Developing Mathematical Reasoning - NCERT Solutions for Class 11 Maths

Introduction to Mathematical reasoning

Mathematical reasoning refers to the ability to think logically and arrive at mathematical conclusions based on some given facts and relationships. As per the class 11 NCERT maths curriculum, developing mathematical reasoning skills is essential for students to become good problem solvers. In this blog, we will explore what mathematical reasoning involves, different techniques used, and some examples from the NCERT solutions for class 11 maths.

What is Mathematical Reasoning?

Mathematical reasoning refers to the logical thinking process used to solve math problems. It involves:

• Analyzing given information
• Identifying relationships between mathematical objects like numbers, shapes, etc.
• Making logical deductions based on these relationships
• Arriving mathematically at the required result or solution

Various Techniques Used

Some common mathematical reasoning techniques outlined in the NCERT maths textbook for class 11 include:

Induction - Making general conclusions based on observation of patterns in examples

Deduction - Logically deducing facts based on certain given facts, rules or premises

Abstraction - Identifying essential features while ignoring inessentials

Algorithmic Thinking - Developing step-by-step algorithms or set of rules to solve problems

Estimation - Making reasonable approximations and guesses of solutions

Illustrations from NCERT Class 11 Maths

Let us look at some examples from the NCERT solutions to see mathematical reasoning in action:

Example 1: Prove that sum of first n odd natural numbers is n^2.

Solution: We observe the pattern -

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

By induction, we can conclude the result.

Example 2: If x is a positive number, prove that x^3 - x is a multiple of 6

Solution: Let x = 2m where m is some integer

x^3 - x = (2m)^3 - 2m

= 8m^3 - 2m

= 2(4m^3 - m)

Since 4m^3 - m is an integer, x^3 - x is a multiple of 2

Also, x^3 - x is a multiple of 3 (deduction)

∴ x^3 - x is a multiple of 6

Applications of Mathematical Reasoning

Some real-life applications include:

• Formulating theoretical mathematics
• Discovering new algorithms
• Developing computer programs
• Cryptography and cybersecurity
• Modeling real world systems
• Data analysis and prediction
• Financial mathematics

Game theory

Mastering mathematical reasoning forms the basis of logical thinking and quantitative reasoning skills. It enables solving complex mathematical problems with multiple steps by applying the techniques outlined in the NCERT class 11 maths curriculum. Regular practice with the NCERT maths solutions helps hone these skills right from high school.

In summary, mathematical reasoning is the backbone of analyzing real world situations mathematically. A thorough understanding of reasoning techniques coupled with solving the illustrative examples given in the NCERT textbook and solutions will go a long way in developing skills for higher level maths and analytical problem solving.