NCERT Exemplar Class 12 Mathematics Solutions & Questions

What is NCERT Exemplar for Class 12 Mathematics?

The NCERT Exemplar for Class 12 Mathematics is a supplementary book designed to help students go beyond the standard textbook exercises. While the NCERT textbook focuses on the fundamental concepts of each topic, the Exemplar questions are more application-oriented, testing the deeper understanding of those concepts.

Key Features of NCERT Exemplar for Class 12 Mathematics:

Higher-level Questions: These questions push students to think critically and apply mathematical concepts in real-life situations.
Variety of Problems: The Exemplar includes multiple types of problems, such as conceptual questions, practice problems, and challenging questions, which are great for exam preparation.
Step-by-Step Solutions: It offers detailed solutions for each question, helping students understand the process and reasoning behind each step.

By solving these problems, students not only get a solid understanding of core mathematical concepts but also prepare themselves for the complexity of entrance exams like JEE Main and JEE Advanced.

Why Should You Focus on NCERT Exemplar for Class 12 Mathematics?

Relevance to Board Exams

The NCERT Exemplar is highly relevant to the Class 12 Mathematics board exam. The problems in this book are designed to align with the syllabus and exam pattern, ensuring that you’re well-prepared for what might appear on your exam paper.

Preparation for Competitive Exams

For students aiming to crack JEE or other competitive exams, the NCERT Exemplar serves as an essential tool. The book features problems similar to those found in competitive exams, especially in JEE Main and JEE Advanced.

Deepens Conceptual Understanding

Solving NCERT Exemplar questions helps deepen your understanding of the subject. It challenges you to think critically and apply mathematical principles to solve problems. This is the key to mastering not just the textbook but also the applications of these concepts in real-world scenarios.

How to Use NCERT Exemplar for Class 12 Mathematics Effectively?

Using the NCERT Exemplar book effectively can make a huge difference in your preparation. Here’s how you can maximize the benefits of solving these questions:

Start by Understanding Core Concepts
- Before diving into the Exemplar questions, ensure you have a strong understanding of the basic concepts from the NCERT textbook.
- Revisit any chapters where you feel less confident, and read through the explanations in the textbook.
Solve Easy Questions First
- Start with the easier questions and gradually work your way up to the more challenging ones. This builds confidence and helps you strengthen your problem-solving skills.
Practice Regularly
- Consistency is key. Set aside a fixed time every day for solving NCERT Exemplar Mathematics questions. Aim for a mix of easy, medium, and difficult questions.
Review Solutions Carefully
- It’s important to go over the step-by-step solutions provided in the Exemplar. Understand each step and the reasoning behind it. This will help you apply the concepts better in future problems.

NCERT Exemplar Class 12 Mathematics Chapter-Wise Questions

Chapter 1: Relations and Functions

This chapter deals with the concepts of relations, functions, and types of functions. The Exemplar questions test your understanding of how to work with different types of relations and their properties.

Sample Question:
Define a function and explain its properties with examples.
Solution: A function is a relation where each element of the domain is mapped to a unique element of the range.

Chapter 2: Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for solving many problems in calculus. The Exemplar questions provide practice on both basic and complex inverse trigonometric functions.

Sample Question:
If $f(x) = \sin^{-1}(x)$ , find the derivative of $f (x)$ .
Solution: The derivative of $f (x)$ is given by $\frac{1}{\sqrt{1-x^2}}$ .

Chapter 3: Matrices and Determinants

Matrices and determinants are key concepts that appear frequently in Class 12 Mathematics. The Exemplar focuses on matrix operations, properties, and solving systems of linear equations.

Sample Question:
Prove that the determinant of a matrix is equal to the determinant of its transpose.
Solution: The determinant of a matrix is the same as its transpose because the operations of row and column swaps do not affect the determinant value.

Chapter 4: Continuity and Differentiability

This chapter is crucial for understanding how functions behave as they approach certain points. Exemplar problems cover both theoretical and application-based questions on continuity and differentiability.

Sample Question:
Prove that the function $\frac{x^2 - 1}{x - 1}$ is continuous at $x = 1$ .
Solution: By factoring and simplifying the expression, we can show that $f (x)$ is continuous at $x = 1$ .

Solved NCERT Exemplar Class 12 Mathematics Questions

Here are some solved problems that can serve as a model for understanding how to approach NCERT Exemplar questions:

Problem 1:
Find the inverse of the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ .
Solution: The inverse is given by $A−1=1det(A)×adj(A)A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)$ .
Problem 2:
Differentiate the function $f(x) = x^3 + 5x^2 + 4x$ .
Solution: The derivative is $f'(x) = 3x^2 + 10x + 4$ .
Problem 3:
Find the probability of drawing a red card from a deck of cards.
Solution: There are 26 red cards in a deck of 52 cards, so the probability is $2652=12\frac{26}{52} = \frac{1}{2}$ .

Tips and Tricks to Solve NCERT Exemplar Mathematics Problems

Work on Time Management: Time yourself while solving problems to simulate exam conditions.
Practice Different Question Types: Don’t just stick to one type of question. Practice as many different question types as possible to get a comprehensive understanding of the subject.
Use the “Reverse Approach”: If you get stuck, try to work backwards. Start with the answer and think about how you would have arrived at it.
Learn from Mistakes: After solving each problem, go over the solution carefully and learn from any mistakes you made.

Importance of Solving NCERT Exemplar for Competitive Exams

Apart from board exams, the NCERT Exemplar also plays a significant role in preparing for competitive exams like JEE Main and JEE Advanced. The problem-solving approach required for these questions sharpens your skills and helps you understand how to approach complex mathematical problems.

Commonly Asked Questions (FAQs)

What is the best way to approach NCERT Exemplar questions?
- Begin with easy questions to build confidence, then move to complex ones. Ensure you understand the reasoning behind each solution.
Are NCERT Exemplar questions enough for Class 12 Mathematics preparation?
- While the NCERT Exemplar is an excellent resource, it's always a good idea to refer to other practice materials as well for comprehensive preparation.
How many times should I practice NCERT Exemplar questions?
- Aim to solve each question at least twice – once as an initial attempt and again after reviewing the solution.

Conclusion

By diligently solving the NCERT Exemplar Class 12 Mathematics questions, you not only prepare for your board exams but also build a strong foundation for competitive exams. Use this guide as your blueprint for tackling the most challenging problems, and remember that consistent practice is the key to success. Happy studying, and good luck with your exams!

NCERT Exemplar Question for Class 12 Mathematics in English

Relations And Functions

Inverse Trigonometric Functions

Matrices

Determinants

Continuity And Differentiability

Application Of Derivatives

Integrals

Application Of Integrals

Differential Equations

Vector Algebra

Three Dimensional Geometry

Linear Programming

Probability

Design Of The Question Paper Set I

Design Of The Question Paper Set II

Answers