RD Sharma Solution- Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 3 - Binary Operations (Ex 3.3) Exercise 3.3
1. Where can I find reliable, step-by-step solutions for RD Sharma Class 12 Maths Chapter 3, Exercise 3.3?
Vedantu provides detailed, expert-verified solutions for every question in RD Sharma Class 12 Maths, Chapter 3 (Binary Operations), Exercise 3.3. Each solution is crafted to align with the CBSE 2025-26 syllabus and marking scheme, ensuring you understand the correct method to solve problems involving commutativity, associativity, and identity elements.
2. What is the correct method to determine if a binary operation '*' is commutative in Exercise 3.3 problems?
To check for commutativity for any binary operation '*' on a set A, as shown in the solutions for Exercise 3.3, you must follow this two-step process:
First, calculate the result of a * b for any two arbitrary elements a, b ∈ A.
Next, calculate the result of b * a.
If a * b = b * a for all possible a and b, the operation is commutative. The RD Sharma solutions clearly demonstrate this by testing the general case.
3. How do the solutions for Exercise 3.3 explain finding the identity element for a binary operation?
The solutions for Exercise 3.3 demonstrate that to find the identity element 'e' for a binary operation '*' on a set A, you need to solve the equation a * e = e * a = a. The value of 'e' must be a member of the set A and must work for every element 'a' in A. Our solutions provide a clear, step-by-step derivation to find 'e' for each specific problem.
4. What is a common mistake when solving problems on identity and inverse elements from Exercise 3.3?
A very common mistake is finding a potential identity element 'e' but failing to verify if 'e' belongs to the given set (e.g., N, Z, Q). For example, if the operation is on the set of natural numbers (N) and you find e = 0, it is not a valid identity element because 0 is not a natural number. Similarly, for an inverse, the calculated inverse element must also belong to the specified set. Our solutions always include this crucial verification step.
5. Why do some binary operations in Ex 3.3 have an identity element but no inverse for certain elements?
This happens because the existence of an inverse for an element 'a' depends on two conditions: first, solving the equation a * b = e (where 'e' is the identity and 'b' is the inverse), and second, the resulting 'b' must belong to the given set. For some operations defined on sets like N (Natural Numbers) or Z (Integers), the calculated inverse might be a fraction or a non-integer, which is not part of the set. Therefore, even with an identity element, an inverse for that specific element does not exist within the set.
6. How do the problems in RD Sharma Ex 3.3 help differentiate between a binary operation and a general function?
The problems in this exercise reinforce a key property: a binary operation on a set A must be closed. This means that for any two elements a and b in set A, the result 'a * b' must also be in A. This closure property is fundamental to the definition of a binary operation itself and is not a mandatory condition for all general functions. The solutions for Exercise 3.3 always operate under this assumption of closure, which is a core concept tested.
7. What key concepts from Binary Operations are covered in the questions of RD Sharma Exercise 3.3?
Exercise 3.3 of RD Sharma Class 12 primarily focuses on testing and applying the core properties of binary operations. The questions require you to:
Determine if an operation is commutative (a * b = b * a).
Check if an operation is associative ((a * b) * c = a * (b * c)).
Find the identity element 'e' for an operation, if it exists.
Calculate the inverse of an element, given an identity element.
Vedantu's solutions cover all these problem types with detailed, easy-to-understand explanations.
