NCERT Solutions for Class 12 Maths
NCERT Solutions for Class 12 Maths - Best Chapter-Wise Answers Download Free PDF
Students preparing for their Class 12 board examinations often struggle with the sheer volume and complexity of the NCERT Maths syllabus, which spans 13 chapters covering everything from Relations and Functions to Probability. The NCERT Solutions for Class 12 Maths provided here are structured chapter-wise and exercise-wise, making it straightforward to locate the exact solution you need without wading through unrelated content. A common mistake students make is skipping the Miscellaneous Exercises at the end of each chapter - these exercises frequently contain the most exam-relevant problems and have appeared in CBSE board papers in previous years. Each solution is written in a step-by-step format that mirrors the answer presentation expected by CBSE examiners, helping students understand exactly where marks are allocated. Chapters like Integrals (Chapter 7) include 11 exercises, making it the largest chapter in the syllabus and one where targeted exercise-wise practice is essential. Whether you are searching for best NCERT Solutions Class 12 Maths PDF download or chapter-specific answers for Determinants, Continuity, or Probability, this resource covers every exercise comprehensively to support thorough board exam preparation.
NCERT Solutions Class 12 Maths Chapter 1 Relations and Functions
Chapter 1 introduces the formal mathematical definitions of relations and functions, including types such as reflexive, symmetric, transitive, and equivalence relations. Students often confuse onto (surjective) and one-one (injective) functions, which are core concepts tested heavily in CBSE exams. The chapter also covers composition of functions and invertible functions. Mastering this chapter is critical because its concepts underpin later chapters like Inverse Trigonometric Functions and Calculus.
- NCERT Solutions: Relations and Functions
- NCERT Solutions (Miscellaneous Exercise) - Relations and Functions
NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions
Chapter 2 deals with the inverse forms of trigonometric functions - sin⁻¹, cos⁻¹, tan⁻¹, and others - along with their domains, ranges, and principal value branches. A frequent error students make is applying trigonometric identities outside the restricted principal value domain, leading to incorrect answers. The chapter includes properties of inverse trigonometric functions that are directly used in integration problems in Chapter 7, making it foundational for the Calculus portion of the syllabus.
- NCERT Solutions Exercise 2.1: Inverse Trigonometric Functions
- NCERT Solutions Exercise 2.2: Inverse Trigonometric Functions
- NCERT Solutions Exercise Miscellaneous: Inverse Trigonometric Functions
NCERT Solutions Class 12 Maths Chapter 3 Matrices
Chapter 3 covers the algebra of matrices, including addition, multiplication, scalar multiplication, and the concept of transpose. One key area where students lose marks is in matrix multiplication - the operation is not commutative (AB ≠ BA in general), and confusing this with scalar multiplication is a common exam error. The chapter's four exercises build progressively, with Exercise 3.4 focusing on finding inverses of matrices using elementary row operations, a technique directly applied in Chapter 4 on Determinants.
- NCERT Solutions Exercise 3.1: Matrices
- NCERT Solutions Exercise 3.2: Matrices
- NCERT Solutions Exercise 3.3: Matrices
- NCERT Solutions Exercise 3.4: Matrices
NCERT Solutions Class 12 Maths Chapter 4 Determinants
Chapter 4 explores determinants of square matrices up to order 3, along with properties of determinants, cofactors, adjoints, and the use of determinants in solving systems of linear equations via Cramer's Rule. Students frequently make sign errors when expanding determinants along a row or column - a minor mistake that can result in complete loss of marks for multi-step problems. The Miscellaneous Exercise in this chapter is especially important for board exam scoring.
- NCERT Solutions: Determinants
- NCERT Solutions Exercise 4.2: Determinants
- NCERT Solutions Exercise 4.3: Determinants
- NCERT Solutions Exercise 4.4: Determinants
- NCERT Solutions Exercise 4.5: Determinants
- NCERT Solutions Miscellaneous Exercise: Determinants
NCERT Solutions Class 12 Maths Chapter 5 Continuity and Differentiability
Chapter 5 is one of the most extensive chapters in the Class 12 syllabus, covering continuity, differentiability, chain rule, implicit differentiation, logarithmic differentiation, and derivatives of parametric functions. The concept of logarithmic differentiation - used to differentiate functions of the form [f(x)]^g(x) - is a technique students frequently omit because it seems indirect, yet it appears regularly in board exams. The chapter contains seven exercises plus a Miscellaneous Exercise, all of which are covered in the solutions below.
