JEE Maths Cheatsheet Cheatsheet

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Calculus — Derivatives & Integrals

HIGHEST WEIGHTAGE

Standard Derivatives

Function f(x)	Derivative f'(x)	Notes
xⁿ	n·xⁿ⁻¹	Power rule; n ∈ ℝ
aˣ	aˣ · ln(a)	Special case: eˣ → eˣ
ln(x)	1/x	Natural logarithm
logₐ(x)	1 / (x · ln(a))	Change of base: logₐ(x) = ln(x)/ln(a)
sin(x)	cos(x)	—
cos(x)	−sin(x)	Note: negative sign
tan(x)	sec²(x)	—
sec(x)	sec(x)·tan(x)	—
cosec(x)	−cosec(x)·cot(x)	Note: negative sign
cot(x)	−cosec²(x)	Note: negative sign
sin⁻¹(x)	1 / √(1 − x²)	Domain: \|x\| ≤ 1
tan⁻¹(x)	1 / (1 + x²)	Domain: all real x

Differentiation Rules

Sum/Difference Ruled/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Product Ruled/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

Quotient Ruled/dx [f(x)/g(x)] = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]²

Chain Ruled/dx [f(g(x))] = f'(g(x)) · g'(x)

Implicit DifferentiationDifferentiate both sides w.r.t. x; collect dy/dx terms

Logarithmic DifferentiationTake ln both sides, then differentiate; useful for y = [f(x)]^g(x)

Leibniz Ruled/dx ∫[a(x) to b(x)] f(t)dt = f(b(x))·b'(x) − f(a(x))·a'(x)

L'Hôpital's RuleIf lim f(x)/g(x) = 0/0 or ∞/∞, then lim = lim f'(x)/g'(x)

Standard Integrals (Indefinite)

Integrand	Integral (∫)	Notes
∫ xⁿ dx	xⁿ⁺¹/(n+1) + C	n ≠ −1
∫ 1/x dx	ln\|x\| + C	—
∫ eˣ dx	eˣ + C	—
∫ aˣ dx	aˣ/ln(a) + C	a > 0, a ≠ 1
∫ sin(x) dx	−cos(x) + C	—
∫ cos(x) dx	sin(x) + C	—
∫ sec²(x) dx	tan(x) + C	—
∫ cosec²(x) dx	−cot(x) + C	—
∫ tan(x) dx	ln\|sec(x)\| + C	—
∫ cot(x) dx	ln\|sin(x)\| + C	—
∫ sec(x) dx	ln\|sec(x) + tan(x)\| + C	—
∫ 1/(x² + a²) dx	(1/a)·tan⁻¹(x/a) + C	—
∫ 1/√(a² − x²) dx	sin⁻¹(x/a) + C	\|x\| < a
∫ 1/√(x² + a²) dx	ln\|x + √(x² + a²)\| + C	—

Integration Techniques

SubstitutionReplace part of integrand with u; ∫ f(g(x))g'(x) dx = ∫ f(u) du

By Parts∫ u dv = uv − ∫ v du; choose u by LIATE: Log, Inverse trig, Algebraic, Trig, Exponential

Partial FractionsDecompose P(x)/Q(x) into simpler fractions; Q(x) must be factorable

Trigonometric substitution√(a²−x²): x = a sin θ; √(a²+x²): x = a tan θ; √(x²−a²): x = a sec θ

Definite integral properties∫[a to b] = −∫[b to a]; ∫[a to b] = ∫[a to c] + ∫[c to b]; ∫[a to b] f(x)dx = ∫[a to b] f(a+b−x)dx

Walli's formula∫[0 to π/2] sinⁿ(x) dx = ∫[0 to π/2] cosⁿ(x) dx = (n−1)/n · (n−3)/(n−2) · ... · 1/2 · π/2 (n even) or 1 (n odd)

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JEE tip:Calculus carries 30–35% weightage in JEE Maths. Master: (1) Application of derivatives (maxima/minima, Rolle's, LMVT), (2) Definite integration with properties, (3) Area under curves, (4) Differential equations. Practice integration by recognition — many integrals can be solved by pattern matching.

