JEE Main Mathematics Topics

1. Application of Derivatives

Important Concepts

• Slope of a Curve: The slope of the normal at any point is equal to the ratio of ordinate and abscissa.
• Tangents and Normals: Equation of tangent line using parametric equations.
• Shortest Distance: Calculating the shortest distance between a line and a curve.
• Orthogonality: Curves are orthogonal if their tangent slopes are negative reciprocals.
• Critical Points: Identify local minima, maxima, and points of inflection.

Key Questions

• Find the curve passing through a point and satisfying given slope conditions.
• Calculate the angle between a line and a circle at points of intersection.
• Determine the shortest distance between a line and a curve.

2. Area Under Curves

Important Concepts

• Graphical Interpretation: Interpretation of areas bounded by different curves.
• Definite Integrals: Calculating areas using integration.
• Volume of Solids: Volume of solids of revolution.
• Closed Regions: Area enclosed by parabolas and lines.

Key Questions

• Calculate the area bounded by curves and coordinate axes.
• Find volumes of solids formed by rotating areas around an axis.

3. Continuity and Differentiability

Important Concepts

• Continuity: Conditions for a function to be continuous.
• Differentiability: Points of non-differentiability and using derivative rules.
• Rolle's Theorem: Application of mean value theorem.

Key Questions

• Find points where functions are not differentiable.
• Use Rolle's theorem to find specific values.

4. Definite Integration

Important Concepts

• Properties of Integrals: Using symmetry and periodic properties.
• Techniques of Integration: Substitution, partial fractions, and by parts.

Key Questions

• Evaluate integrals using different techniques.
• Find integral values for given functions.

5. Differential Equations

Important Concepts

• Order and Degree: Understanding order and degree of differential equations.
• Solving Differential Equations: Techniques including separation of variables.
• Application of Integration: Applying initial conditions to find particular solutions.

Key Questions

• Solve differential equations with given initial conditions.
• Find the orthogonal trajectory of given curves.

6. Sets and Relations

Important Concepts

• Types of Relations: Reflexive, symmetric, and transitive properties.
• Operations on Sets: Union, intersection, and complement of sets.

Key Questions

• Analyze sets with given properties and operations.
• Determine the number of elements in set operations.

7. Functions

Important Concepts

• Domain and Range: Determining the domain and range for functions.
• One-One and Onto: Identifying injective and surjective functions.
• Inverse of Functions: Calculating inverse functions.

Key Questions

• Find the domain and range of functions involving logs and trigonometric functions.
• Determine whether a function is one-one, onto, or both.

This summary covers the main topics found in the JEE Main Mathematics syllabus, focusing on application techniques, key concepts, and example questions for practice. For detailed problem-solving and proofs, refer to textbooks and practice problems.