Chapter 1 of 13

Andhra Pradesh (BIEAP)Class 9 Mathematics★ AP Board (BSEAP) Class 9 Mathematics · ~80 marks · Board: AP Board

Mathematics — Real Numbers, Polynomials, Geometry & Statistics (AP Class 9)

Complete AP Class 9 Mathematics: REAL NUMBERS (rational, irrational, decimal expansions). POLYNOMIALS (degree, zeroes, Remainder & Factor Theorems). COORDINATE GEOMETRY. LINES and ANGLES. TRIANGLES. QUADRILATERALS. SURFACE AREAS and VOLUMES. STATISTICS. BSEAP Class 9 Mathematics with AP-specific examples.

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By the end of this chapter you'll be able to…

1Classify real numbers; prove irrationality of √2; identify terminating vs recurring decimals
2Apply Remainder Theorem and Factor Theorem; use algebraic identities for factorisation
3Plot points in the Cartesian plane; identify quadrant and axis positions
4Prove and apply parallel line theorems (corresponding, alternate, co-interior angles)
5Apply congruence criteria (SSS, SAS, ASA, AAS, RHS) to triangles
6State properties of parallelogram, rhombus, rectangle, square, trapezium
7Calculate surface area and volume of cube, cuboid, cylinder, cone, sphere, hemisphere
8Calculate mean (three methods), median, and mode for grouped and ungrouped data

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Why this chapter matters

This is the complete AP Class 9 Mathematics guide, covering all major units: Real Numbers, Polynomials, Coordinate Geometry, Lines and Angles, Triangles, Quadrilaterals, Surface Areas and Volumes, and Statistics. Class 9 Mathematics lays the algebraic and geometric foundation for Class 10. Mastery of this guide prepares a student for all questions across the AP Class 9 Mathematics paper — including number theory, algebraic identities, geometric proofs, mensuration formulas, and statistics.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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AP Class 9 Mathematics

1. Real Numbers

Rational Numbers — Numbers that CAN be expressed as p/q (p,q ∈ Z, q≠0)

Terminating decimals: Denominator has ONLY 2 and/or 5 as prime factors. Example: 3/8 = 0.375 (8 = 2³).
Non-terminating RECURRING decimals: Denominator has prime factors OTHER than 2 and 5. Example: 1/3 = 0.333... (3 is not 2 or 5).

Irrational Numbers — CANNOT be expressed as p/q

√2, √3, √5, π. Proof that √2 is irrational (by contradiction — assume p/q in simplest form → contradiction).

Real Numbers on the Number Line

'Every real number corresponds to EXACTLY one point on the number line — and every point corresponds to EXACTLY one real number.'

2. Polynomials

Key Definitions

Term	Definition	Example
Degree	HIGHEST power of the variable	3x⁴+2x²: degree = 4
Zero of polynomial	Value of x for which p(x)=0	p(2)=0 → 2 is a zero
Constant polynomial	Degree 0	p(x)=5
Zero polynomial	ALL coefficients = 0	p(x)=0 (degree undefined)

Remainder Theorem

When p(x) is divided by (x−a), the remainder = p(a). 'This is POWERFUL — you don't need to do long division. Just EVALUATE the polynomial at x=a.'

Factor Theorem

(x−a) is a factor of p(x) IF AND ONLY IF p(a) = 0. 'The converse of the Remainder Theorem. If p(a)=0 → (x−a) is a FACTOR.'

Algebraic Identities

(a+b)² = a²+2ab+b²
(a−b)² = a²−2ab+b²
a²−b² = (a+b)(a−b)
(x+a)(x+b) = x²+(a+b)x+ab
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca
a³+b³ = (a+b)(a²−ab+b²)
a³−b³ = (a−b)(a²+ab+b²)

Factorisation Methods

Method	When to Use
Common Factor	All terms share a factor
Grouping	Group terms with common factors
Splitting Middle Term	For quadratics: find two numbers whose product = ac and sum = b

3. Coordinate Geometry

The Cartesian Plane

x-axis (horizontal). y-axis (vertical). Origin O(0,0).
Quadrants: I (+,+). II (−,+). III (−,−). IV (+,−).