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Coordinate Geometry Formulas

15–20% WEIGHTAGE

Straight Line — Formulas

Distanced = √[(x₂−x₁)² + (y₂−y₁)²]

Section formula (internal)P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)) — ratio m:n

MidpointM = ((x₁+x₂)/2, (y₁+y₂)/2)

Centroid of triangleG = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

Area of triangle|1/2 · [x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)]|; determinant method

IncenterI = ((ax₁+bx₂+cx₃)/(a+b+c), (ay₁+by₂+cy₃)/(a+b+c)) — a,b,c are side lengths opposite vertices

Slope of linem = (y₂−y₁)/(x₂−x₁) = tan θ

Equation formsPoint-slope: y−y₁ = m(x−x₁); Slope-intercept: y = mx + c; Two-point form; Intercept form: x/a + y/b = 1

General formAx + By + C = 0; slope = −A/B; y-intercept = −C/B; x-intercept = −C/A

Angle between linestan θ = |(m₁ − m₂)/(1 + m₁·m₂)|

Parallel linesm₁ = m₂; distance between parallel lines Ax+By+C₁=0 and Ax+By+C₂=0: |C₁−C₂|/√(A²+B²)

Perpendicular linesm₁ · m₂ = −1

Foot of perpendicularFrom (x₁,y₁) to Ax+By+C=0: use formula involving (h,k) = (x₁−Aλ, y₁−Bλ) where λ = (Ax₁+By₁+C)/(A²+B²)

Circle — Key Formulas

Standard form(x−h)² + (y−k)² = r²; center (h,k), radius r

General formx² + y² + 2gx + 2fy + c = 0; center (−g,−f), radius = √(g²+f²−c)

Diameter form(x−x₁)(x−x₂) + (y−y₁)(y−y₂) = 0; endpoints of diameter (x₁,y₁), (x₂,y₂)

Tangent at (x₁,y₁)xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0

Length of tangentFrom external point (x₁,y₁): L = √(S₁) where S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c

Tangent conditionFor line y = mx + c to be tangent: c = ±r√(1+m²); for general line Ax+By+C=0: |C|/√(A²+B²) = r

Common tangents0: one circle inside other; 1: internally tangent; 2: intersecting; 3: externally tangent; 4: separate circles

Conic Sections — Properties

Conic	Standard Equation	Eccentricity (e)	Key Properties
Circle	x² + y² = r²	e = 0	Set of points equidistant from center; no foci/directrix
Parabola	y² = 4ax	e = 1	Focus (a,0); Directrix x = −a; Latus rectum = 4a; Axis: x-axis
Ellipse	x²/a² + y²/b² = 1 (a > b)	e < 1; e² = 1 − b²/a²	Foci (±ae, 0); Latus rectum = 2b²/a; Sum of distances = 2a
Hyperbola	x²/a² − y²/b² = 1	e > 1; e² = 1 + b²/a²	Foci (±ae, 0); Latus rectum = 2b²/a; Difference of distances = 2a; Asymptotes: y = ±(b/a)x
Rectangular Hyperbola	xy = c²	e = √2	Asymptotes are coordinate axes; rotated 45°; equilateral hyperbola

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Coordinate geometry shortcuts: (1) Use parametric forms for parabola (at², 2at) and ellipse (a cos θ, b sin θ). (2) Normal to parabola y²=4ax in slope form: y = mx − 2am − am³. (3) Chord of contact from (x₁,y₁) to circle/parabola/ellipse/hyperbola — just replace x²→xx₁, y²→yy₁ in standard equation.

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Trigonometry Identities

ESSENTIAL FORMULAS

Fundamental Trigonometric Identities

Pythagoreansin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ

Reciprocalcosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ

Quotienttan θ = sin θ / cos θ; cot θ = cos θ / sin θ

Double anglesin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2tan θ/(1−tan²θ)

Triple anglesin 3θ = 3sin θ − 4sin³θ; cos 3θ = 4cos³θ − 3cos θ; tan 3θ = (3t−t³)/(1−3t²) where t = tan θ

Half anglesin(θ/2) = √[(1−cos θ)/2]; cos(θ/2) = √[(1+cos θ)/2]; tan(θ/2) = sin θ/(1+cos θ) = (1−cos θ)/sin θ

