Chapter 1: Sets
**Brief Description**: Introduces the concept of sets, their representation, and operations, forming the foundation for mathematical reasoning and problem-solving.
**Key Topics/Subtopics**:
- Types of sets (finite, infinite, empty, singleton).
- Subsets, power set, universal set.
- Set operations (union, intersection, complement, difference).
- Venn diagrams, applications of sets.
**Important Formulas/Concepts**:
- n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
- Power set: If a set has n elements, its power set has 2^n elements.
- De Morgan’s laws: (A ∪ B)’ = A’ ∩ B’, (A ∩ B)’ = A’ ∪ B’.
**Exercise Details**:
- Exercises: 6 + 1 (miscellaneous).
- Total Questions: ~35–40, including theoretical and application-based problems.
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Chapter 2: Relations and Functions
**Brief Description**: Explores relations and functions, key to understanding mappings and their applications in calculus and coordinate geometry.
**Key Topics/Subtopics**:
- Cartesian product, relations.
- Types of relations (reflexive, symmetric, transitive).
- Functions: injective, surjective, bijective.
- Domain, codomain, range.
**Important Formulas/Concepts**:
- Number of relations from set A (m elements) to set B (n elements): 2^(m×n).
- A function f: A → B is bijective if it is both injective and surjective.
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~30, with problems on types of functions and relations.
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Chapter 3: Trigonometric Functions
**Brief Description**: Covers trigonometric ratios, identities, and equations, essential for geometry, physics, and engineering.
**Key Topics/Subtopics**:
- Angles (degree, radian measure).
- Trigonometric ratios, identities (e.g., sin²θ + cos²θ = 1).
- Trigonometric equations, general solutions.
- Sum-to-product, product-to-sum formulas.
**Important Formulas/Concepts**:
- sin(A ± B) = sinA cosB ± cosA sinB.
- General solution of sinθ = k: θ = nπ + (-1)^n k, n ∈ Z.
- 1 radian = 180°/π.
**Exercise Details**:
- Exercises: 4 + 1 (miscellaneous).
- Total Questions: ~50, including identities and equation-solving.
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Chapter 4: Principle of Mathematical Induction
**Brief Description**: Introduces mathematical induction, a powerful proof technique for establishing statements about natural numbers.
**Key Topics/Subtopics**:
- Steps of induction: base case, induction hypothesis, induction step.
- Applications to divisibility, inequalities, and series.
**Important Formulas/Concepts**:
- Induction principle: If P(1) is true and P(k) implies P(k+1), then P(n) is true for all n ≥ 1.
**Exercise Details**:
- Exercises: 1.
- Total Questions: ~25, proof-based problems.
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Chapter 5: Complex Numbers and Quadratic Equations
**Brief Description**: Explores complex numbers and their applications in solving quadratic equations, crucial for algebra and engineering.
**Key Topics/Subtopics**:
- Complex numbers: real, imaginary parts, modulus, argument.
- Algebraic operations, conjugate, polar form.
- Quadratic equations with complex roots.
- Argand plane, geometric representation.
**Important Formulas/Concepts**:
- z = a + ib, modulus |z| = √(a² + b²), argument = tan⁻¹(b/a).
- Quadratic formula: x = [-b ± √(b² - 4ac)]/(2a), for complex roots when D < 0.
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~30, with numerical and graphical problems.
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Chapter 6: Linear Inequalities
**Brief Description**: Discusses solving linear inequalities and their graphical representation, useful in optimization and economics.
**Key Topics/Subtopics**:
- Algebraic solutions of linear inequalities.
- Inequalities with two variables, graphical solutions.
- Systems of inequalities.
- Applications (e.g., feasible regions).
**Important Formulas/Concepts**:
- If a > b, then a + c > b + c, ac > bc (for c > 0).
- Graphical solution: Shaded region represents solution set.
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~30, with algebraic and graphical problems.
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Chapter 7: Permutations and Combinations
**Brief Description**: Covers counting principles, permutations, and combinations, key for probability and discrete mathematics.
**Key Topics/Subtopics**:
- Fundamental principle of counting.
- Permutations (arrangements), factorial notation.
- Combinations (selections).
- Applications in real-life problems.
**Important Formulas/Concepts**:
- Permutations: P(n,r) = n!/(n-r)!.
- Combinations: C(n,r) = n!/[r!(n-r)!].
- C(n,r) = C(n, n-r).
**Exercise Details**:
- Exercises: 4 + 1 (miscellaneous).
- Total Questions: ~40, numerical-heavy.
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Chapter 8: Binomial Theorem
**Brief Description**: Introduces the binomial theorem for expanding expressions and finding coefficients, useful in algebra and probability.
**Key Topics/Subtopics**:
- Binomial expansion for positive integral index.
- General and middle terms.
- Binomial theorem for any index (introduction).
- Applications in approximations.
**Important Formulas/Concepts**:
- (a + b)^n = Σ [C(n,r) a^(n-r) b^r], r = 0 to n.
- General term: T_(r+1) = C(n,r) a^(n-r) b^r.
- Middle term: For n even, at r = n/2; for n odd, at r = (n±1)/2.
**Exercise Details**:
- Exercises: 2 + 1 (miscellaneous).
- Total Questions: ~25, with expansion and coefficient problems.
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Chapter 9: Sequences and Series
**Brief Description**: Explores arithmetic and geometric progressions, their sums, and applications in finance and physics.