- NCERT Solutions: Continuity & Differentiability
- NCERT Solutions Exercise 5.2: Continuity & Differentiability
- NCERT Solutions Exercise 5.3: Continuity & Differentiability
- NCERT Solutions Exercise 5.4: Continuity & Differentiability
- NCERT Solutions Exercise 5.5: Continuity & Differentiability
- NCERT Solutions Exercise 5.6: Continuity & Differentiability
- NCERT Solutions Exercise 5.7: Continuity & Differentiability
- NCERT Solutions Miscellaneous Exercise: Continuity and Differentiability
NCERT Solutions Class 12 Maths Chapter 6 Application of Derivatives
Chapter 6 applies differentiation to real-world problems: finding rates of change, determining intervals of increasing and decreasing functions, locating tangents and normals to curves, and identifying local maxima and minima. A common board exam scenario involves finding the maximum volume of a box formed by cutting squares from corners of a rectangular sheet - a classic optimization problem that requires applying the first and second derivative tests carefully. Exercise 6.5 covers this optimization content in full.
- NCERT Solutions: Application of Derivatives
- NCERT Solutions: Exercise 6.2 - Application of Derivative
- NCERT Solutions: Exercise 6.3 - Application of Derivative
- NCERT Solutions: Exercise 6.4 - Application of Derivative
- NCERT Solutions: Exercise 6.5 - Application of Derivative
- NCERT Solutions Miscellaneous Exercise: Application of Derivative
NCERT Solutions Class 12 Maths Chapter 7 Integrals
Chapter 7 is the largest chapter in the Class 12 Maths syllabus with 11 exercises, covering indefinite and definite integrals, integration by substitution, integration by parts, integration using partial fractions, and definite integrals using limit of a sum. Students consistently lose marks by applying the integration by parts formula (ILATE rule) to the wrong function first, resulting in integrals that become progressively more complex rather than simpler. All 11 exercises and the Miscellaneous Exercise are linked below.
- NCERT Solutions Exercise 7.1: Integrals
- NCERT Solutions Exercise 7.2: Integrals
- NCERT Solutions Exercise 7.3: Integrals
- NCERT Solutions Exercise 7.4: Integrals
- NCERT Solutions Exercise 7.5: Integrals
- NCERT Solutions Exercise 7.6: Integrals
- NCERT Solutions Exercise 7.7: Integrals
- NCERT Solutions Exercise 7.8: Integrals
- NCERT Solutions Exercise 7.9: Integrals
- NCERT Solutions Exercise 7.10: Integrals
- NCERT Solutions: Integrals (Miscellaneous)
NCERT Solutions Class 12 Maths Chapter 8 Application of Integrals
Chapter 8 uses definite integration to calculate areas bounded by curves, lines, and the coordinate axes. The chapter specifically focuses on areas using the concept of the area under a curve and between two curves. Students often make the error of subtracting in the wrong direction when computing the area between two curves - the upper curve must always be subtracted from the lower curve within the correct limits. Exercise 8.1 deals with standard area problems while Exercise 8.2 covers areas between two curves.
- NCERT Solutions - Exercise 8.1: Application of Integral
- NCERT Solutions - Exercise 8.2: Application of Integrals
- NCERT Solutions - Exercise Miscellaneous: Application of Integral
- NCERT Solutions Miscellaneous Exercise: Application of Integrals
NCERT Solutions Class 12 Maths Chapter 9 Differential Equations
Chapter 9 introduces ordinary differential equations, covering the order and degree of a differential equation, methods of solving separable equations, homogeneous differential equations, and linear first-order differential equations using the integrating factor method. A frequent conceptual error is confusing the order (highest derivative present) with the degree (power of the highest derivative), especially when the equation contains radicals or fractions before it is simplified. The Miscellaneous Exercise contains problems combining multiple solution methods.
- NCERT Solutions: Differential Equations
- NCERT Solutions - Exercise 9.3: Differential Equations
- NCERT Solutions - Exercise 9.4: Differential Equations
- NCERT Solutions - Exercise 9.5: Differential Equations
- NCERT Solutions - Exercise Miscellaneous: Differential Equations
NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra
Chapter 10 introduces vectors as mathematical quantities with both magnitude and direction, covering vector addition, scalar (dot) product, and vector (cross) product. Students commonly lose marks by using the cross product formula incorrectly - the result of a cross product is a vector perpendicular to both operands, not a scalar, and the order of multiplication matters because a × b = -(b × a). The chapter has four exercises, with Exercises 10.3 and 10.4 focusing on dot and cross products respectively.