Sum/Difference (sin)sin(A±B) = sin A cos B ± cos A sin B

Sum/Difference (cos)cos(A±B) = cos A cos B ∓ sin A sin B

Product-to-sum2 sin A cos B = sin(A+B) + sin(A−B); 2 cos A cos B = cos(A+B) + cos(A−B)

Sum-to-productsin C + sin D = 2 sin[(C+D)/2] cos[(C−D)/2]; cos C + cos D = 2 cos[(C+D)/2] cos[(C−D)/2]

Standard Angle Values

Angle	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	∞ (undefined)
cosec	∞	2	√2	2/√3	1
sec	1	2/√3	√2	2	∞ (undefined)
cot	∞	√3	1	1/√3	0

Inverse Trigonometry — Domain & Range

Function	Domain	Range (Principal Value)	Key Identity
sin⁻¹(x)	[−1, 1]	[−π/2, π/2]	sin⁻¹(x) + cos⁻¹(x) = π/2
cos⁻¹(x)	[−1, 1]	[0, π]	cos⁻¹(−x) = π − cos⁻¹(x)
tan⁻¹(x)	(−∞, ∞)	(−π/2, π/2)	tan⁻¹(x) + cot⁻¹(x) = π/2
cosec⁻¹(x)	R − (−1, 1)	[−π/2, 0) ∪ (0, π/2]	cosec⁻¹(x) = sin⁻¹(1/x)
sec⁻¹(x)	R − (−1, 1)	[0, π/2) ∪ (π/2, π]	sec⁻¹(x) = cos⁻¹(1/x)
cot⁻¹(x)	(−∞, ∞)	(0, π)	cot⁻¹(−x) = π − cot⁻¹(x)

Solving Trigonometric Equations

General solution (sin θ = sin α)θ = nπ + (−1)ⁿ α; n ∈ ℤ

General solution (cos θ = cos α)θ = 2nπ ± α; n ∈ ℤ

General solution (tan θ = tan α)θ = nπ + α; n ∈ ℤ

sin θ = 0θ = nπ; n ∈ ℤ

cos θ = 0θ = (2n+1)π/2; n ∈ ℤ

sin²θ = sin²αθ = nπ ± α

cos²θ = cos²αθ = nπ ± α

tan²θ = tan²αθ = nπ ± α

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Trigonometry pattern:JEE loves compound angle questions mixed with inverse trigonometry. Remember: sin⁻¹(x) + cos⁻¹(x) = π/2, tan⁻¹(x) + cot⁻¹(x) = π/2, tan⁻¹(x) + tan⁻¹(y) = tan⁻¹((x+y)/(1−xy)) if xy < 1. Practice simplifying inverse trig expressions.

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Probability & Statistics

9–12% WEIGHTAGE

Probability — Core Formulas

Classical probabilityP(A) = n(A)/n(S); where n(A) = favorable outcomes, n(S) = total outcomes

ComplementP(A') = 1 − P(A)

Addition ruleP(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Multiplication ruleP(A ∩ B) = P(A) · P(B|A) = P(B) · P(A|B)

Independent eventsP(A ∩ B) = P(A) · P(B); P(B|A) = P(B)

Conditional probabilityP(B|A) = P(A ∩ B) / P(A); P(A|B) = P(A ∩ B) / P(B)

Total probabilityP(B) = P(B|A₁)P(A₁) + P(B|A₂)P(A₂) + ... + P(B|Aₙ)P(Aₙ) where {Aᵢ} partition S

Binomial probabilityP(X=r) = ⁿCᵣ · pʳ · qⁿ⁻ʳ; q = 1 − p; n trials, r successes

Discrete Probability Distributions

Distribution	PMF/Formula	Mean (μ)	Variance (σ²)	Use Case
Bernoulli	P(X=1) = p; P(X=0) = q	p	pq	Single trial, 2 outcomes (success/fail)
Binomial	P(X=r) = ⁿCᵣ·pʳ·qⁿ⁻ʳ	np	npq	n independent Bernoulli trials
Poisson	P(X=r) = e⁻λ·λʳ/r!; λ = np (when n→∞, p→0)	λ	λ	Rare events in fixed interval; λ < 5
Geometric	P(X=r) = qʳ⁻¹·p	1/p	q/p²	Trials until first success

