Triangles, circles, constructions |
| 7 | Arithmetic | Percentage, interest, ratio |
| 8 | Statistics | Mean, median, mode, SD |
| 9-10 | Full Revision | Past papers, mock tests |
Conclusion#
Class 9 Mathematics is about building solid fundamentals. Don't just memorize formulas—understand the derivations. Practice 10-15 problems daily, focusing on weak areas. Regular geometry construction practice is essential. With consistent effort, you'll build a strong foundation for SEE.
The only way to learn mathematics is to do mathematics.
Good luck with your Class 9 Mathematics!
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