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Class 12 Mathematics NCERT Exemplar Solutions
Chapter 1: Relations and Functions
Explore the concept of relations and functions between sets, understanding domain, co-domain, and range. Study one-one, onto, and invertible functions with examples. For instance, if $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$. Visualize these using mapping diagrams and graphical interpretations.
Chapter 2: Inverse Trigonometric Functions
Understand the inverse of trigonometric functions like $\sin^{-1}x$, $\cos^{-1}x$, $\tan^{-1}x$, and their principal values. Learn the fundamental identities, such as $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$, and how inverse functions are represented graphically.
Chapter 3: Matrices
Matrices form the foundation of linear algebra. Learn operations like addition, multiplication, and transpose of matrices, along with concepts of identity and inverse matrices. For example, if $A = \begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}$, find $A^{-1}$ using the formula $A^{-1} = \frac{1}{|A|}\text{adj}(A)$.
Chapter 4: Determinants
Study determinants of order $2$ and $3$, properties, and their use in solving linear equations through Cramer's rule. For instance, if $|A| = 0$, the system of equations has either no or infinitely many solutions.
Chapter 5: Continuity and Differentiability
Learn the meaning of continuity at a point using the limit concept: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$. Understand the derivative of composite and implicit functions using the chain rule and explore exponential and logarithmic differentiation.
Chapter 6: Application of Derivatives
Apply derivatives to problems involving rate of change, increasing/decreasing functions, and maxima–minima. For example, if $y = x^3 - 3x^2 + 2$, then $\frac{dy}{dx} = 3x^2 - 6x$, helping identify critical points where $\frac{dy}{dx} = 0$.
Chapter 7: Integrals
Learn about integration as the inverse process of differentiation. Practice standard results like $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ and explore methods such as substitution, by parts, and partial fractions.
Chapter 8: Application of Integrals
Apply integration to calculate area under curves and area between two curves. For example, the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$ is $A = \int_0^1 (x - x^2)\,dx = \frac{1}{6}$.
Chapter 9: Differential Equations
Understand the formation and solution of differential equations. For example, the equation $\frac{dy}{dx} + Py = Q$ can be solved using the integrating factor $e^{\int P\,dx}$.
Chapter 10: Vector Algebra
Explore vector quantities and operations such as addition, scalar and vector products. For two vectors $\vec{a}$ and $\vec{b}$, the dot product is given by $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$.
Chapter 11: Three Dimensional Geometry
Learn equations of lines and planes in 3D space, the distance between them, and the angle between two lines using vector methods. For example, the cosine of the angle between two lines is $\cos\theta = \frac{|\vec{a_1}\cdot\vec{a_2}|}{|\vec{a_1}||\vec{a_2}|}$.
Chapter 12: Linear Programming
Understand formulation of linear programming problems (LPPs) and their graphical solutions involving optimization of a linear objective function, e.g. maximize $Z = 3x + 4y$ subject to given constraints.
Chapter 13: Probability
Study conditional probability, total probability theorem, and Bayes’ theorem: $P(A|B) = \frac{P(A \cap B)}{P(B)}$. Apply these concepts to solve real-life problems involving dependent events.
Design of the Question Paper - Set I
This section offers a model design of the official CBSE Mathematics question paper, helping students understand the pattern, marks distribution, and question weightage.
Design of the Question Paper - Set II
Provides an alternate paper design and additional practice problems for improved exam readiness.