Formula sheetClass 11 and 12
JEE Maths Formula Sheet
The results you are expected to know without deriving them in the exam hall, from quadratic roots to shortest distances between skew lines. Every line here comes up in JEE Main papers year after year.
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01Algebra
- Quadratic roots x = [−b ± √(b2 − 4ac)]/2a; α + β = −b/a, αβ = c/a
- Nature of roots D > 0 real and distinct; D = 0 equal; D < 0 complex
- Arithmetic progression an = a + (n − 1)d, Sn = (n/2)[2a + (n − 1)d]
- Geometric progression an = arn−1, Sn = a(rn − 1)/(r − 1), S∞ = a/(1 − r) for |r| < 1
- Means AM ≥ GM ≥ HM for positive numbers
- Sums Σn = n(n + 1)/2, Σn2 = n(n + 1)(2n + 1)/6, Σn3 = [n(n + 1)/2]2
- Binomial theorem (x + y)n = Σ nCr xn−ryr; general term Tr+1 = nCr xn−ryr
- Logarithms log(xy) = log x + log y; logax = logbx/logba
02Counting and probability
- Permutations nPr = n!/(n − r)!
- Combinations nCr = n!/[r!(n − r)!]; nCr + nCr−1 = n+1Cr
- Circular arrangements (n − 1)!
- Addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Conditional probability P(A|B) = P(A ∩ B)/P(B)
- Bayes' theorem P(Ai|B) = P(B|Ai)P(Ai) / Σ P(B|Aj)P(Aj)
- Random variable mean μ = Σxipi, variance σ2 = Σxi2pi − μ2; binomial: mean np, variance npq
03Complex numbers
- Forms z = x + iy = r(cosθ + i sinθ) = reiθ, |z| = √(x2 + y2)
- Conjugate z z̄ = |z|2
- De Moivre (cosθ + i sinθ)n = cos nθ + i sin nθ
- Cube roots of unity 1, ω, ω2; 1 + ω + ω2 = 0, ω3 = 1
- Triangle inequality |z1 + z2| ≤ |z1| + |z2|
04Trigonometry
- Identities sin2θ + cos2θ = 1; 1 + tan2θ = sec2θ; 1 + cot2θ = cosec2θ
- Compound angles sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB
- tan(A ± B) (tanA ± tanB)/(1 ∓ tanA tanB)
- Double angle sin2A = 2sinA cosA; cos2A = 1 − 2sin2A = 2cos2A − 1; tan2A = 2tanA/(1 − tan2A)
- Triple angle sin3A = 3sinA − 4sin3A; cos3A = 4cos3A − 3cosA
- Sum to product sinC + sinD = 2sin[(C + D)/2]cos[(C − D)/2]; cosC + cosD = 2cos[(C + D)/2]cos[(C − D)/2]
- Sine and cosine rules a/sinA = b/sinB = c/sinC = 2R; c2 = a2 + b2 − 2ab cosC
- General solutions sinθ = sinα ⇒ θ = nπ + (−1)nα; cosθ = cosα ⇒ θ = 2nπ ± α; tanθ = tanα ⇒ θ = nπ + α
05Coordinate geometry
- Distance √[(x2 − x1)2 + (y2 − y1)2]
- Section formula [(mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)]
- Lines y = mx + c; y − y1 = m(x − x1); tanθ = |(m1 − m2)/(1 + m1m2)|
- Point to line distance |ax1 + by1 + c|/√(a2 + b2)
- Area of a triangle ½|x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|
- Circle x2 + y2 + 2gx + 2fy + c = 0, centre (−g, −f), r = √(g2 + f2 − c)
- Parabola y2 = 4ax focus (a, 0), directrix x = −a, latus rectum 4a, tangent y = mx + a/m
- Ellipse x2/a2 + y2/b2 = 1 (a > b) e = √(1 − b2/a2), foci (±ae, 0), latus rectum 2b2/a
- Hyperbola x2/a2 − y2/b2 = 1 e = √(1 + b2/a2), asymptotes y = ±(b/a)x
06Limits and derivatives
- Standard limits sin x/x → 1; (ex − 1)/x → 1; ln(1 + x)/x → 1 as x → 0; (1 + 1/x)x → e as x → ∞
- Power limit limx→a (xn − an)/(x − a) = nan−1
- Derivatives xn → nxn−1; ex → ex; ln x → 1/x; sin x → cos x; cos x → −sin x; tan x → sec2x
- Inverse trig sin−1x → 1/√(1 − x2); tan−1x → 1/(1 + x2)
- Rules (uv)' = u'v + uv'; (u/v)' = (u'v − uv')/v2; dy/dx = (dy/du)(du/dx)
- Maxima and minima f'(x) = 0 with f''(x) < 0 gives a maximum, f''(x) > 0 a minimum
- Mean value theorem f'(c) = [f(b) − f(a)]/(b − a) for some c in (a, b)
07Integrals
- Standard forms ∫xndx = xn+1/(n + 1) + C (n ≠ −1); ∫dx/x = ln|x| + C; ∫axdx = ax/ln a + C
- Trigonometric ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec2x dx = tan x + C; ∫tan x dx = ln|sec x| + C
- Special forms ∫dx/(x2 + a2) = (1/a)tan−1(x/a) + C; ∫dx/√(a2 − x2) = sin−1(x/a) + C
- More special forms ∫dx/(x2 − a2) = (1/2a)ln|(x − a)/(x + a)| + C; ∫dx/√(x2 ± a2) = ln|x + √(x2 ± a2)| + C
- By parts ∫u dv = uv − ∫v du (choose u by ILATE)
- Useful result ∫ex[f(x) + f'(x)]dx = exf(x) + C
- Definite integrals ∫ab f(x)dx = ∫ab f(a + b − x)dx
- Odd and even ∫−aa f(x)dx = 0 if f is odd, 2∫0a f(x)dx if f is even
08Differential equations
- Linear dy/dx + Py = Q; integrating factor e∫P dx; y·IF = ∫Q·IF dx + C
- Homogeneous substitute y = vx
- Variables separable collect x and y terms on opposite sides, then integrate
09Vectors and 3D
- Dot product a·b = |a||b|cosθ = a1b1 + a2b2 + a3b3
- Cross product |a × b| = |a||b|sinθ
- Projection of a on b (a·b)/|b|
- Scalar triple product [a b c] = a·(b × c); zero if coplanar
- Line r = a + λb; angle between lines cosθ = |b1·b2|/(|b1||b2|)
- Plane (JEE Advanced) ax + by + cz = d; point to plane |ax1 + by1 + cz1 − d|/√(a2 + b2 + c2)
- Skew lines, shortest distance |(a2 − a1)·(b1 × b2)|/|b1 × b2|
- Direction cosines l2 + m2 + n2 = 1
10Matrices, determinants, statistics
- Transpose and inverse (AB)T = BTAT; A−1 = adj(A)/|A| when |A| ≠ 0
- Determinants |AB| = |A||B|; |kA| = kn|A|; |adj A| = |A|n−1 (n × n)
- Cramer's rule x = Dx/D; a unique solution when D ≠ 0
- Variance σ2 = Σ(x − x̄)2/n = Σx2/n − x̄2; coefficient of variation = (σ/x̄) × 100
How to use this sheet
- Derive two results from each section once by hand, so you can rebuild them if memory fails.
- Keep this sheet beside you while doing previous papers; mark every formula you had to look up.
- Before the exam, rewrite the marked lines from memory on a blank page.
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