Chapter 1: Relations and Functions
**Brief Description**: Builds on Class 11 concepts, focusing on types of relations and functions, their properties, and applications in mappings.
**Key Topics/Subtopics**:
- Types of relations: reflexive, symmetric, transitive, equivalence.
- Types of functions: one-one, onto, bijective, inverse.
- Composition of functions.
- Binary operations.
**Important Formulas/Concepts**:
- Composition: (f ∘ g)(x) = f(g(x)).
- A function f: A → B is invertible if it is bijective.
- Equivalence relation: Must be reflexive, symmetric, and transitive.
**Exercise Details**:
- Exercises: 4 + 1 (miscellaneous).
- Total Questions: ~35, with problems on function properties and operations.
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Chapter 2: Inverse Trigonometric Functions
**Brief Description**: Introduces inverse trigonometric functions, their properties, and applications in solving equations, crucial for calculus and geometry.
**Key Topics/Subtopics**:
- Definition, domain, range, and graphs of inverse trigonometric functions.
- Principal values of sin⁻¹, cos⁻¹, tan⁻¹, etc.
- Properties and identities.
- Simplification of expressions.
**Important Formulas/Concepts**:
- sin⁻¹(-x) = -sin⁻¹(x), tan⁻¹(x) + tan⁻¹(y) = tan⁻¹[(x + y)/(1 - xy)] (xy < 1).
- 2sin⁻¹(x) = sin⁻¹(2x√(1 - x²)).
- Domain of sin⁻¹(x): [-1, 1], Range: [-π/2, π/2].
**Exercise Details**:
- Exercises: 2 + 1 (miscellaneous).
- Total Questions: ~25, with simplification and identity problems.
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Chapter 3: Matrices
**Brief Description**: Covers matrices, their operations, and properties, essential for linear algebra and computer science.
**Key Topics/Subtopics**:
- Types of matrices (row, column, square, diagonal).
- Matrix operations: addition, multiplication, transpose.
- Elementary operations, inverse of a matrix.
- Applications in solving systems of equations.
**Important Formulas/Concepts**:
- Matrix multiplication: AB ≠ BA (non-commutative).
- Inverse: A⁻¹ exists if det(A) ≠ 0.
- (AB)⁻¹ = B⁻¹A⁻¹.
**Exercise Details**:
- Exercises: 4 + 1 (miscellaneous).
- Total Questions: ~40, with numerical and application-based problems.
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Chapter 4: Determinants
**Brief Description**: Explores determinants, their properties, and applications in solving equations and geometry.
**Key Topics/Subtopics**:
- Determinant of square matrices (2×2, 3×3).
- Properties of determinants.
- Adjoint and inverse of a matrix.
- Applications: area of triangle, solving linear equations.
**Important Formulas/Concepts**:
- Determinant of 2×2 matrix: det(A) = ad - bc.
- Area of triangle: (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|.
- Cramer’s rule: For ax + by = e, cx + dy = f, x = Δ₁/Δ, y = Δ₂/Δ.
**Exercise Details**:
- Exercises: 5 + 1 (miscellaneous).
- Total Questions: ~40, with determinant calculations and applications.
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Chapter 5: Continuity and Differentiability
**Brief Description**: Extends calculus concepts, focusing on continuity, differentiability, and their applications, critical for analyzing functions.
**Key Topics/Subtopics**:
- Continuity of a function, types of discontinuities.
- Differentiability, derivatives of composite, implicit, and inverse functions.
- Rolle’s theorem, Mean Value Theorem.
- Logarithmic differentiation, parametric differentiation.
**Important Formulas/Concepts**:
- Continuity: lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a).
- Derivative of implicit function: dy/dx via implicit differentiation.
- Rolle’s theorem: If f(a) = f(b), then f’(c) = 0 for some c ∈ (a, b).
**Exercise Details**:
- Exercises: 8 + 1 (miscellaneous).
- Total Questions: ~50, with differentiation and theorem-based problems.
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Chapter 6: Application of Derivatives
**Brief Description**: Applies derivatives to real-world problems like optimization, rates of change, and curve sketching.
**Key Topics/Subtopics**:
- Rate of change of quantities.
- Increasing/decreasing functions.
- Tangents, normals, approximations.
- Maxima and minima, optimization problems.
**Important Formulas/Concepts**:
- Derivative as rate of change: dy/dx.
- Condition for maxima: f’(c) = 0, f’’(c) < 0.
- Equation of tangent: y - y₁ = f’(x₁)(x - x₁).
**Exercise Details**:
- Exercises: 6 + 1 (miscellaneous).
- Total Questions: ~45, with application-based numericals.
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Chapter 7: Integrals
**Brief Description**: Introduces integration as the inverse of differentiation, foundational for calculus and physics.
**Key Topics/Subtopics**:
- Indefinite integrals, standard forms.
- Integration by substitution, partial fractions, by parts.
- Definite integrals, properties.
