Mathematics Part I

NCERT Class 12 Maths Chapter 1 covers Relations and Functions, explaining types of relations (reflexive, symmetric, transitive, equivalence), types of functions (one-one, onto, bijective), composition of functions, and invertible functions.

• 1A relation R in set A is reflexive if (a, a) ∈ R for every a ∈ A, symmetric if (a, b) ∈ R implies (b, a) ∈ R, and transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R.
• 2An equivalence relation is one that is simultaneously reflexive, symmetric, and transitive; it partitions the set into mutually disjoint equivalence classes.
• 3A function f : X → Y is one-one (injective) if f(x₁) = f(x₂) implies x₁ = x₂, and onto (surjective) if every element of Y is the image of some element in X.
• 4A bijective function is both one-one and onto; for a finite set X, a function f : X → X is one-one if and only if it is onto — a property that does not hold for infinite sets.
• 5The composition gof of functions f : A → B and g : B → C is defined as gof(x) = g(f(x)); composition is not commutative in general (gof ≠ fog).

NCERT Class 12 Maths Chapter 2 covers Inverse Trigonometric Functions — the restrictions on domains and ranges of trigonometric functions that make their inverses well-defined, along with principal value branches and key properties used in calculus and engineering.

• 1Inverse trigonometric functions exist only when the domain of the original function is restricted to make it one-one and onto; the chosen restricted interval is called the principal value branch.
• 2Principal value branch domains and ranges: sin⁻¹: [−1,1] → [−π/2, π/2]; cos⁻¹: [−1,1] → [0,π]; tan⁻¹: ℝ → (−π/2, π/2); cot⁻¹: ℝ → (0,π); sec⁻¹: ℝ−(−1,1) → [0,π]−{π/2}; cosec⁻¹: ℝ−(−1,1) → [−π/2, π/2]−{0}.
• 3sin⁻¹x must not be confused with (sin x)⁻¹ = 1/sin x; the superscript −1 on an inverse trig function denotes the inverse function, not a reciprocal.
• 4The graph of y = sin⁻¹x (or any inverse trig function) is the reflection of the original graph across the line y = x; the principal value branch is highlighted as the standard output.
• 5Key identities hold within principal value domains: sin(sin⁻¹x) = x for x ∈ [−1,1] and sin⁻¹(sinx) = x for x ∈ [−π/2, π/2], with analogous results for all six functions.

Class 12 Maths Chapter 3 covers Matrices — an ordered rectangular array of numbers or functions — including types of matrices, matrix operations (addition, scalar multiplication, matrix multiplication), transpose, symmetric and skew-symmetric matrices, and invertible matrices.

• 1A matrix of order m × n has m rows and n columns and mn elements total; element aij lies in the ith row and jth column.
• 2Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B; the product C = AB has order m × p if A is m × n and B is n × p.
• 3Matrix multiplication is not commutative in general: AB ≠ BA even when both products are defined.
• 4The transpose A′ of an m × n matrix is the n × m matrix with rows and columns interchanged; key property: (AB)′ = B′A′.
• 5A square matrix A is symmetric if A′ = A, and skew-symmetric if A′ = −A; every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.

NCERT Class 12 Maths Chapter 4 covers Determinants — a number associated with every square matrix that determines properties such as uniqueness of solutions to linear equations, with applications in finding areas of triangles and solving systems of equations using the matrix method.

• 1The determinant of a 2×2 matrix A = [[a11,a12],[a21,a22]] is defined as a11·a22 − a21·a12; only square matrices have determinants.
• 2A 3×3 determinant can be expanded along any row or column — all six expansions yield the same value; expanding along the row or column with the most zeros simplifies calculation.
• 3The minor Mij of element aij is the determinant obtained by deleting the ith row and jth column; the cofactor is Aij = (−1)^(i+j) · Mij.
• 4The adjoint of matrix A is the transpose of the cofactor matrix; A⁻¹ = (1/|A|) · adj A, which exists if and only if |A| ≠ 0 (non-singular matrix).
• 5Area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) equals (1/2)|det([x1,y1,1; x2,y2,1; x3,y3,1])|; three points are collinear when this determinant is zero.

Class 12 Maths Chapter 5 — Continuity and Differentiability — covers the formal definitions of continuity and differentiability, the chain rule, derivatives of inverse trigonometric, exponential, and logarithmic functions, logarithmic differentiation, parametric differentiation, and second-order derivatives.

