Fundamental Theorem of Arithmetic
Prime factorisation trees, uniqueness theorem, power of 2 & 5 trailing zero termination, and composite expressions.
HCF & LCM Using Prime Factorisation
Exponent-based HCF & LCM determination, HCF × LCM = a × b verification identity, and word problems.
Proof of Irrationality
Rigorous proof by contradiction for single radicals (√2, √3, √5), linear combinations (3 + 2√5), and reciprocals (1/√2).
Zeroes of a Polynomial
Algebraic definition of zeroes, linear transposition, quadratic zero-product property, geometrical x-intercepts, and degree bounds.
Relationship Between Zeroes & Coefficients
Vieta's identities (α + β = -b/a, αβ = c/a), forming quadratic polynomials k(x² - Sx + P), unknown parameters, and transformed zeroes.
Graphical Method & Consistency Conditions
Testing consistency via ratio comparison a1/a2, b1/b2, c1/c2 for intersecting, parallel, and coincident lines.
Solving Linear Systems by Substitution Method
Expressing one variable in terms of the other and solving 2-variable linear systems systematically.
Solving Linear Systems by Elimination Method
Multiplying equations by suitable constants to eliminate one variable and isolate solutions.
Real-Life Word Problems on Linear Equations
Translating age, speed-distance-time, fraction, and geometry problems into simultaneous linear equations.
Standard Form & Checking Quadratic Nature
Writing equations in standard form ax² + bx + c = 0 and verifying whether given values are roots.
Solving Quadratic Equations by Factorisation
Middle-term splitting, algebraic factor pairs, grouping terms, and finding roots with zero-product rule.
Solving by Quadratic Formula (Sridhar Acharya)
Applying x = (-b ± √D) / (2a) to find rational, irrational, and radical roots.
Discriminant & Nature of Roots
Evaluating D = b² - 4ac to determine real distinct, equal, or no real roots and finding unknown constant k.
Applied Word Problems on Quadratic Equations
Modeling consecutive integers, right-angled triangles, uniform speed, and time-work setups into quadratics.
General Term (nth Term) of an AP
Identifying first term a, common difference d, checking AP series, and finding an = a + (n - 1)d.
Finding nth Term from the End & AP Parameters
Reverse AP analysis, finding term positions from the last term, and calculating a and d from two known terms.
Sum of First n Terms of an AP
Applying Sn = n/2 [2a + (n - 1)d] and Sn = n/2 [a + l] to compute series sums.
Relationship Between an and Sn
Finding general term an = Sn - Sn-1, first term, common difference, and verifying series from given sum formulas.
Daily Life Word Problems on AP
Savings plans, installment repayments, tiered production counts, and staircase step heights.
Basic Proportionality Theorem (Thales Theorem)
Statement, formal geometric proof, and proportional side calculations for lines parallel to one side.
Converse of Basic Proportionality Theorem
Proving lines parallel when given proportional segment ratios in a triangle.
Criteria for Similarity of Triangles (AA, SSS, SAS)
Applying AA, SSS, and SAS similarity conditions to solve heights, shadows, and geometric ratio proofs.
Distance Formula & Geometric Applications
Calculating distance between points, testing collinearity, and classifying isosceles, equilateral, right triangles and quadrilaterals.
Section Formula for Internal Division
Finding coordinates of points dividing a line segment in ratio m1 : m2 internally.
Midpoint Formula & Trisection of Line Segments
Calculating midpoints and coordinates of points trisecting (1:2 and 2:1) a line segment.
Finding Division Ratio k:1 & Axis Intercepts
Determining the ratio in which coordinate axes (x-axis, y-axis) or given points divide a line segment.
Trigonometric Ratios in a Right Triangle
Defining sin, cos, tan, cot, sec, cosec with respect to acute angles and finding unknown side lengths.
Trigonometric Ratios of Specific Angles (0°, 30°, 45°, 60°, 90°)
Evaluating numerical trigonometric expressions using standard values table.
Fundamental Trigonometric Identities
Proving trigonometric identities using sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.
Heights and Distances with Single Angle
Angle of elevation and depression in single right-angled triangle word problems (towers, trees, poles).
Heights and Distances with Two Angles
Multi-triangle setups with varying observation points, heights of buildings, towers, and flying kites/balloons.
Tangent is Perpendicular to Radius at Point of Contact
Theorem statement, proof, and finding radii or hypotenuse lengths via Pythagorean theorem.
Lengths of Tangents from an External Point
Proof that lengths of tangents from an external point are equal and angle relationship subtended at center.
Circumscribed & Inscribed Polygon Proofs
Proving AB + CD = AD + BC for circumscribed quadrilaterals and right-angled triangle incircle radii.
