79 atoms behind every NDA Maths question

If α, β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. Never solve; use structure. Cross-chapter into Complex Numbers, M&D, Properties of Triangle, Trig Identities, and Sequence & Series — and 43% HARD overall, third-toughest principle in the bank.

Whenever you need a minimum of a sum or maximum of a product under a constraint, AM-GM is the lever. Spans Sequence & Series, Application of Derivatives, Trig Identities and Logarithms; questions disguise the inequality across chapter lines.

If a, b, c are in AP, then 2b = a + c. The bank disguises this across Lines (collinearity, family of lines), Logarithms, M&D, P&C, Probability, Properties of Triangle, Inverse Trig, and Trig Identities — nine chapters of cross-chapter reach.

Multiplicative counterpart to AP. Pairs with AM-GM in the bank's highest-yield compound recipe.

Sₙ for AP, GP, and special telescoping series. Knowing the closed forms saves 2-3 minutes per question.

Stronger than AM-GM alone — chains three means with equality iff all values equal.

The pre-Vieta toolkit. The trick is recognising structure before brute-forcing.

If α, β are roots and Sₙ = αⁿ + βⁿ, then Sₙ₊₁ = (α + β)Sₙ − αβ·Sₙ₋₁.

Pairs with AP/GP and trig equations frequently. Memorise three; derive the rest.

Binary numbers, factorial divisibility, Legendre's formula. Rarely tested but easy when present.

sin 2A = 2 sin A cos A and cousins. 44% HARD overall — second-hardest principle in the bank after Sine/Cosine rules. Splits across Trig Identities Multi/Half-Angle and Properties of Triangle.

The base trig identity that unlocks double angle, product-to-sum, and most identity manipulation. Tagged across Trig Identities + Trig Equations.

a/sin A = 2R and c² = a² + b² − 2ab cos C. The hardest principle in the live bank — 57% of tagged questions are HARD. Drives every "solve the triangle" problem; also surfaces in Height & Distance and inside trig-laden determinants and in-circle problems.

Unlocks tan A + tan B + tan C = tan A · tan B · tan C and similar projection identities.

1 + tan² = sec², 1 + cot² = csc². The substrate of every trig manipulation.

2 sin A cos B = sin(A+B) + sin(A−B) and cousins. Used to telescope or factor trig sums.

Standard values at 30°/45°/60°/90° plus sign by quadrant. The lowest-level skill but tested everywhere.

Principal-range rules + sum/difference of inverse trig. The chapter is short and high-yield.

Pairs with Vieta in the ω-Vieta compound. Beyond the named subtopic, ω appears explicitly inside Complex Numbers' modulus problems, M&D's special determinants, and Quadratic Equation questions where x² + x + 1 = 0 unlocks ω³ = 1 simplifications.

|z|² = z·z̄, arg z, polar form. The base of every complex-number question.

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Tested rarely but elegant.

(√2 + 1)ⁿ + (√2 − 1)ⁿ is always an integer because the irrational parts cancel. NDA loves this.

Differentiability ⇒ continuity (not the converse). Modulus and greatest-integer are the standard counter-examples. Spans Limits & Continuity, Differentiation, and Functions.

lim sin x / x = 1, lim (1 + 1/x)ˣ = e, indeterminate forms via L'Hôpital.

Left limit = right limit = f(c). Piecewise problems pair this with modulus.

f'(x) = 0 at critical points, f''(x) tells you max vs min. AM-GM is often the shorter alternative.

∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx. Reduces many hard-looking integrals to plug-and-add.

(f(g(x)))' = f'(g(x))·g'(x). Plus log-differentiation for products of powers.

When y is given via parameter t or implicitly via F(x, y) = 0.

[x] and |x| at integers — left and right limits diverge. NDA loves the edge cases.

Algebraic, trig, and composite substitutions. Practice ~20 standard forms.

Rational integrands decompose. NDA-style: assume coefficients, compare numerators.

Pattern recognition — if the integrand is e^x times f + f', the integral is e^x · f.

∫₋ₐᵃ f(x)dx = 0 if f odd, 2∫₀ᵃ if even. Spotting the symmetry is the trick.