Statistics — Measures of Central Tendency

Mean (ungrouped)x̄ = Σxᵢ / n

Mean (grouped)x̄ = Σfᵢxᵢ / Σfᵢ

Median (ungrouped)Middle value when data is sorted; n odd: (n+1)/2th term; n even: average of n/2 and (n+2)/2th

Median (grouped)L + [(N/2 − cf)/f] · h; where L = lower class boundary of median class

Mode (grouped)L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] · h

Empirical relation3 Median = Mode + 2 Mean (approximate)

Statistics — Measures of Dispersion

RangeMaximum − Minimum

Mean deviation (MD)MD = Σ|xᵢ − x̄| / n (or Σfᵢ|xᵢ − x̄| / N for grouped)

Variance (σ²)σ² = Σ(xᵢ − x̄)² / n = Σxᵢ²/n − x̄²

Standard deviation (σ)σ = √(variance)

Variance shortcutσ² = (Σfᵢdᵢ²/N − (Σfᵢdᵢ/N)²) · h²; using step deviation method

Coefficient of variationCV = (σ/x̄) × 100%; used for comparing variability of different datasets

Quartile deviationQD = (Q3 − Q1)/2; semi-interquartile range

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Probability tricks:(1) For "at least one" problems, use complement: P(at least 1) = 1 − P(none). (2) Binomial approximation to Poisson when n ≥ 20, p ≤ 0.05. (3) For permutation problems with restrictions, handle constraints first, then arrange the rest. (4) Total probability + Bayes' theorem are combined frequently in JEE.

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Matrices & Determinants

HIGH SCORING

Matrix Operations & Properties

Addition(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ; Commutative: A + B = B + A; Associative: (A+B)+C = A+(B+C)

Multiplication(AB)ᵢⱼ = Σ Aᵢₖ · Bₖⱼ; NOT commutative: AB ≠ BA generally; Associative: (AB)C = A(BC)

Transpose(Aᵉ)ᵢⱼ = Aⱼᵢ; (AB)ᵉ = BᵉAᵉ; (Aᵉ)ᵉ = A; (A+B)ᵉ = Aᵉ + Bᵉ

Symmetric matrixA = Aᵉ; skew-symmetric: A = −Aᵉ; diagonal elements of skew-symmetric = 0

InverseA⁻¹ exists iff |A| ≠ 0; A⁻¹ = adj(A)/|A|; (AB)⁻¹ = B⁻¹A⁻¹

IdempotentA² = A; e.g., projection matrices

NilpotentAⁿ = 0 for some n; smallest such n = index of nilpotency

OrthogonalAᵀA = I (or AAᵀ = I); A⁻¹ = Aᵉ; |A| = ±1

InvolutoryA² = I; A⁻¹ = A

Tracetr(A) = Σ Aᵢᵢ (sum of diagonal); tr(AB) = tr(BA); tr(A+B) = tr(A) + tr(B)

Determinant Formulas

2×2 determinant|a b; c d| = ad − bc

3×3 determinanta(ei−fh) − b(di−fg) + c(dh−eg) (Sarrus rule or cofactor expansion)

Propertiesdet(Aᵉ) = det(A); det(AB) = det(A)·det(B); det(kA) = kⁿ·det(A) for n×n matrix

Singular matrixdet(A) = 0; No inverse exists; system may have no solution or infinite solutions

Row operationsSwapping rows: sign change; Scaling row by k: det multiplied by k; Adding multiple of row to another: no change

Adjugate/Adjointadj(A)ᵢⱼ = Cⱼᵢ (cofactor transpose); A · adj(A) = |A| · I

Cramer's RuleFor Ax = b: xᵢ = |Aᵢ|/|A| where Aᵢ is A with column i replaced by b

Systems of Linear Equations — Matrix Method

Condition	Number of Solutions	Type	Geometric Interpretation
\|A\| ≠ 0 (consistent)	Unique solution	Independent system	Three planes intersect at one point
\|A\| = 0, (adj A)B ≠ 0	No solution	Inconsistent system	Three planes form a triangular prism
\|A\| = 0, (adj A)B = 0	Infinitely many solutions	Dependent system	Three planes coincide or intersect along a line

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Matrices tricks:(1) For symmetric + skew-symmetric decomposition: A = (A+Aᵉ)/2 + (A−Aᵉ)/2. (2) Rank of matrix = maximum number of linearly independent rows/columns. (3) Cayley-Hamilton theorem: every matrix satisfies its characteristic equation. (4) Eigenvalues: |A−λI| = 0.

