Question

42. The operation * is defined over the set of real numbers “R” by a * b = a + b + 1/2ab. What is 6 * (2 * 5) a. 28 b. 0 e. 13 d. 54​

Answer 1

Step-by-step explanation:

A binary operation ∗ on A is associative if ∀a,b,c∈A,

(a∗b)∗c=a∗(b∗c) and

A binary operation ∗ on A is commutative if ∀a,b∈A,

a∗b=b∗a

a∗b=

4

a+b

is commutative as:

a∗b=b∗a

⇒

4

a+b

=

4

b+a

⇒

4

a+b

=

4

a+b

which is true.

a∗b=

4

a+b

is not associative as:

(a∗b)∗c=a∗(b∗c)

⇒(

4

a+b

)∗c=a∗(

4

b+c

)

⇒

4

4

a+b

+c

=

4

a+

4

b+c

⇒

16

a+b+4c

=

16

4a+b+c

which is not true.

Hence, the binary operation ∗ is commutative but not associative.

Answer 2

The correct option is D. 54

Step-by-step explanation:

According to the given question,

$a *b = a + b+ \frac{1}{2}ab$

$6 * (2*5) = 6* ( 2+5+\frac{1}{2}$ × $2$ × $5)$

$= 6*12$

$= 6+12+\frac{1}{2}$ × $6$ × $12$

$= 6+12+36$

$= 54$

⇒ The correct option is (d) 54.