**Key Topics/Subtopics**:
- Arithmetic progression (AP), geometric progression (GP).
- Sum of n terms of AP and GP.
- Arithmetic mean (AM), geometric mean (GM).
- Special series (e.g., Σn, Σn²).
**Important Formulas/Concepts**:
- AP nth term: a_n = a + (n-1)d.
- Sum of AP: S_n = n/2 [2a + (n-1)d].
- GP nth term: a_n = ar^(n-1), Sum: S_n = a(1-r^n)/(1-r) (r ≠ 1).
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~40, numerical-heavy.
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Chapter 10: Straight Lines
**Brief Description**: Introduces coordinate geometry, focusing on straight lines, their equations, and properties, essential for graphing and engineering.
**Key Topics/Subtopics**:
- Slope of a line, angle between lines.
- Forms of line equations: slope-intercept, point-slope, intercept form.
- Distance from a point to a line.
- Family of lines.
**Important Formulas/Concepts**:
- Slope: m = (y₂ - y₁)/(x₂ - x₁).
- Distance between point (x₀, y₀) and line ax + by + c = 0: |ax₀ + by₀ + c|/√(a² + b²).
- General form: ax + by + c = 0.
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~35, with graphical and numerical problems.
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Chapter 11: Conic Sections
**Brief Description**: Covers curves like circles, parabolas, ellipses, and hyperbolas, key for physics (e.g., orbits) and engineering.
**Key Topics/Subtopics**:
- Circle: standard and general forms.
- Parabola, ellipse, hyperbola: standard equations, foci, directrix.
- Eccentricity, latus rectum.
- Applications in geometry.
**Important Formulas/Concepts**:
- Circle: (x-h)² + (y-k)² = r².
- Parabola: y² = 4ax (vertex at origin).
- Ellipse: x²/a² + y²/b² = 1.
- Hyperbola: x²/a² - y²/b² = 1.
**Exercise Details**:
- Exercises: 4 + 1 (miscellaneous).
- Total Questions: ~40, with equation and property-based problems.
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Chapter 12: Introduction to Three Dimensional Geometry
**Brief Description**: Introduces 3D geometry, focusing on coordinates, distances, and section formulas, useful in physics and computer graphics.
**Key Topics/Subtopics**:
- Coordinates in 3D space.
- Distance between two points.
- Section formula (internal, external division).
- Direction cosines, direction ratios.
**Important Formulas/Concepts**:
- Distance: √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].
- Section formula: [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n)].
- Direction cosines: cosα = x/r, cosβ = y/r, cosγ = z/r, where r = √(x² + y² + z²).
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~20, numerical-based.
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Chapter 13: Limits and Derivatives
**Brief Description**: Introduces calculus through limits and derivatives, foundational for analyzing rates of change and motion.
**Key Topics/Subtopics**:
- Concept of limits, left-hand and right-hand limits.
- Algebra of limits.
- Derivatives, first principle of differentiation.
- Derivatives of standard functions (polynomials, trigonometric).
**Important Formulas/Concepts**:
- Limit: lim(x→a) f(x) = L if f(x) approaches L as x approaches a.
- Derivative: f’(x) = lim(h→0) [f(x+h) - f(x)]/h.
- d/dx (x^n) = nx^(n-1), d/dx (sin x) = cos x.
**Exercise Details**:
- Exercises: 2 + 1 (miscellaneous).
- Total Questions: ~35, with limit evaluation and differentiation.
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Chapter 14: Mathematical Reasoning
**Brief Description**: Focuses on logical reasoning and mathematical statements, enhancing critical thinking and proof skills.
**Key Topics/Subtopics**:
- Statements, connectives (and, or, not).
- Implications, contrapositive, converse.
- Validating statements, truth tables.
- Quantifiers (for all, there exists).
**Important Formulas/Concepts**:
- Negation: ~p (not p).
- Implication: p → q (if p, then q).
- Contrapositive: ~q → ~p.
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~20, logical and statement-based.
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Chapter 15: Statistics
**Brief Description**: Introduces measures of central tendency and dispersion, key for data analysis in science and economics.
**Key Topics/Subtopics**:
- Mean, median, mode.
- Measures of dispersion: range, mean deviation, variance, standard deviation.
- Analysis of frequency distributions.
**Important Formulas/Concepts**:
- Mean: x̄ = Σx_i/n.
- Variance: σ² = Σ(x_i - x̄)²/n.
- Standard deviation: σ = √(variance).
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~25, numerical-based.
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Chapter 16: Probability
**Brief Description**: Covers basic probability concepts, essential for statistics, physics, and decision-making.
**Key Topics/Subtopics**:
- Random experiments, sample space, events.
- Probability of an event, addition rule.
- Mutually exclusive and exhaustive events.
- Conditional probability (introduction).
**Important Formulas/Concepts**:
- Probability: P(A) = n(A)/n(S), where n(S) = total outcomes.
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
- For mutually exclusive events: P(A ∩ B) = 0.
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~25, with probability calculations.
- **Assessment**: Each chapter includes exercises with numericals (Chapters 5–13, 15–16), theoretical questions (Chapters 1–2, 4, 14), and graphical problems (Chapters 6, 10–11). NCERT Exemplar problems are recommended for advanced practice.
- **Study Tips**: Focus on algebraic manipulations in Chapters 3, 5, 7–9; coordinate geometry in Chapters 10–12; and calculus basics in Chapter 13. Practice proofs in Chapter 4 and logical reasoning in Chapter 14.