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3D Geometry

VISUAL + FORMULA

Distance & Direction Formulas in 3D

Distance between pointsd = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

Direction cosinesl = cos α, m = cos β, n = cos γ; l² + m² + n² = 1

Direction ratiosa, b, c proportional to l, m, n; l = a/√(a²+b²+c²), etc.

Angle between linescos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁²+b₁²+c₁²) · √(a₂²+b₂²+c₂²)]

Parallel linesa₁/a₂ = b₁/b₂ = c₁/c₂ (direction ratios proportional)

Perpendicular linesa₁a₂ + b₁b₂ + c₁c₂ = 0

Shortest distance (skew lines)d = |[d⃗ · (b⃗₁ × b⃗₂)]| / |b⃗₁ × b⃗₂| where d⃗ joins a point on each line, b⃗ = direction vectors

Equations of Lines & Planes

Entity	Equation	Key Terms
Line (vector)	r⃗ = a⃗ + λb⃗	a⃗ = position vector of a point; b⃗ = direction vector; λ = parameter
Line (Cartesian)	(x−x₁)/a = (y−y₁)/b = (z−z₁)/c	(x₁,y₁,z₁) = point on line; (a,b,c) = direction ratios
Line through 2 points	(x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−y₁) = (z−z₁)/(z₂−z₁)	Two points P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂)
Plane (general)	Ax + By + Cz = D	Normal vector n⃗ = (A, B, C); D/n⃗ = distance from origin (signed)
Plane (point-normal)	A(x−x₀) + B(y−y₀) + C(z−z₀) = 0	Plane through (x₀,y₀,z₀) with normal (A,B,C)
Plane (3 points)	Equation using determinant \|x−x₁, y−y₁, z−z₁; x₂−x₁, ...\| = 0	Passes through 3 non-collinear points
Plane (2 lines)	Special cases: parallel, intersecting, coplanar condition	Coplanar: [d⃗ · (b⃗₁ × b⃗₂)] = 0
Angle between planes	cos θ = \|n⃗₁ · n⃗₂\| / (\|n⃗₁\| · \|n⃗₂\|)	n⃗₁, n⃗₂ = normals of the two planes

Plane Relationships & Important Results

Distance from point to planed = |Ax₁ + By₁ + Cz₁ − D| / √(A² + B² + C²)

Angle between line and planesin θ = |Al + Bm + Cn| / (√(A²+B²+C²) · √(l²+m²+n²))

Bisector planesTwo angle bisectors of planes π₁ and π₂: |Ax+By+Cz−D|/√(A²+B²+C²) = ±|A'x+B'y+C'z−D'|/√(A'²+B'²+C'²)

Line of intersectionDirection = n⃗₁ × n⃗₂; find a point satisfying both plane equations

Coplanarity conditionFour points or two lines are coplanar if certain determinant = 0

Image of point in planeFoot of perpendicular + reflection: Q = P − 2·(distance along normal)·n̂

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3D Geometry approach: Use vector approach for complex problems — often simpler than Cartesian. For two skew lines, shortest distance formula using cross product is faster. For plane problems, always find the normal vector first.

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Complex Numbers

8–10% WEIGHTAGE

Complex Number — Basics & Forms

Standard formz = a + ib; a = real part (Re(z)), b = imaginary part (Im(z))

Polar formz = r(cos θ + i sin θ) = r·e^(iθ); r = |z| = √(a²+b²), θ = arg(z) = tan⁻¹(b/a)

Euler's forme^(iθ) = cos θ + i sin θ; e^(iπ) + 1 = 0 (Euler's identity)

Conjugatez̄ = a − ib; |z| = |z̄|; z·z̄ = |z|² = a² + b²; z + z̄ = 2a; z − z̄ = 2ib

Modulus|z₁ · z₂| = |z₁| · |z₂|; |z₁/z₂| = |z₁|/|z₂|; |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality)

Argumentarg(z₁·z₂) = arg(z₁) + arg(z₂); arg(z₁/z₂) = arg(z₁) − arg(z₂); principal value: (−π, π]

