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Class 12 Mathematics NCERT Solutions (Non-Rationalised)
Chapter 1 Relations and Functions
Relations and Functions form the foundation of higher mathematics. This chapter helps you explore ordered pairs, domain, co-domain, and range with real-life examples. Understand types of functions such as one-one, onto, and many-one mappings, and learn operations on functions like addition and composition. These NCERT Class 12 Maths Solutions provide a clear conceptual link between algebra and calculus.
Chapter 2 Inverse Trigonometric Functions
Dive into inverse trigonometric functions and their properties. Understand how to represent them graphically, and apply formulas such as $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$. Learn principal values and their importance in solving complex trigonometric equations. This chapter is essential for mastering problems involving periodicity and transformations.
Chapter 3 Matrices
Matrices are one of the most powerful tools in algebra. This chapter explains types of matrices, matrix addition, multiplication, and scalar operations. You’ll also learn to find transpose and inverse of a matrix and apply them in solving systems of equations using $AX = B$. These concepts are useful in computer graphics, physics, and data analysis.
Chapter 4 Determinants
Understand the properties of determinants and their role in solving equations. Learn to calculate determinants of orders up to $3 \times 3$ and apply Cramer's Rule for finding solutions of linear systems. Determinants also help determine whether a set of vectors is linearly independent.
Chapter 5 Continuity and Differentiability
Master the fundamentals of continuity and differentiability using rigorous definitions. Learn to differentiate composite, implicit, and inverse functions. Understand the significance of derivatives in terms of rate of change and tangent slopes, a key concept in calculus where $\frac{dy}{dx}$ represents instantaneous change.
Chapter 6 Application of Derivatives
Apply derivatives in real-world contexts like finding maxima and minima, tangents and normals to curves, and understanding motion through velocity and acceleration. Problems on increasing and decreasing functions and rate of change make calculus more practical and intuitive.
Chapter 7 Integrals
Explore the concept of integration as the inverse of differentiation. Learn basic formulas like $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ and techniques such as substitution, partial fractions, and integration by parts. Integration connects algebraic methods with geometric interpretations like finding areas under curves.
Chapter 8 Application of Integrals
Apply integration to find areas under curves and between two curves. This chapter strengthens your understanding of geometry through calculus, bridging abstract math with visual concepts. Learn to calculate $\int_a^b f(x)\,dx$ to determine total accumulation or area between limits.
Chapter 9 Differential Equations
Differential Equations form the heart of modern science and engineering. Learn the formation, order, and degree of equations, and explore general and particular solutions. Master standard forms like $\frac{dy}{dx} + Py = Q$ and understand their applications in modeling growth, decay, and motion.
Chapter 10 Vector Algebra
Discover the world of vectors — directed line segments representing quantities with both magnitude and direction. Learn vector addition, scalar and vector products, and applications in 3D geometry. Understand formulas like $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ and $\vec{a} \times \vec{b}$ for cross product.
Chapter 11 Three Dimensional Geometry
Study the geometry of space using direction cosines, direction ratios, and equations of lines and planes. Learn formulas for the shortest distance between lines and angle between two planes. These NCERT Class 12 Maths Solutions make 3D visualization and problem-solving easier.
Chapter 12 Linear Programming
Learn the basics of Linear Programming and optimization. Understand how to represent constraints graphically and find feasible regions. Apply the graphical method to solve maximization and minimization problems that arise in business and economics.
Chapter 13 Probability
Delve into advanced probability concepts like conditional probability and Bayes’ theorem. Learn how to compute probabilities using formulas such as $P(A|B) = \frac{P(A \cap B)}{P(B)}$. Explore random variables, expectation, and probability distributions — essential for data science and statistics.