Plotting Points

A point (x,y) is plotted by: x units along the x-axis. y units parallel to the y-axis.

Key Formulas

Distance from x-axis = |y|. Distance from y-axis = |x|.

4. Lines and Angles

Angle Types: Acute (<90°). Right (=90°). Obtuse (90°–180°). Straight (=180°). Reflex (180°–360°).

Angle Pairs

Relationship	Property	Example
Complementary	Sum = 90°	30°+60°
Supplementary	Sum = 180°	110°+70°
Vertically Opposite	EQUAL	When two lines intersect
Adjacent	Share common arm and vertex	Neighbouring angles

Parallel Lines and a Transversal

Corresponding angles = EQUAL. Alternate interior angles = EQUAL. Interior angles on the same side of transversal = SUPPLEMENTARY (sum = 180°).

5. Triangles

Types: By Sides (Equilateral, Isosceles, Scalene). By Angles (Acute, Right, Obtuse).

Congruence Criteria

SSS (Side-Side-Side). SAS (Side-Angle-Side). ASA (Angle-Side-Angle). AAS (Angle-Angle-Side). RHS (Right angle-Hypotenuse-Side).

Key Properties

Angle sum = 180°. Exterior angle = sum of two interior OPPOSITE angles.
Larger SIDE → opposite Larger ANGLE.

6. Quadrilaterals

Shape	Key Properties
Parallelogram	Opposite sides ∥ & =. Diagonals BISECT each other.
Rhombus	All sides =. Diagonals ⟂.
Rectangle	All angles 90°. Diagonals =.
Square	All properties of rhombus AND rectangle.
Trapezium	ONE pair of ∥ sides.

Angle Sum of Quadrilateral = 360°.

7. Surface Areas and Volumes

Solid	Surface Area	Volume
Cube	6s²	s³
Cuboid	2(lb+bh+hl)	lbh
Cylinder	2πr(r+h)	πr²h
Cone	πr(r+l) [l=√(r²+h²)]	⅓πr²h
Sphere	4πr²	(4/3)πr³
Hemisphere	3πr²	(2/3)πr³

8. Statistics

Data Collection and Presentation

Primary data (collected directly). Secondary (from existing sources). Frequency distribution tables. Histograms (continuous data — bars TOUCH). Frequency polygons.

Measures of Central Tendency

Mean: X̄ = Σx/n (ungrouped). X̄ = Σfx/Σf (grouped).
Median: Middle value when ordered.
Mode: Most frequent value.

Exam Strategy

Unit	Approx. Marks
Number System & Algebra	17-20
Geometry (Lines, Triangles, Quadrilaterals)	15-18
Coordinate Geometry	6-8
Mensuration	13-15
Statistics	8-10

Common Mistakes to Avoid

Forgetting to mention rational numbers must have q≠0.
Confusing Remainder Theorem (divide by x−a, remainder = p(a)) with Factor Theorem (p(a)=0 → x−a is a factor).
Mixing SA and volume formulas for cone vs cylinder — the ⅓ factor for cone volume is commonly missed.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Algebra — Real Numbers and Polynomials

RATIONAL NUMBERS: p/q where q≠0. Terminating decimal ↔ denominator has ONLY 2s and 5s as prime factors. Non-terminating recurring ↔ denominator has OTHER prime factors. IRRATIONAL: √2, √3, π, e — cannot be expressed as p/q. PROOF √2 IS IRRATIONAL: Assume √2 = p/q (simplest form). Squaring: 2 = p²/q² → p² = 2q² → p² is even → p is even → p = 2m. Then 4m² = 2q² → q² = 2m² → q is even. But p and q are both even → CONTRADICTION (assumed simplest form). ALGEBRAIC IDENTITIES: (a+b)² = a²+2ab+b². (a−b)² = a²−2ab+b². a²−b² = (a+b)(a−b). (a+b+c)² = a²+b²+c²+2ab+2bc+2ca. (a+b)³ = a³+3a²b+3ab²+b³. a³+b³ = (a+b)(a²−ab+b²). a³−b³ = (a−b)(a²+ab+b²). REMAINDER THEOREM: Divide p(x) by (x−a) → remainder = p(a). FACTOR THEOREM: p(a)=0 ↔ (x−a) is a factor of p(x). FACTORISATION: Common factor. Grouping. Splitting middle term (find two numbers with product=ac, sum=b for ax²+bx+c).

