Consider a binary operation * on N defined as a * b = a³ + b³. Choose the correct answer.
A. Is * both associative and commutative?
B. Is * commutative but not associative?
C. Is * associative but not commutative?
D. Is * neither commutative nor associative?

Solution:

The commutative property deals with the arithmetic operations of addition and multiplication. It means that changing the order or position of numbers while adding or multiplying them does not change the end result.

On N, operation * is defined as a * b = a³ + b³.

For all a, b ∈ N

a * b = a³ + b³

= b³ + a³

= b * a

Operation * is commutative.

(1 * 2) * 3 = (1³ + 2³) * 3

= (1 + 8) * 3 = 9 * 3 = 9³ + 3³

= 729 + 27 = 756

1 * (2 * 3) = 1 * (2³ + 3³ )

= 1 * (8 + 27) = 1 * 35 = 1³ + 35³

= 1 + 42875

= 42876

⇒ ( 1* 2) * 3 ≠ 1 * (2 * 3)

Operation * is not associative.

Therefore, Operation * is commutative, but not associative.

The correct answer is B

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.4 Question 13

Consider a binary operation * on N defined as a * b = a³ + b³. Choose the correct answer. A. Is * both associative and commutative? B. Is * commutative but not associative? C. Is * associative but not commutative? D. Is * neither commutative nor associative?

Summary:

Let us consider a binary operation * on N defined as a * b = a³ + b³. Operation * is commutative, but not associative. The correct answer is B