De Moivre's[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)

n-th rootsz^(1/n) = r^(1/n) · [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]; k = 0, 1, ..., n−1; n roots equally spaced on circle

Complex Number — Properties & Geometry

Square root of a+ib± [√((|z|+a)/2) + i·(b/|b|)·√((|z|−a)/2)]; use formula for quick calculation

Cube roots of unity1, ω, ω²; ω³ = 1; 1 + ω + ω² = 0; ω² = ω̄ = 1/ω; each = e^(2kπi/3)

Fourth roots of unity±1, ±i; i⁰ = 1, i¹ = i, i² = −1, i³ = −i, i⁴ = 1 (cycle of 4)

RotationMultiplying by e^(iα) rotates by angle α counterclockwise about origin

Geometric representationz = x + iy plotted as point (x,y) in Argand plane; |z| = distance from origin

Circle in complex planearg((z−z₁)/(z−z₂)) = α = angle subtended; |z − z₀| = r = circle center z₀, radius r

Perpendicular bisectorarg((z−z₁)/(z−z₂)) = ±π/2; or |z−z₁| = |z−z₂|

Equation of circlezz̄ − z̄z₀ − z z̄₀ + z₀z̄₀ = r²; or |z − z₀| = r

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Complex number shortcuts: (1) Use ω properties for rapid calculation: ω³ = 1, 1+ω+ω² = 0. (2) For |z₁ − z₂|, interpret as distance between points z₁ and z₂ in Argand plane. (3) Locus problems become geometry problems — triangle inequality, perpendicular bisector, angle bisector.

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Sequences & Series Shortcuts

QUICK REVISION

Arithmetic Progression (AP)

General forma, a+d, a+2d, ..., a+(n−1)d; a = first term, d = common difference

n-th termTₙ = a + (n−1)d

Sum of n termsSₙ = n/2 · [2a + (n−1)d] = n/2 · [a + l] where l = last term

AM of two numbersAM = (a + b)/2; 3 numbers in AP: a−d, a, a+d

Sum propertySₙ / n = Average of first and last term = Average of all terms

Useful resultsIf Tₚ = q, Tᵧ = r, Tᵣ = p then p, q, r are in AP; n·Tₙ = S₂ₙ − S₂ₙ₋₁

Geometric Progression (GP)

General forma, ar, ar², ..., arⁿ⁻¹; a = first term, r = common ratio

n-th termTₙ = a · rⁿ⁻¹

Sum of n terms (r ≠ 1)Sₙ = a(rⁿ − 1)/(r − 1)

Sum to infinity (|r| < 1)S∞ = a / (1 − r)

GM of two numbersGM = √(ab); 3 numbers in GP: a/r, a, ar

ImportantIf GP has all positive terms, AM ≥ GM ≥ HM (AM–GM–HM inequality)

Harmonic Progression (HP) & Special Series

HP definitionReciprocals of AP terms; if a, b, c are in AP then 1/a, 1/b, 1/c are in HP

HM of two numbersHM = 2ab/(a + b)

n-th term of HP1/Tₙ = 1/a + (n−1)d where d = common difference of corresponding AP

AM ≥ GM ≥ HM(a+b)/2 ≥ √(ab) ≥ 2ab/(a+b); equality when a = b

Sum of squaresΣn² = n(n+1)(2n+1)/6; Σn = n(n+1)/2; Σn³ = [n(n+1)/2]²

Arithmetico-Geometric (AGP)a, (a+d)r, (a+2d)r², ...; sum using multiply & subtract method