Setting up the right integral with correct limits is 80% of the work; computing is mechanical.

Order = highest derivative, degree = highest power once free of fractional/derivative forms.

Separation of variables + integrating-factor for first-order linear ODE.

d² = Σ(xᵢ − yᵢ)². Trivial but appears everywhere.

Point dividing a segment in m:n. Centroid, in-centre, circumcentre all build from this.

Point-slope, two-point, intercept forms. Family of lines through a point.

Centre + radius from general form via completing the square.

Focus, directrix, latus rectum, focal chord. The vocabulary is tested as much as the math.

x²/a² + y²/b² = 1 with e² = 1 − b²/a². Sum of focal distances = 2a.

x²/a² − y²/b² = 1 with e² = 1 + b²/a². Asymptotes slope ±b/a.

Direction cosines square-sum to 1 — the 3D unit-vector identity.

Vector form, Cartesian form, foot of perpendicular, distance between skew lines.

Coordinate-geometry questions about polygons (area, centroid, type of triangle).

Angle between vectors, projection, perpendicularity test.

Area, volume, coplanarity test via determinant.

r = (1−t)a + tb for the line; (mb + na)/(m+n) for the section.

ΣC(n,r) = 2ⁿ, C(n,r) = C(n,n−r), Pascal's rule. Spans Binomial Theorem and P&C primarily, plus M&D / Sets / Statistics questions where the identity is the key step.

Standard arrangements, plus boys-girls-together, no-two-X-adjacent, vowel constraints.

n!/(r!(n−r)!). Compute small cases by hand; recognise C(n,r) = C(n, n−r).

Digit-arrangement counting with constraints (no zero in lead, even, prime, etc.).

How many lines/triangles can be drawn from n points (no 3 collinear).

n(A∪B) = n(A) + n(B) − n(A∩B) and the three-set generalisation. The complement form 1 − P(none) is the other face of the same identity, used heavily in Binomial Distribution and Probability via Counting questions.

P(A|B) = P(A ∩ B) / P(B). Watch for the "given that" framing — many classical-looking questions are conditional.

The base. Sample-space construction is usually the actual work.

Independence vs mutual exclusivity — students confuse them. Independence multiplies; ME adds.

Compute P(A∪B), P(A∩B), P(Aᶜ) given various relations.

When 'at least one X' is the question, complement is almost always faster.

P(X=k) = C(n,k)p^k q^(n-k); mean = np, variance = npq. One chapter, two formulas.

Same skeleton, different surface. Recognise the sample-space template.

Mean, median, mode for grouped and ungrouped data. The single highest-yield subtopic for a weak student.

σ² = E[(X − μ)²]. Mean deviation about mean vs median is a common trap.

CV = σ/μ × 100. Used to compare variability across data sets of different scales.

y = bx + a, with b = r·σ_y/σ_x. Two regression lines intersect at (x̄, ȳ).

Class boundaries, frequency density (when widths differ), ogive.

Piecewise splitting at zero. The principle behind the 2023 modulus spike — broadest cross-chapter reach in the bank (10 chapters tagged), plus invocations across App of Derivatives, Apps of Integration, Functions, Linear Inequalities, Probability, Quad Eq and Sets & Relations.

De Morgan, distributivity, symmetric difference. Read each statement carefully.

Domain restrictions from √, log, 1/x; range from value-set analysis.

(f ∘ g)(x) and f⁻¹. The composition order matters; inverse exists only for bijections.

f(x + 1) = ... or 4f(x) − f(1/x) = ... — solve for the unknown function.

[x] behaviour at integers, in limits, in integrals. Edge-case heavy.

Equivalence relation = R, S, T. Each property tested with concrete relations.

Largest single-chapter principle. Cofactor expansion + row/column operations + special determinant patterns (trig, complex, polynomial).

A⁻¹ = adj(A) / det(A). Adjoint properties are tested directly (e.g., adj(adj(A)) = det(A)^(n−2) · A).

Each type has 1-2 defining properties; questions test which one applies.

Δ, Δₓ, Δᵧ, Δ_z. Consistency conditions are the most tested aspect.