EXAM SHORTCUTS: p(a) = 0 → (x−a) is a factor. Use Factor Theorem before attempting long division. For a³±b³: remember the middle sign changes relative to (a±b). Identity (a+b+c)² appears in AP Class 9 — memorise it separately. √2 irrationality proof by contradiction is a standard 3-mark proof question.

Geometry and Mensuration

COORDINATE GEOMETRY: Quadrants I(+,+), II(−,+), III(−,−), IV(+,−). x-axis: y=0. y-axis: x=0. LINES AND ANGLES: Complementary = 90°. Supplementary = 180°. Vertically opposite = equal. Parallel lines + transversal: corresponding = equal, alternate interior = equal, co-interior (same-side interior) = 180°. TRIANGLES: Angle sum = 180°. Exterior angle = sum of opposite interior angles. Congruence: SSS, SAS, ASA, AAS, RHS. Larger side → opposite larger angle. QUADRILATERALS: Angle sum = 360°. Parallelogram: opp sides ∥ & =, diagonals bisect each other. Rhombus: all sides equal, diagonals ⊥ bisect. Rectangle: all 90°, diagonals equal. Square: all properties of both. CIRCLE THEOREMS: Angle at centre = 2 × angle at circumference. Angle in semicircle = 90°. Perpendicular from centre bisects chord. Equal chords equidistant from centre. Cyclic quad: opposite angles sum to 180°. Same segment angles equal. MENSURATION FORMULAS: Cube: SA = 6s², V = s³. Cuboid: SA = 2(lb+bh+hl), V = lbh. Cylinder: CSA = 2πrh, TSA = 2πr(r+h), V = πr²h. Cone: l = √(r²+h²), CSA = πrl, TSA = πr(r+l), V = ⅓πr²h. Sphere: SA = 4πr², V = (4/3)πr³. Hemisphere: CSA = 2πr², TSA = 3πr², V = (2/3)πr³. STATISTICS: Class mark xᵢ = (lower + upper)/2. Mean (direct) = Σfx/Σf. Mean (assumed mean) = A + Σfd/Σf. Mean (step deviation) = A + h(Σfu/Σf). Median (grouped) = L + [(N/2−CF)/f]×h. Modal class = highest frequency class.

MENSURATION KEY: Cone and sphere formulas use π. Cone volume has ⅓. Sphere volume has 4/3. Hemisphere is exactly half the sphere. CIRCLES: Angle in semicircle = 90° (NOT 180°). Cyclic quad opposite angles = 180° (opposite NOT adjacent). STATISTICS: CF in median formula = before the median class. Histogram bars TOUCH. Bar graph has GAPS.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting the ⅓ factor in cone volume and using πr²h instead of ⅓πr²h

✓ The volume of a CONE is exactly ONE-THIRD of the volume of a CYLINDER with the same base radius and height: V_cone = ⅓πr²h (NOT πr²h). V_cylinder = πr²h. Hemisphere: V = (2/3)πr³ (= half of sphere = half of (4/3)πr³). A useful check: if you fill a cone with water and pour it into a cylinder of the same radius and height, it takes EXACTLY THREE CONES to fill the cylinder. Always check the formula sheet: Cone → ⅓, Sphere → (4/3), Hemisphere → (2/3). These fractions are the most commonly missed in AP Class 9 mensuration.

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Mathematics — Real Numbers, Polynomials, Geometry & Statistics (AP Class 9)?