AGP sum (n terms)Sₙ = a/(1−r) − [a+(n−1)d]rⁿ/(1−r) + dr(1−rⁿ⁻¹)/(1−r)²

Sum of first n natural numbersΣn = n(n+1)/2 = T(n+1, 2) — triangular number

Means & Special Sequences

Sequence Type	n terms	Sum Formula	Key Identity
AP	a, a+d, a+2d, ...	Sₙ = n/2 [2a+(n−1)d]	If a, b, c in AP: 2b = a + c
GP	a, ar, ar², ...	Sₙ = a(rⁿ−1)/(r−1); S∞ = a/(1−r)	If a, b, c in GP: b² = ac
HP	1/a, 1/(a+d), ...	No simple sum formula; use AP of reciprocals	If a, b, c in HP: 2/b = 1/a + 1/c
AGP	a, (a+d)r, (a+2d)r², ...	Use S = S − rS method	Combines AP & GP patterns
Σn²	1, 4, 9, 16, ...	n(n+1)(2n+1)/6	Sum of squares of first n naturals
Σn³	1, 8, 27, 64, ...	[n(n+1)/2]²	Cube of sum = sum of cubes
Fibonacci	1, 1, 2, 3, 5, 8, ...	Fₙ = Fₙ₋₁ + Fₙ₋₂	No closed sum form; ratio Fₙ₊₁/Fₙ → φ (golden ratio)

💡

Series shortcuts for JEE:(1) Method of differences: if Tₙ = f(n) − f(n−1), then Sₙ telescopes to f(n) − f(0). (2) AGP: always use multiply-by-r-and-subtract technique. (3) Use AM-GM-HM inequality for finding minimum values. (4) For series like Σ 1/n(n+1) = 1 − 1/(n+1) (telescoping).

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Function f(x)

Derivative f'(x)

Notes

xⁿ

n·xⁿ⁻¹

Power rule; n ∈ ℝ

aˣ

aˣ · ln(a)

Special case: eˣ → eˣ

ln(x)

1/x

Natural logarithm

logₐ(x)

1 / (x · ln(a))

Change of base: logₐ(x) = ln(x)/ln(a)

sin(x)

cos(x)

—

cos(x)

−sin(x)

Note: negative sign

tan(x)

sec²(x)

—

sec(x)

sec(x)·tan(x)

—

cosec(x)

−cosec(x)·cot(x)

Note: negative sign

cot(x)

−cosec²(x)

Note: negative sign

sin⁻¹(x)

1 / √(1 − x²)

Domain: |x| ≤ 1

tan⁻¹(x)

1 / (1 + x²)

Domain: all real x

Integrand

Integral (∫)

Notes

∫ xⁿ dx

xⁿ⁺¹/(n+1) + C

n ≠ −1

∫ 1/x dx

ln|x| + C

—

∫ eˣ dx

eˣ + C

—

∫ aˣ dx

aˣ/ln(a) + C

a > 0, a ≠ 1

∫ sin(x) dx

−cos(x) + C

—

∫ cos(x) dx

sin(x) + C

—

∫ sec²(x) dx

tan(x) + C

—

∫ cosec²(x) dx

−cot(x) + C

—

∫ tan(x) dx

ln|sec(x)| + C

—

∫ cot(x) dx

ln|sin(x)| + C

—

∫ sec(x) dx

ln|sec(x) + tan(x)| + C

—

∫ 1/(x² + a²) dx

(1/a)·tan⁻¹(x/a) + C

—

∫ 1/√(a² − x²) dx

sin⁻¹(x/a) + C

|x| < a

∫ 1/√(x² + a²) dx

ln|x + √(x² + a²)| + C

—

Conic

Standard Equation

Eccentricity (e)

Key Properties

Circle

x² + y² = r²

e = 0

Set of points equidistant from center; no foci/directrix

Parabola

y² = 4ax

e = 1

Focus (a,0); Directrix x = −a; Latus rectum = 4a; Axis: x-axis

Ellipse

x²/a² + y²/b² = 1 (a > b)

e < 1; e² = 1 − b²/a²

Foci (±ae, 0); Latus rectum = 2b²/a; Sum of distances = 2a

Hyperbola

x²/a² − y²/b² = 1

e > 1; e² = 1 + b²/a²

Foci (±ae, 0); Latus rectum = 2b²/a; Difference of distances = 2a; Asymptotes: y = ±(b/a)x

Rectangular Hyperbola

xy = c²

e = √2

Asymptotes are coordinate axes; rotated 45°; equilateral hyperbola

Angle

0°

30°

45°

60°

90°

sin

1/2

1/√2

√3/2

cos

√3/2

1/√2

1/2

tan

1/√3

√3

∞ (undefined)

cosec

∞

√2

2/√3

sec

2/√3

√2

∞ (undefined)

cot

∞

√3

1/√3

Function

Domain

Range (Principal Value)

