Sets, relations & functions
- n(A∪B) = n(A) + n(B) − n(A∩B).
- Functions: one-one, onto, bijection.
- Composition (gof)(x) = g(f(x)).
- Inverse exists iff bijection.
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JEE Main · Cheatsheet
Mathematics Quick Revision
Complete formula bank — algebra, trig, calculus, coordinate, vectors, prob.
Quadratic & complex numbers
Roots
x = (−b ± √(b²−4ac))/2a
Sum/Prod
α+β=−b/a, αβ=c/a
Nature
D>0 real distinct; D=0 equal; D<0 imag
Euler
eⁱᶿ = cosθ + i sinθ
De Moivre
(cosθ+i sinθ)ⁿ = cos nθ + i sin nθ
|z|
√(x²+y²)
Sequences & series
AP nth
a + (n−1)d
AP sum
n/2 (2a + (n−1)d)
GP nth
arⁿ⁻¹
GP sum
a(rⁿ−1)/(r−1)
GP inf (|r|<1)
a/(1−r)
Σn
n(n+1)/2
Σn²
n(n+1)(2n+1)/6
Σn³
[n(n+1)/2]²
Binomial theorem
Expansion
(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ
General term
Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ
Middle term
n even → (n/2+1)th; odd → 2 middle
Sum of coeffs
put a=b=1
Permutations & combinations
nPr
n!/(n−r)!
nCr
n!/(r!(n−r)!)
nCr + nCr₋₁
n+1Cr
Circular
(n−1)!
Identical objs
n!/(p!q!r!)
Trigonometry
Identity
sin²+cos²=1; 1+tan²=sec²
sin(A±B)
sinA cosB ± cosA sinB
cos(A±B)
cosA cosB ∓ sinA sinB
tan(A±B)
(tanA±tanB)/(1∓tanA tanB)
2A
sin2A=2sinA cosA; cos2A=1−2sin²A
Sine rule
a/sinA = b/sinB = c/sinC = 2R
Cosine rule
a² = b² + c² − 2bc cosA
Area △
½ab sinC = √(s(s−a)(s−b)(s−c))
Inverse trig & properties
- sin⁻¹x + cos⁻¹x = π/2.
- tan⁻¹x + cot⁻¹x = π/2.
- tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy<1.
- sin⁻¹(sin x) = x for x ∈ [−π/2, π/2].
Straight lines & circles
Line
y − y₁ = m(x − x₁)
Slope-intercept
y = mx + c
Distance from line
|ax₀+by₀+c|/√(a²+b²)
Angle between
tanθ = |(m₁−m₂)/(1+m₁m₂)|
Circle
x²+y²+2gx+2fy+c=0; centre(−g,−f)
Tangent length
√(x₁²+y₁²+2gx₁+2fy₁+c)
Conic sections
Parabola
y² = 4ax; focus (a,0); directrix x=−a
Ellipse
x²/a²+y²/b²=1; e=√(1−b²/a²)
Hyperbola
x²/a²−y²/b²=1; e=√(1+b²/a²)
Latus rectum ellipse
2b²/a
Limits, continuity & differentiation
lim sin x/x
1 (x→0)
lim (1+x)^(1/x)
e (x→0)
lim (a^x−1)/x
ln a
d/dx xⁿ
nxⁿ⁻¹
d/dx sinx
cosx
d/dx eˣ
eˣ
d/dx lnx
1/x
Chain
dy/dx = dy/du · du/dx
Product
(uv)' = u'v + uv'
Applications of derivatives
- Tangent slope = f'(x); normal slope = −1/f'(x).
- Maxima/minima: f'(x)=0 & sign change; 2nd derivative test.
- Rolle: f(a)=f(b) ⇒ ∃c f'(c)=0.
- LMVT: f'(c) = (f(b)−f(a))/(b−a).
- Increasing: f'(x)>0; concave up f''(x)>0.
- Approximation: f(x+Δx) ≈ f(x) + f'(x)Δx.
Integration
∫xⁿ
xⁿ⁺¹/(n+1)+C
∫1/x
ln|x|+C
∫eˣ
eˣ+C
∫sec²x
tanx+C
∫dx/√(a²−x²)
sin⁻¹(x/a)+C
∫dx/(a²+x²)
(1/a)tan⁻¹(x/a)+C
By parts
∫u dv = uv − ∫v du
Definite
∫ₐᵇ f = F(b)−F(a)
Tip: ILATE: Inverse, Log, Alg, Trig, Exp — choose u.
Differential equations
- Variable separable: f(y)dy = g(x)dx.
- Linear (dy/dx + Py = Q): IF = e^∫P dx; y·IF = ∫Q·IF dx.
- Homogeneous: substitute y = vx.
- Order = highest derivative; degree = power of highest.
Vectors & 3D
Dot
a·b = |a||b|cosθ
Cross
|a×b| = |a||b|sinθ; direction ⟂ both
Scalar triple
[a b c] = a·(b×c) (volume)
Line
r = a + λb
Plane
r·n = d
Dist pt-plane
|ax₀+by₀+cz₀+d|/√(a²+b²+c²)
Matrices & determinants
- |A^T|=|A|; |kA|=kⁿ|A| for n×n.
- |AB|=|A|·|B|.
- A⁻¹ = adj(A)/|A|.
- System AX=B: unique if |A|≠0.
- Cramer: xᵢ = |Aᵢ|/|A|.
Probability & statistics
P(A∪B)
P(A)+P(B)−P(A∩B)
Conditional
P(A|B) = P(A∩B)/P(B)
Bayes
P(A|B)=P(B|A)P(A)/ΣP(B|Aᵢ)P(Aᵢ)
Binomial
P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ
Mean/Var (Bin)
np, npq
Mean (grouped)
Σfᵢxᵢ/Σfᵢ
Variance
Σf(x−x̄)²/N
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