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Latest Class 12 Mathematics NCERT Solutions
Chapter 1 – Relations and Functions
Relations and Functions form the foundation of advanced mathematical reasoning. In this chapter, students explore how ordered pairs connect two sets through relations and how specific mappings form functions. Key topics include types of relations (reflexive, symmetric, transitive), domain, co-domain, and range. You’ll also study composition of functions and invertible functions, preparing you for higher-level calculus and real-world applications in computing and logic.
Chapter 2 – Inverse Trigonometric Functions
Deepen your understanding of trigonometry by studying the inverse trigonometric functions. Learn their domains and ranges, principal values, and graphical representations. This chapter helps you connect trigonometric and algebraic expressions while building skills essential for integration techniques and geometric analysis.
Chapter 3 – Matrices
Enter the world of matrices — a compact and powerful way to represent data and linear equations. Learn about types of matrices (square, diagonal, identity, zero), matrix addition and multiplication, and scalar operations. Explore transpose, symmetric, and skew-symmetric matrices and see how matrices form the backbone of computer graphics, transformations, and engineering models.
Chapter 4 – Determinants
This chapter focuses on determinants, which play a vital role in solving linear equations and transformations. Learn the properties of determinants, computation using minors and cofactors, and how to find the inverse of a matrix. Practical applications include solving systems of equations through Cramer’s Rule and understanding geometric transformations.
Chapter 5 – Continuity and Differentiability
Build a deeper grasp of calculus with this chapter on continuity and differentiability. Learn how a function behaves smoothly across its domain, explore chain rule, implicit differentiation, and logarithmic differentiation. The chapter also introduces exponential and logarithmic functions with differentiation techniques used in physics and economics.
Chapter 6 – Application of Derivatives
This chapter applies derivatives in real-world situations. Learn how to determine increasing and decreasing functions, find maxima and minima, and analyze tangents and normals to curves. These concepts are widely used in optimization problems and engineering design.
Chapter 7 – Integrals
Master the concept of integration, the reverse process of differentiation. Understand indefinite and definite integrals, properties of integrals, and techniques such as substitution, partial fractions, and integration by parts. The chapter culminates in the Fundamental Theorem of Calculus, linking differentiation and integration elegantly.
Chapter 8 – Application of Integrals
See how integration applies to geometry and real-life problems. Learn to compute the area under curves, area between two curves, and volumes of solids of revolution. This chapter connects abstract calculus with tangible results in physics and engineering design.
Chapter 9 – Differential Equations
Understand the formation and solution of differential equations, a cornerstone of mathematical modeling. Study order and degree, variables separable method, homogeneous equations, and linear differential equations. Learn how these equations describe growth and decay, motion, and heat transfer.
Chapter 10 – Vector Algebra
Develop an understanding of vectors as quantities having both magnitude and direction. Study vector addition, scalar (dot) and vector (cross) products, and their geometrical interpretations. Learn how vector algebra simplifies complex problems in physics and engineering mechanics.
Chapter 11 – Three Dimensional Geometry
Explore 3D geometry involving direction cosines, direction ratios, and equations of lines and planes. Learn to find angles between lines and planes, distance between points, and intersections. These skills are fundamental for spatial visualization in engineering and architecture.
Chapter 12 – Linear Programming
Understand the powerful concept of optimization using linear inequalities and graphical methods. Learn how to formulate and solve real-life problems involving maximum profit or minimum cost, forming a link between mathematics and economics.
Chapter 13 – Probability
Dive into advanced probability theory, including conditional probability, Bayes’ theorem, and random variables. Learn about probability distributions such as binomial and normal distributions. This chapter bridges pure mathematics and statistics—crucial for research, data analysis, and AI models.