1 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

1 questions~2 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•REAL NUMBERS: N ⊂ W ⊂ Z ⊂ Q ⊂ R. Rational = terminating or recurring decimal. Irrational = non-terminating non-recurring. √2, √3, π are irrational. Proof of √2 irrationality: by contradiction (assume rational, derive p and q both even, contradicting coprimeness).
•POLYNOMIALS — REMAINDER & FACTOR THEOREMS: Remainder when p(x) is divided by (x−a) = p(a). (x−a) is a factor ⟺ p(a) = 0. Cubic factorisation: find one zero by trial, divide to get quadratic, factorise quadratic.
•ALGEBRAIC IDENTITIES (must memorise): (a+b)², (a−b)², a²−b², (x+a)(x+b), (a+b+c)², (a+b)³, (a−b)³, a³+b³=(a+b)(a²−ab+b²), a³−b³=(a−b)(a²+ab+b²). Notice alternating signs in cubic factorisation brackets.
•COORDINATE GEOMETRY: Quadrants I(+,+), II(−,+), III(−,−), IV(+,−). x-axis: y=0. y-axis: x=0. Origin (0,0) is on both axes. Abscissa = x, Ordinate = y.
•LINES AND ANGLES: Vertically opposite = equal. Linear pair = 180°. Parallel lines + transversal: CORRESPONDING = equal (F), ALTERNATE INTERIOR = equal (Z), CO-INTERIOR = 180° (C). Triangle angle sum = 180°. Exterior angle = sum of two non-adjacent interiors.
•TRIANGLES — CONGRUENCE: SSS, SAS (included angle), ASA (included side), AAS, RHS (right triangle only). INVALID: SSA, AAA. After proving congruence, use CPCT to derive equal parts. Isosceles theorem: equal sides → equal opposite angles.
•QUADRILATERALS: angle sum = 360°. PARALLELOGRAM: opposite sides/angles equal, consecutive angles supplementary, diagonals bisect each other. RECTANGLE: diagonals EQUAL. RHOMBUS: diagonals PERPENDICULAR. SQUARE: both equal and perpendicular. MID-POINT THEOREM: line joining midpoints of two sides ∥ third side and = half of it.
•CIRCLE THEOREMS (6): (1) Angle at centre = 2× angle at circumference. (2) Angle in semicircle = 90°. (3) Perpendicular from centre bisects chord. (4) Equal chords equidistant from centre. (5) Cyclic quad: opposite angles sum 180°. (6) Same-segment angles equal.
•MENSURATION FORMULAS: Cube V=s³, SA=6s². Cuboid V=lbh, SA=2(lb+bh+hl). Cylinder V=πr²h, CSA=2πrh, TSA=2πr(r+h). Cone V=⅓πr²h, CSA=πrl, TSA=πr(r+l), where l=√(r²+h²). Sphere V=(4/3)πr³, SA=4πr². Hemisphere V=(2/3)πr³, CSA=2πr², TSA=3πr².
•STATISTICS: Mean — direct Σfx/Σf, assumed mean A+Σfd/Σf, step deviation A+h(Σfu/Σf). Median (grouped) = L + [(N/2 − CF)/f] × h. Modal class = highest frequency class. Histogram bars TOUCH; bar graph has GAPS.

Andhra Pradesh (BIEAP) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Engineering and STEM career foundation

AP Class 9 Mathematics is the foundation for EVERY engineering and science career. Algebra and identities are used in equation solving across all of physics and engineering. Geometry (triangles, circles) is essential for civil/mechanical engineering and architecture. Mensuration is used daily by civil engineers in AP's growing infrastructure sector. Statistics is foundational for data science and analytics — Hyderabad's IT industry employs thousands of data scientists. Mastery of Class 9 mathematics opens doors to ALL technical careers.

Government job exam preparation

Almost every government job exam in AP (APPSC Group 1/2/3, Banks, SSC, Railways, Police) tests basic mathematics — exactly the Class 9 syllabus. Statistical reasoning, mensuration calculations, algebraic manipulation, geometric problem-solving — all appear in competitive exam mathematics sections. Mastering Class 9 mathematics is the first step toward competitive exam success in AP — used by lakhs of aspirants.