Area of Sector and Length of Arc (60°, 90°, 120°)
Calculating sector area (θ/360 × πr²) and arc length (θ/360 × 2πr) for clock hands, car wipers, and brooches.
Area of Minor & Major Segments of a Circle
Finding segment area = Area of sector - Area of triangle for central angles 60°, 90°, 120°.
Surface Area of Combinations of Solids
Calculating total outer surface area for combinations of cubes, cylinders, cones, and hemispheres (toys, tents, capsules).
Volume of Combinations of Solids
Computing volume of complex combined solids without overlapping gaps (gulab jamun syrup, test tubes, silos).
Mean of Grouped Data (Direct & Assumed Mean Methods)
Finding class marks xi, direct mean (Σfixi / Σfi), and assumed mean method (a + Σfidi / Σfi).
Mode of Grouped Data
Identifying modal class and applying Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] × h.
Median of Grouped Data
Forming cumulative frequency table, finding N/2 median class, and applying Median = l + [((N/2 - cf)/f)] × h.
Empirical Relationship & Missing Frequencies
Using 3 Median = Mode + 2 Mean and determining unknown frequencies f1, f2 from given median/mean.
Classical Theoretical Probability & Complementary Events
Computing P(E) = n(E)/n(S), sure events (P=1), impossible events (P=0), and P(not E) = 1 - P(E).
Sample Spaces & Events: Coins, Dice, and Cards
Outcomes for tossing 1/2/3 coins, rolling 1 or 2 dice, and well-shuffled 52-card standard deck suits and face cards.
Real-Life Probability Applications & Compound Events
Defective bulb batches, colored marbles drawn from bags, target dartboards, and game spinning wheels.
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Prerequisite Microlearning — Skill Foundations
17 foundational modules • 74 microtopics sequenced from arithmetic to proofs. Clear these first and every Class 10 chapter becomes comfortable.
M7 · Systematic Factor Pairs & Middle-Term Splitting
Sign signature rules, √N divisor stopping bounds, signed factor pair table generation, and 4-term grouping mastery.
M1 · Factors, Multiples & Divisibility
Primes & factorisation trees, divisibility rules, HCF/LCM meaning, and even/odd parity laws for proof readiness.
M2 · Fractions, Ratios & Percent Sense
Fraction operations, simplest-form & conversions, ratio comparisons, cross-multiplication, and unitary method.
M3 · Squares, Roots & Surds
Squares to 25², surd simplification & rationalisation, and Pythagorean triples (3-4-5, 5-12-13) to automaticity.
M4 · Powers & Exponent Laws
Product, quotient & power rules, zero/negative exponents, and prime-power decomposition of numbers.
M5 · Linear Equation in One Variable
Transposition & balancing, solving ax + b = cx + d, fractional/negative coefficients, and checking solutions.
M6 · Algebraic Identities & Manipulation
(a±b)² and a²−b² fluency, substitution & evaluation, sign rules with negatives, and formula rearrangement.
M8 · Word-Problem Translation Patterns
Sentence-to-equation workflow, speed-distance-time, cost/age/area setups, consecutive integers, and AP story patterns.
M9 · Cartesian Plane Fluency
Plotting (x, y) & quadrants, reading intercepts from graphs, and visually distinguishing parallel, intersecting & coincident lines.
M10 · Pythagoras & Right-Triangle Anatomy
Theorem & converse, hypotenuse identification, altitude/base vocabulary, and the slant-height relation l² = r² + h².
M11 · Parallel-Line Angle Facts
Alternate, corresponding & co-interior angles, vertically opposite pairs, and elevation ↔ depression as alternate angles.
M12 · Ratio in Geometry: Similarity Foundations
Scale factor intuition, corresponding sides in proportion, solving x/(x+3) = 4/5 style equations, and the midpoint theorem.
M13 · Circle Anatomy & Fraction-of-Circle Thinking
Radius/chord/arc/tangent vocabulary, sector vs segment distinction, the 60°-is-⅙ model, and π fluency.
M14 · Solid Vocabulary Recap (Class 9)
Cone/cylinder/sphere/hemisphere CSA-TSA-volume recall, cone-slant ↔ sector link, and cm³↔litre↔m³ conversions.
M15 · Data Tables & Averages
Frequency tables & class intervals, class marks, cumulative frequency reading, and mean/mode/median on raw data.
M16 · If–Then Thinking & Proof Format
Given/to-prove/proof structure, contradiction intuition, congruence rules (SSS, SAS, ASA, RHS), and Class 9 circle facts.
M17 · Counting & Chance Intuition
Sample spaces, coin/dice outcomes, 52-card deck structure, complementary events, and informal "and" vs "or".