- NCERT Solutions: Exercise - Vector Algebra
- NCERT Solutions: Exercise 10.2 - Vector Algebra
- NCERT Solutions: Exercise 10.3 - Vector Algebra
- NCERT Solutions: Exercise 10.4 - Vector Algebra
- NCERT Solutions Miscellaneous Exercise: Vector Algebra
NCERT Solutions Class 12 Maths Chapter 11 Three Dimensional Geometry
Chapter 11 extends coordinate geometry into three dimensions, covering direction cosines, direction ratios, equations of lines and planes in 3D space, and the angle between two planes or between a line and a plane. A specific area of difficulty for students is deriving the shortest distance between two skew lines - lines that are neither parallel nor intersecting - which requires a formula involving the cross product of direction vectors. The chapter has two exercises and a Miscellaneous Exercise.
- NCERT Solutions: Three Dimensional Geometry
- NCERT Solutions: Exercise 11.1 - Three Dimensional Geometry
- NCERT Solutions: Exercise 11.2 - Three Dimensional Geometry
- NCERT Solutions Miscellaneous Exercise: Three Dimensional Geometry
NCERT Solutions Class 12 Maths Chapter 12 Linear Programming
Chapter 12 applies mathematical optimization techniques to real-world scenarios such as maximizing profit or minimizing cost subject to a set of linear constraints. The graphical method is used to identify the feasible region, and the optimal value always occurs at one of the corner points of this region - a fact students sometimes forget, instead trying to evaluate the objective function over the entire region. The chapter includes problems modelling manufacturing, transportation, and diet planning scenarios.
- NCERT Solutions - Linear Programming (Exercise 12.1)
- NCERT Solutions - Linear Programming (Exercise 12.2)
NCERT Solutions Class 12 Maths Chapter 13 Probability
Chapter 13 covers conditional probability, the multiplication theorem, independent events, Bayes' theorem, random variables, probability distributions, and the Binomial distribution. Bayes' theorem is a specific area where students frequently make errors by inverting the conditional probability - confusing P(A|B) with P(B|A) - leading to completely wrong answers. This chapter contributes a significant portion of marks in CBSE board exams, with questions based on real-world scenarios like medical testing and quality control.
- NCERT Solutions - Exercise 13.1: Probability
- NCERT Solutions - Exercise 13.1: Probability (Additional)
- NCERT Solutions - Exercise 13.2: Probability
- NCERT Solutions - Exercise 13.3: Probability
- NCERT Solutions Miscellaneous Exercise: Probability
Best NCERT Solutions for Class 12 Maths: Chapter-Wise Tips for Scoring Full Marks in CBSE Board Exams
Scoring above 90 in CBSE Class 12 Maths requires more than reading through solutions - it demands understanding the specific marking scheme and structuring answers the way CBSE examiners expect. For example, in Calculus chapters like Integration and Differential Equations, showing every intermediate step is mandatory; skipping a substitution step can cost you one mark even when the final answer is correct. In the Algebra section, particularly Matrices and Determinants, questions worth 5 marks are almost always broken into sub-parts, so working systematically through each operation protects partial credit. For the best Class 12 Maths NCERT Solutions PDF results, students should solve each Miscellaneous Exercise independently before checking the solution, since these exercises blend concepts from the entire chapter and mirror the style of CBSE long-answer questions. Students targeting 95+ marks should note that Chapter 7 (Integrals) and Chapter 13 (Probability) together account for a disproportionately large share of board paper marks, making them the highest-return chapters for focused revision.
Class 12 Maths NCERT Solutions PDF: How to Use Chapter-Wise Exercises for Effective Exam Preparation
The most effective way to use NCERT Solutions Class 12 Maths chapter-wise PDF resources is to attempt each exercise problem independently first, then cross-reference the solution only to verify your method and identify where your reasoning diverged. In chapters like Application of Derivatives (Chapter 6), students who rely on solutions passively often cannot reproduce optimization problems under timed conditions because they have memorized the answer without internalizing the differentiation strategy. For Continuity and Differentiability (Chapter 5), it is particularly useful to work through Exercise 5.1 before attempting 5.2, since the concept of differentiability is built directly on the definition of continuity. Similarly, in Vector Algebra (Chapter 10), ensuring clarity on scalar product before moving to vector product prevents the common error of treating cross product results as scalars. Three Dimensional Geometry (Chapter 11) requires students to visualize equations of lines and planes, and drawing a rough 3D sketch before solving each problem significantly reduces errors in identifying direction cosines and ratios. Consistent exercise-by-exercise engagement with this material is what separates students who achieve distinction-level scores from those who fall just short.
FAQs on NCERT Solutions for Class 12 Maths
| 1. How do I solve integration problems using substitution method in Class 12 Maths? |  |
| 2. What's the difference between definite and indefinite integrals for board exam preparation? | |
| 3. Why do I keep getting continuity and differentiability problems wrong on my Class 12 tests? | |
| 4. How can I quickly identify which differentiation rule to use for composite functions? | |
| 5. What are the key properties of matrices and determinants I must memorise for Class 12 exams? | |