Daily life mathematical literacy

Every adult uses Class 9 mathematics: calculating loan EMIs (algebra), measuring rooms (mensuration), comparing prices (ratio/percentages), reading election poll statistics (mean/median/mode), understanding interest rates, computing fuel efficiency. AP citizens make better financial and personal decisions when they have mathematical literacy. The mathematics studied in Class 9 is not just for exams — it is essential life skill for navigating modern Indian society.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

FORMULA SHEET in first 5 min: write all major formulas (mensuration, identities, mean/median, circle theorems) at top of rough work page. Saves time and prevents formula confusion later.
MARK MAXIMISATION: do MENSURATION first (highest marks per minute if formulas are memorised). Then ALGEBRA (Factor Theorem, identities — quick wins). Then GEOMETRY proofs (need careful work). Finally STATISTICS (table construction is time-consuming).
DIAGRAMS for ALL geometry: never solve a geometry problem without drawing the figure. Label given quantities. Right angles marked. Auxiliary lines shown if needed. Diagrams earn structure marks even with arithmetic errors.
STATE THE THEOREM you are using: 'By Pythagoras theorem,' 'By the Mid-Point Theorem,' 'By cyclic quadrilateral property,' 'By the Factor Theorem.' Citing the theorem before applying it earns method marks.
Show ALL working steps: in a 4-mark problem, each major step (formula, substitution, simplification, final answer) is roughly 1 mark. Skipping steps to save time often costs more marks than it saves.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research the link between Class 9 Mathematics and the four Indian Mathematical Olympiad subject areas: ALGEBRA (polynomials, identities), GEOMETRY (triangles, circles, mid-point theorem extensions), NUMBER THEORY (rational/irrational numbers, divisibility), COMBINATORICS (counting, probability). Each Olympiad area is rooted in the conceptual foundations of Class 9 mathematics.
Investigate Indian mathematicians who shaped mathematics: Aryabhata (5th century — sine table, place value), Brahmagupta (7th century — zero, negative numbers, quadratic formula), Bhaskara II (12th century — calculus precursors, Pythagoras proof), Ramanujan (20th century — number theory, infinite series). Indian mathematics has a 2500-year continuous tradition. Research their specific contributions.
Explore the History of Mathematics — the development of mathematical concepts over millennia. Greek geometry (Euclid, 300 BCE), Indian arithmetic and algebra (Aryabhata, Brahmagupta), Arabic algebra (Al-Khwarizmi, 9th century), European calculus (Newton, Leibniz, 17th century), modern abstract algebra (Galois, 19th century). Research how mathematics is a continuous human endeavour built by many cultures.
Research GIRL'S PERFORMANCE IN MATHEMATICS — historically, girls have been underrepresented in mathematics due to societal stereotypes (not ability). AP has made significant progress in girls' STEM education. Research Indian women mathematicians (Shakuntala Devi 'Human Computer,' Mangala Narlikar, Sujatha Ramdorai) and how AP can encourage more girls into mathematics and STEM careers.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

AP Board SSC (Class 10) — MathematicsVery High — every Class 10 chapter directly extends a Class 9 chapter; mastery of Class 9 is essential for Class 10

JEE Main and AdvancedVery High — algebra, geometry, mensuration, and statistics are major JEE topics; Class 9 provides the foundational skills

NTSE (Mathematics)Very High — all major Class 9 topics appear in NTSE Stage I and II mathematics

AP EAPCETVery High — Class 9 mathematics is the foundation for all EAPCET Mathematics chapters

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

(1) MENSURATION (~13-15 marks): All formulas for cube, cuboid, cylinder, cone, sphere, hemisphere. Slant height l = √(r²+h²) for cones. Volume formulas have factors (⅓ for cone, 4/3 for sphere, ⅔ for hemisphere). (2) ALGEBRA — Polynomials and Identities (~10-12 marks): Remainder/Factor Theorems, cubic factorisation, all 8 algebraic identities (especially a³±b³). (3) CIRCLES (~8-10 marks): 6 theorems, especially angle at centre and cyclic quadrilateral. Pythagoras for chord-radius. (4) TRIANGLES + QUADRILATERALS (~12-14 marks): congruence proofs (SSS/SAS/ASA/AAS/RHS) with CPCT, Mid-Point Theorem, parallelogram properties. (5) STATISTICS (~8-10 marks): three mean methods, grouped median, histogram. These 5 areas total ~55 marks of the 80-mark paper.

