Formula sheetClass 12
CBSE Class 12 Maths Formula Sheet
The Class 12 NCERT formulas on a few pages, from inverse trigonometric ranges to Bayes' theorem, laid out for the last month before the boards.
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01Relations and inverse trigonometry
- Equivalence relation reflexive, symmetric and transitive
- Functions one-one (injective), onto (surjective), bijective = both
- Principal value ranges sin−1x in [−π/2, π/2]; cos−1x in [0, π]; tan−1x in (−π/2, π/2)
- Identities sin−1x + cos−1x = π/2; tan−1x + cot−1x = π/2
02Matrices and determinants
- Transpose (AB)' = B'A'; A is symmetric if A' = A, skew-symmetric if A' = −A
- Every square matrix A = ½(A + A') + ½(A − A')
- Inverse A−1 = adj(A)/|A|, defined when |A| ≠ 0
- Determinants |AB| = |A||B|; |kA| = kn|A| for an n × n matrix
- Area of a triangle ½ |x1 y1 1; x2 y2 1; x3 y3 1| (determinant)
- Linear system AX = B unique solution X = A−1B when |A| ≠ 0
03Continuity and derivatives
- Continuity at a limx→a f(x) = f(a)
- Derivatives sin−1x → 1/√(1 − x2); cos−1x → −1/√(1 − x2); tan−1x → 1/(1 + x2)
- Exponential and log ax → ax ln a; logax → 1/(x ln a)
- Logarithmic differentiation for y = uv, take logs before differentiating
- Parametric dy/dx = (dy/dt)/(dx/dt)
- Applications rate of change dy/dt; increasing where f' > 0; maximum where f' = 0 and f'' < 0
04Integrals
- Standard forms ∫xndx = xn+1/(n + 1) + C; ∫dx/x = ln|x| + C; ∫exdx = ex + C; ∫sin x dx = −cos x + C; ∫sec2x dx = tan x + C
- Special forms ∫dx/(a2 + x2) = (1/a)tan−1(x/a) + C; ∫dx/√(a2 − x2) = sin−1(x/a) + C
- By parts ∫u v dx = u∫v dx − ∫[u'∫v dx]dx
- Fundamental theorem ∫ab f(x)dx = F(b) − F(a)
- Properties ∫0a f(x)dx = ∫0a f(a − x)dx; ∫−aa f = 0 for odd f
- Area under a curve ∫ab y dx (simple curves, lines, circles, parabolas, ellipses)
05Differential equations
- Order and degree order = highest derivative; degree = its power when the equation is a polynomial in derivatives
- Variables separable separate and integrate both sides
- Homogeneous put y = vx
- Linear dy/dx + Py = Q; IF = e∫P dx; y·IF = ∫Q·IF dx + C
06Vectors and 3D lines
- Magnitude |a| = √(a12 + a22 + a32)
- Dot product a·b = |a||b|cosθ
- Cross product |a × b| = |a||b|sinθ; area of parallelogram = |a × b|
- Projection of a on b (a·b)/|b|
- Direction cosines l2 + m2 + n2 = 1
- Line r = a + λb; Cartesian (x − x1)/a = (y − y1)/b = (z − z1)/c
- Angle between lines cosθ = |b1·b2|/(|b1||b2|)
- Shortest distance (skew lines) |(b1 × b2)·(a2 − a1)|/|b1 × b2|
07Linear programming and probability
- LPP the optimum of a bounded feasible region lies at a corner point; check every corner
- Conditional probability P(E|F) = P(E ∩ F)/P(F)
- Multiplication rule P(E ∩ F) = P(E)P(F|E)
- Independent events P(E ∩ F) = P(E)P(F)
- Total probability P(A) = Σ P(Ei)P(A|Ei)
- Bayes' theorem P(Ei|A) = P(Ei)P(A|Ei) / Σ P(Ej)P(A|Ej)
- Random variable mean E(X) = Σxipi
How to use this sheet
- Calculus is about half the paper: practise integration every day in the final month.
- In LPP, draw neatly and label every corner point with its coordinates.
- Read probability questions twice and write the events in words before using formulas.
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