Class 12 Maths: Chapter-wise Guide and Formula Sheet
Class 12 mathematics builds on algebra and calculus. This guide gives a short explanation of every major chapter with the formulas you must know. Revise it a day before the exam and test yourself with the 50-question quiz.
Relations and Functions
Know reflexive, symmetric and transitive relations, and the difference between one-one, onto and bijective functions. A function has an inverse only if it is bijective.
Key formulas and facts:
- Relations from A to B: 2^(mn)
- Functions from A to B: n(B)^n(A)
- Bijections on n elements: n!
Inverse Trigonometric Functions
Always stay within the principal value branch. Learn the domain and range table, since most mistakes come from using the wrong branch.
Key formulas and facts:
- sin⁻¹x: [−π/2, π/2]
- cos⁻¹x: [0, π]
- tan⁻¹x: (−π/2, π/2)
- sin⁻¹x + cos⁻¹x = π/2
Matrices and Determinants
Matrix multiplication needs matching inner orders, and it is not commutative. Determinants help find the area of triangles, the inverse and the solution of linear equations.
Key formulas and facts:
- (AB)′ = B′A′
- |kA| = kⁿ|A| for an n×n matrix
- A⁻¹ = adj(A) / |A|
- Area = ½|determinant of vertices|
Continuity and Differentiability
A function is continuous at a point if its left limit, right limit and value are equal. Master the chain rule, logarithmic differentiation and parametric forms.
Key formulas and facts:
- d/dx (xⁿ) = nxⁿ⁻¹
- d/dx (sin x) = cos x; d/dx (cos x) = −sin x
- d/dx (tan x) = sec²x; d/dx (log x) = 1/x
- d/dx (eˣ) = eˣ
Applications of Derivatives
Used for rate of change, tangents, increasing and decreasing functions and maxima and minima. For optimisation, form one function of one variable, then use f′(x) = 0 and the second derivative test.
Key formulas and facts:
- Slope of tangent = dy/dx
- Increasing if f′(x) > 0
- Maxima if f″(x) < 0; minima if f″(x) > 0
Integrals
Integration is the reverse of differentiation. Practise substitution, parts and partial fractions, and use properties of definite integrals to save time.
Key formulas and facts:
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
- ∫(1/x) dx = log|x| + C
- ∫eˣ dx = eˣ + C
- ∫ udv = uv − ∫ vdu
- ∫ₐᵇ f(x)dx = F(b) − F(a)
Differential Equations
Find the order and degree, then solve by separating variables or the linear form with integrating factor.
Key formulas and facts:
- Linear: dy/dx + Py = Q
- Integrating factor = e^(∫P dx)
Vectors and Three Dimensional Geometry
Vectors follow component rules, and the dot and cross products give angle and area. In 3D, direction cosines and ratios describe lines.
Key formulas and facts:
- |a| = √(x² + y² + z²)
- a·b = |a||b|cos θ
- |a × b| = |a||b| sin θ
- l² + m² + n² = 1
Linear Programming and Probability
Draw the feasible region, find its corner points and test the objective function. In probability, learn conditional probability, Bayes' theorem and binomial distribution.
Key formulas and facts:
- P(A|B) = P(A∩B)/P(B)
- Independent: P(A∩B) = P(A)P(B)
- Bayes: P(E|A) = P(E)P(A|E)/ΣP(Eᵢ)P(A|Eᵢ)