AP Class 9 Mathematics paper: 80 marks in 150 minutes (typically). SUGGESTED TIME ALLOCATION: (1) Read full paper (5 min). (2) Objective/short answers (Part A, ~15 marks): 20 min. (3) Short answer questions (Part B, ~25 marks): 35 min. (4) Long answer questions (Part C, ~40 marks): 75 min — about 12-15 min per 4-5 mark question. (5) Review (15 min). PRIORITY in Part C: do MENSURATION numericals first (fastest marks-per-minute if formulas are memorised). Then ALGEBRA (Factor Theorem applications). Then GEOMETRY proofs (require careful step-by-step work — leave for fresh mind at start of long-answer section). Finally STATISTICS (table construction takes time).

TIGHTLY INTERCONNECTED: REAL NUMBERS (√n irrationality, exponents) underlies POLYNOMIALS (roots include irrationals like √2). POLYNOMIAL identities support FACTORISATION and ALGEBRA throughout. COORDINATE GEOMETRY introduces the plane that LINES AND ANGLES live in. LINES AND ANGLES is the foundation for TRIANGLES (angle sum, parallel lines proofs). TRIANGLES extends to QUADRILATERALS (via diagonal-divided triangles) and CIRCLES (inscribed triangles). MENSURATION uses all geometry (right triangles for cone slant heights, circular bases). STATISTICS analyses real-world data using all of the above. Every chapter builds on previous ones — this is why mathematics must be studied SEQUENTIALLY, not in isolation.

Class 9 is the year mathematics transitions from CALCULATION to PROOF. In earlier classes, mathematics was about computing answers. Class 9 introduces RIGOROUS REASONING — the idea that mathematical statements need to be PROVED from accepted axioms and previously proved theorems. This is the foundation of all higher mathematics (Class 11-12 calculus, JEE/EAPCET, college mathematics, research). PROOFS develop: (1) logical thinking — each step must follow from the previous. (2) The discipline of showing WORK rather than just answers. (3) Mathematical maturity — understanding that mathematics is a deductive system. AP Class 9 introduces the major proof techniques: by contradiction (√2 irrationality), by construction (auxiliary lines in Triangle proofs), by congruence (CPCT applications). Mastering these proof techniques is more important than mastering any specific theorem — proofs train the mathematical mind.

Direct extensions: (1) REAL NUMBERS → Class 10 Real Numbers (Euclid's algorithm, LCM/HCF, irrationality proofs). (2) POLYNOMIALS → Class 10 Polynomials (zeroes, division algorithm) and Quadratic Equations. (3) LINEAR EQUATIONS IN 2 VARS → Class 10 Pair of Linear Equations (graphical, substitution, elimination methods). (4) TRIANGLES → Class 10 Similar Triangles (Thales, BPT, area ratios). (5) CIRCLES → Class 10 Tangents and Secants (tangent perpendicular to radius). (6) MENSURATION → Class 10 Surface Areas and Volumes (combination of solids, frustum). (7) STATISTICS → Class 10 Statistics (mean by all methods, median, mode, ogive). (8) COORDINATE GEOMETRY → Class 10 Coordinate Geometry (distance formula, section formula). Class 9 is foundational; Class 10 extends and applies these concepts. Strong Class 9 fundamentals make Class 10 manageable.

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Polynomials and Factorisation

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Coordinate Geometry

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Linear Equations in Two Variables

LINEAR EQUATIONS — ax+by+c=0. SOLUTIONS as ordered pairs (x,y) — infinitely many. GRAPHING a linear equation as a STRAIGHT LINE. x=k and y=k as special cases. Applications and worked examples. AP Board Class 9 Mathematics.

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