• 1A function f is continuous at c if lim(x→c) f(x) = f(c); the greatest integer function [x] is discontinuous at every integer.
• 2Every differentiable function is continuous, but the converse is false — f(x) = |x| is continuous at 0 but not differentiable there.
• 3The chain rule states df/dx = (dv/dt)·(dt/dx) for composite f = v∘u with t = u(x), enabling differentiation of functions like sin(x²).
• 4Derivatives of inverse trig functions: d/dx(sin⁻¹x) = 1/√(1−x²), d/dx(cos⁻¹x) = −1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²).
• 5The natural exponential function satisfies d/dx(eˣ) = eˣ, and d/dx(log x) = 1/x; logarithmic differentiation handles [u(x)]^v(x) forms.

NCERT Class 12 Maths Chapter 6 covers the Application of Derivatives, teaching students how to use derivatives to find rates of change, determine increasing or decreasing intervals of functions, and locate maximum and minimum values using the First and Second Derivative Tests.

• 1The derivative dy/dx represents the rate of change of y with respect to x; the Chain Rule connects rates of change through an intermediate variable t.
• 2A function f is increasing on (a, b) if f'(x) > 0 for each x in (a, b), and decreasing if f'(x) < 0 for each x in (a, b).
• 3A critical point is any point c where f'(c) = 0 or f is not differentiable; critical points are candidates for local maxima or minima.
• 4First Derivative Test: if f'(x) changes from positive to negative at c, then c is a local maximum; if it changes from negative to positive, c is a local minimum; no sign change means c is a point of inflexion.
• 5Second Derivative Test: if f'(c) = 0 and f''(c) < 0, then c is a local maximum; if f''(c) > 0, it is a local minimum; if f''(c) = 0, the test fails and the First Derivative Test must be used.

NCERT Class 12 Mathematics Appendix 1 covers six methods of proof — three direct (straight forward, mathematical induction, proof by cases) and three indirect (contradiction, contrapositive, counterexample) — with fully worked examples for each.

• 1A proof is a chain of deductive arguments where each statement is justified by a definition, axiom, or previously established proposition using allowed logical rules.
• 2Direct proofs have three forms: straight forward (step-by-step implication chain), mathematical induction (base case + inductive step), and proof by cases/exhaustion (split hypothesis into sub-cases r, s, t and prove each implies q).
• 3Mathematical induction rests on the axiom: if 1∈S and k∈S implies k+1∈S for a subset S of ℕ, then S = ℕ.
• 4Indirect proofs have three forms: proof by contradiction (assume negation, derive contradiction), proof by contrapositive (prove ~q⇒~p instead of p⇒q), and proof by counterexample.
• 5The contrapositive of p⇒q is ~q⇒~p, formed by interchanging hypothesis and conclusion and negating both; it is logically equivalent to the original conditional.

NCERT Class 12 Maths Appendix 2 teaches Mathematical Modelling — the structured process of converting real-life physical situations into mathematical form, solving them, and validating the results against observations. It covers the principles, a five-step modelling procedure, and seven worked examples using matrices, linear programming, and differential equations.

• 1Mathematical modelling converts a physical situation into mathematics by introducing parameters/variables and applying known physical laws and symbols.
• 2The five-step modelling procedure is: identify the situation → formulate the model → find the mathematical solution → interpret and compare with observations → accept or revise the assumptions.
• 3Eight principles underlie every model: identify the need, list parameters/variables, identify available data, state assumptions, identify governing physical principles, specify equations and calculations, design consistency and utility tests, and identify parameters that can improve the model.
• 4Example 1 models finding a tower's height using the angle of elevation α and observer height h, giving H = h + l tan α; when the tower foot is inaccessible a second angle of depression β provides l = h cot β.
• 5Examples 2–4 use matrix multiplication and elementary row operations to determine raw-material requirements and optimal production quantities (x = 20 units of P1, y = 35 of P2, z = 5 of P3) for a three-product, three-raw-material firm.

Free PDF of official NCERT answers for Class 12 Mathematics Part I, covering all exercises across Chapters 1–6 — from Relations & Functions through Application of Derivatives. No sign-up required.

• 1Chapter 1 — Relations & Functions: answers for Exercises 1.1 and 1.2 (reflexivity, symmetry, transitivity, injectivity, surjectivity) and Miscellaneous Exercise.
• 2Chapter 2 — Inverse Trigonometric Functions: answers for Exercises 2.1 and 2.2 (principal value branches, identities) and Miscellaneous Exercise.
• 3Chapter 3 — Matrices: answers for Exercises 3.1 through 3.4 (order, operations, transpose, symmetric/skew-symmetric) and Miscellaneous Exercise.
• 4Chapter 4 — Determinants: answers for Exercises 4.1 through 4.5 (minors, cofactors, adjoint, inverse, solving linear systems) and Miscellaneous Exercise.
• 5Chapter 5 — Continuity & Differentiability: answers for Exercises 5.1 through 5.7 (continuity, implicit/parametric/logarithmic differentiation, second-order derivatives) and Miscellaneous Exercise, plus Supplementary Material (Theorem 5 — derivatives of ex and loge x).