Class 12 Mathematics Formulas
1. Relations and Functions
- Types of relations: Reflexive, symmetric, transitive, equivalence
- Composition of functions: (f o g)(x) = f(g(x))
- Inverse function: f^-1 exists if f is bijective
2. Inverse Trigonometric Functions
- sin^-1 x: Domain [-1,1], range [-pi/2, pi/2]
- cos^-1 x: Domain [-1,1], range [0, pi]
- tan^-1 x: Domain R, range (-pi/2, pi/2)
- Properties: sin^-1(-x) = -sin^-1 x, and similar identities
3. Matrices
- Matrix operations: Addition, scalar multiplication, matrix multiplication
- Transpose: A^T
- Inverse: A^-1 such that A A^-1 = I
- Determinant of 2x2: |a b; c d| = ad - bc
4. Determinants
- Determinant of 3x3: a1(b2c3 - b3c2) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2)
- Properties: |AB| = |A||B|, |A^T| = |A|
- Cramer's rule: xi = |Ai| / |A|
5. Continuity and Differentiability
- Continuity: lim x->a f(x) = f(a)
- Differentiability: f'(a) exists
- Rules: Chain rule, product rule, quotient rule
- Rolle's theorem: If f(a) = f(b) and f is continuous and differentiable, then f'(c) = 0 for some c in (a,b)
- Mean value theorem: f'(c) = [f(b) - f(a)]/(b-a)
6. Application of Derivatives
- Increasing/decreasing: f'(x) > 0 increasing, f'(x) < 0 decreasing
- Maxima/minima: f'(x) = 0 and f''(x) > 0 minimum, f''(x) < 0 maximum
- Rate of change: dy/dx
7. Integrals
- Indefinite integral: integral f(x) dx = F(x) + C
- Definite integral: integral from a to b f(x) dx = F(b) - F(a)
- Fundamental theorem: d/dx integral from a to x f(t) dt = f(x)
- Integration by parts: integral u dv = uv - integral v du
- Integration by substitution: integral f(g(x)) g'(x) dx = integral f(u) du
8. Application of Integrals
- Area under curve: integral from a to b f(x) dx
- Area between curves: integral from a to b |f(x) - g(x)| dx
9. Differential Equations
- Order and degree: Describe the structure of the differential equation
- General solution: Contains arbitrary constants
- Particular solution: Satisfies initial conditions
- Separable variables: dy/dx = f(x)g(y) implies integral dy/g(y) = integral f(x) dx
10. Vector Algebra
- Vector addition: a + b
- Scalar multiplication: k a
- Dot product: a . b = |a||b| cos theta
- Cross product: a x b
- Scalar triple product: [a b c] = a . (b x c)
11. Three Dimensional Geometry
- Direction cosines: l = cos alpha, m = cos beta, n = cos gamma
- Equation of line: (x - x1)/l = (y - y1)/m = (z - z1)/n
- Equation of plane: ax + by + cz + d = 0
- Distance from point to plane: |ax1 + by1 + cz1 + d|/sqrt(a^2 + b^2 + c^2)
12. Linear Programming
- Objective function: Z = ax + by
- Constraints: Linear inequalities
- Feasible region: Intersection of constraint regions
- Optimal solution: Corner points of feasible region
13. Probability
- Random variable: Function from sample space to real numbers
- Probability distribution: P(X = xi) for each xi
- Mean: E(X) = sum xi P(X = xi)
- Variance: Var(X) = E(X^2) - [E(X)]^2
- Binomial distribution: P(X = k) = C(n,k) p^k (1-p)^(n-k)
- Poisson distribution: P(X = k) = e^-lambda lambda^k / k!
- Normal distribution: f(x) = (1/(sigma sqrt(2pi))) e^(-(x-mu)^2/(2sigma^2))