Key Identity

sin⁻¹(x)

[−1, 1]

[−π/2, π/2]

sin⁻¹(x) + cos⁻¹(x) = π/2

cos⁻¹(x)

[−1, 1]

[0, π]

cos⁻¹(−x) = π − cos⁻¹(x)

tan⁻¹(x)

(−∞, ∞)

(−π/2, π/2)

tan⁻¹(x) + cot⁻¹(x) = π/2

cosec⁻¹(x)

R − (−1, 1)

[−π/2, 0) ∪ (0, π/2]

cosec⁻¹(x) = sin⁻¹(1/x)

sec⁻¹(x)

R − (−1, 1)

[0, π/2) ∪ (π/2, π]

sec⁻¹(x) = cos⁻¹(1/x)

cot⁻¹(x)

(−∞, ∞)

(0, π)

cot⁻¹(−x) = π − cot⁻¹(x)

Distribution

PMF/Formula

Mean (μ)

Variance (σ²)

Use Case

Bernoulli

P(X=1) = p; P(X=0) = q

Single trial, 2 outcomes (success/fail)

Binomial

P(X=r) = ⁿCᵣ·pʳ·qⁿ⁻ʳ

npq

n independent Bernoulli trials

Poisson

P(X=r) = e⁻λ·λʳ/r!; λ = np (when n→∞, p→0)

Rare events in fixed interval; λ < 5

Geometric

P(X=r) = qʳ⁻¹·p

1/p

q/p²

Trials until first success

Condition

Number of Solutions

Type

Geometric Interpretation

|A| ≠ 0 (consistent)

Unique solution

Independent system

Three planes intersect at one point

|A| = 0, (adj A)B ≠ 0

No solution

Inconsistent system

Three planes form a triangular prism

|A| = 0, (adj A)B = 0

Infinitely many solutions

Dependent system

Three planes coincide or intersect along a line

Entity

Equation

Key Terms

Line (vector)

r⃗ = a⃗ + λb⃗

a⃗ = position vector of a point; b⃗ = direction vector; λ = parameter

Line (Cartesian)

(x−x₁)/a = (y−y₁)/b = (z−z₁)/c

(x₁,y₁,z₁) = point on line; (a,b,c) = direction ratios

Line through 2 points

(x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−y₁) = (z−z₁)/(z₂−z₁)

Two points P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂)

Plane (general)

Ax + By + Cz = D

Normal vector n⃗ = (A, B, C); D/n⃗ = distance from origin (signed)

Plane (point-normal)

A(x−x₀) + B(y−y₀) + C(z−z₀) = 0

Plane through (x₀,y₀,z₀) with normal (A,B,C)

Plane (3 points)

Equation using determinant |x−x₁, y−y₁, z−z₁; x₂−x₁, ...| = 0

Passes through 3 non-collinear points

Plane (2 lines)

Special cases: parallel, intersecting, coplanar condition

Coplanar: [d⃗ · (b⃗₁ × b⃗₂)] = 0

Angle between planes

cos θ = |n⃗₁ · n⃗₂| / (|n⃗₁| · |n⃗₂|)

n⃗₁, n⃗₂ = normals of the two planes

Sequence Type

n terms

Sum Formula

Key Identity

a, a+d, a+2d, ...

Sₙ = n/2 [2a+(n−1)d]

If a, b, c in AP: 2b = a + c

a, ar, ar², ...

Sₙ = a(rⁿ−1)/(r−1); S∞ = a/(1−r)

If a, b, c in GP: b² = ac

1/a, 1/(a+d), ...

No simple sum formula; use AP of reciprocals

If a, b, c in HP: 2/b = 1/a + 1/c

AGP

a, (a+d)r, (a+2d)r², ...

Use S = S − rS method

Combines AP & GP patterns

Σn²

1, 4, 9, 16, ...

n(n+1)(2n+1)/6

Sum of squares of first n naturals

Σn³

1, 8, 27, 64, ...

[n(n+1)/2]²

Cube of sum = sum of cubes

Fibonacci

1, 1, 2, 3, 5, 8, ...

Fₙ = Fₙ₋₁ + Fₙ₋₂

No closed sum form; ratio Fₙ₊₁/Fₙ → φ (golden ratio)