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Class 11 Mathematics NCERT Exemplar Solutions
Chapter 1 Sets
Dive deep into the concept of sets, one of the fundamental ideas in mathematics. Learn about representation of sets, types of sets (finite, infinite, equal, null, universal, etc.), and operations such as union $(A \cup B)$, intersection $(A \cap B)$, and complement $(A')$. Understand how Venn diagrams visually represent relationships between sets and solve problems using set identities and De Morgan’s laws.
Chapter 2 Relations and Functions
Explore ordered pairs, Cartesian products, and the idea of relations as subsets of a Cartesian product. Learn about different types of relations (reflexive, symmetric, transitive) and how functions map elements from one set to another. Study domain, co-domain, range, and special types of functions such as one-one, onto, and inverse functions. Represent functions algebraically and graphically.
Chapter 3 Trigonometric Functions
Understand trigonometric functions as extensions of trigonometric ratios to any angle. Learn how to define $\sin \theta$, $\cos \theta$, and $\tan \theta$ on the unit circle. Explore periodicity, principal values, and trigonometric identities like $\sin^2 \theta + \cos^2 \theta = 1$. Also study graphs of trigonometric functions and transformations involving amplitude and phase shifts.
Chapter 4 Principle of Mathematical Induction
Master the principle of mathematical induction, a powerful tool for proving statements involving natural numbers. Understand the base case and inductive step, and learn to prove formulas such as $\sum\limits_{k=1}^{n} k = \frac{n(n+1)}{2}$ and $\sum\limits_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.
Chapter 5 Complex Numbers and Quadratic Equations
Learn about complex numbers of the form $z = a + ib$, their representation on the Argand plane, and operations like addition and multiplication. Explore modulus $|z|$ and argument $\arg(z)$. Study quadratic equations with real and complex roots using the discriminant $\Delta = b^2 - 4ac$ and understand the concept of conjugates and polar form of complex numbers.
Chapter 6 Linear Inequalities
Understand inequalities involving one and two variables. Learn to represent solutions on a number line and in the Cartesian plane. Solve linear inequalities such as $ax + b > 0$ and systems of inequalities using graphical methods and algebraic manipulation.
Chapter 7 Permutations and Combinations
Learn the fundamental counting principle and its applications. Understand permutations as arrangements where order matters and combinations where order does not. Solve problems involving $nPr = \frac{n!}{(n-r)!}$ and $nCr = \frac{n!}{r!(n-r)!}$, applying these concepts in probability and counting scenarios.
Chapter 8 Binomial Theorem
Study the expansion of $(a + b)^n$ using the binomial theorem: $(a + b)^n = \sum\limits_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$. Learn about binomial coefficients, their recursive property $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$, and applications in algebra and combinatorics.
Chapter 9 Sequence and Series
Understand arithmetic and geometric progressions (A.P. and G.P.). Learn formulas for the $n^{th}$ term $a_n = a + (n-1)d$ and the sum of the first $n$ terms $S_n = \frac{n}{2}[2a + (n-1)d]$. Explore geometric progressions where $a_n = ar^{n-1}$ and $S_n = a\frac{r^n - 1}{r - 1}$, and apply these concepts to real-world problems.
Chapter 10 Straight Lines
Study coordinate geometry concepts of straight lines. Learn slope $(m)$, point-slope form $(y - y_1 = m(x - x_1))$, two-point form, intercept form, and normal form of a line. Understand conditions for parallelism and perpendicularity of lines, and the angle between two lines using $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$.
Chapter 11 Conic Sections
Explore conic sections—parabola, ellipse, and hyperbola—formed by intersecting a plane with a double cone. Learn standard equations like parabola $y^2 = 4ax$, ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, and hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Study their foci, directrices, and eccentricities.
Chapter 12 Introduction to Three Dimensional Geometry
Understand the coordinate system in three dimensions $(x, y, z)$, distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$, section formula, and the midpoint formula. Visualize 3D geometry concepts for real-world applications in physics and engineering.
Chapter 13 Limits and Derivatives
Get introduced to calculus by understanding limits, continuity, and derivatives. Learn to find $\lim_{x \to a} f(x)$ and the derivative $\frac{dy}{dx}$ using the first principle $\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Explore differentiation rules for polynomials, trigonometric, exponential, and logarithmic functions.
Chapter 14 Mathematical Reasoning
Study the logic behind mathematical arguments and proofs. Learn about statements, negations, compound statements, implications, and equivalence. Understand reasoning techniques such as direct, contrapositive, and contradiction proofs used in problem solving.
Chapter 15 Statistics
Learn statistical measures such as mean $(\bar{x}) = \frac{\sum\limits f_i x_i}{\sum\limits f_i}$, median, and mode. Understand how to represent data graphically and analyze frequency distributions for better data interpretation.
Chapter 16 Probability
Understand the theoretical approach to probability. Learn key terms like experiment, outcome, sample space $(S)$, and event $(E)$. Use the probability formula $P(E) = \frac{n(E)}{n(S)}$ to solve real-life and mathematical problems.