- Fundamental Theorem of Calculus.
**Important Formulas/Concepts**:
- Integration by parts: ∫u dv = uv - ∫v du.
- Definite integral: ∫(a to b) f(x) dx = F(b) - F(a).
- Properties: ∫(a to b) f(x) dx = ∫(a to b) f(a+b-x) dx.
**Exercise Details**:
- Exercises: 7 + 1 (miscellaneous).
- Total Questions: ~60, with integration techniques and definite integrals.
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Chapter 8: Application of Integrals
**Brief Description**: Applies integrals to compute areas under curves and between curves, useful in geometry and physics.
**Key Topics/Subtopics**:
- Area under simple curves.
- Area between two curves.
- Area bounded by curves and lines.
**Important Formulas/Concepts**:
- Area under curve: ∫(a to b) f(x) dx.
- Area between curves: ∫(a to b) [f(x) - g(x)] dx, where f(x) ≥ g(x).
**Exercise Details**:
- Exercises: 2 + 1 (miscellaneous).
- Total Questions: ~20, with area calculation problems.
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Chapter 9: Differential Equations
**Brief Description**: Covers differential equations, their formation, and solutions, essential for modeling physical systems.
**Key Topics/Subtopics**:
- Order and degree of differential equations.
- General and particular solutions.
- First-order equations: variable separable, homogeneous, linear.
- Applications in growth and decay.
**Important Formulas/Concepts**:
- Linear DE: dy/dx + Py = Q, solution: y·(IF) = ∫Q·(IF) dx, IF = e^∫P dx.
- Variable separable: ∫f(y) dy = ∫g(x) dx.
**Exercise Details**:
- Exercises: 4 + 1 (miscellaneous).
- Total Questions: ~35, with equation-solving problems.
Chapter 10: Vector Algebra
**Brief Description**: Introduces vectors, their operations, and applications, key for physics and 3D geometry.
**Key Topics/Subtopics**:
- Types of vectors (unit, zero, collinear).
- Dot product, cross product.
- Scalar triple product, vector triple product.
- Applications in geometry.
**Important Formulas/Concepts**:
- Dot product: a·b = |a||b|cosθ.
- Cross product: |a × b| = |a||b|sinθ.
- Scalar triple product: [a b c] = a·(b × c).
**Exercise Details**:
- Exercises: 5 + 1 (miscellaneous).
- Total Questions: ~35, with vector operations and applications.
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Chapter 11: Three Dimensional Geometry
**Brief Description**: Extends coordinate geometry to 3D, focusing on lines and planes, crucial for engineering and physics.
**Key Topics/Subtopics**:
- Direction cosines, direction ratios.
- Equation of a line in 3D (cartesian, vector form).
- Equation of a plane, angle between planes.
- Distance between points, lines, planes.
**Important Formulas/Concepts**:
- Line equation: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c.
- Plane: ax + by + cz + d = 0.
- Distance from point to plane: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).
**Exercise Details**:
- Exercises: 3 + 1 (miscellaneous).
- Total Questions: ~30, with 3D geometry problems.
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Chapter 12: Linear Programming
**Brief Description**: Introduces optimization using linear programming, applicable in economics and operations research.
**Key Topics/Subtopics**:
- Linear programming problems (LPP).
- Objective function, constraints.
- Graphical method for solving LPP.
- Feasible and infeasible regions.
**Important Formulas/Concepts**:
- Objective function: Maximize/Minimize Z = ax + by.
- Constraints: Linear inequalities define feasible region.
**Exercise Details**:
- Exercises: 2 + 1 (miscellaneous).
- Total Questions: ~20, with graphical and optimization problems.
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Chapter 13: Probability
**Brief Description**: Advances probability concepts, including conditional probability and random variables, key for statistics and data science.
**Key Topics/Subtopics**:
- Conditional probability, Bayes’ theorem.
- Independent events, multiplication rule.
- Random variables, probability distributions.
- Mean and variance of a distribution.
**Important Formulas/Concepts**:
- Bayes’ theorem: P(A|B) = P(B|A)P(A)/P(B).
- P(A ∩ B) = P(A)P(B) (for independent events).
- Mean: E(X) = ΣxP(x), Variance: Var(X) = E(X²) - [E(X)]².
**Exercise Details**:
- Exercises: 5 + 1 (miscellaneous).
- Total Questions: ~40, with probability and distribution problems.
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- **Assessment**: Each chapter includes exercises with numericals (Chapters 3–9, 11–13), theoretical questions (Chapters 1–2, 10), and graphical problems (Chapters 8, 11–12). NCERT Exemplar problems are recommended for advanced practice.
- **Study Tips**: Focus on calculus (Chapters 5–9) for JEE/NEET, matrices and determinants (Chapters 3–4) for linear algebra, and vectors/3D geometry (Chapters 10–11) for physics applications. Practice probability (Chapter 13) thoroughly for competitive exams.
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