This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Binary Operations”.

1. Let a binary operation ‘*’ be defined on a set A. The operation will be commutative if ________

a) a*b=b*a

b) (a*b)*c=a*(b*c)

c) (b ο c)*a=(b*a) ο (c*a)

d) a*b=a

View Answer

Explanation: A binary operation ‘*’ defined on a set A is said to be commutative only if a*b=b*a, ∀a, b∈A.

If (a*b)*c=a*(b*c), then the operation is said to associative ∀ a, b∈ A.

If (b ο c)*a=(b*a) ο (c*a), then the operation is said to be distributive ∀ a, b, c ∈ A.

2. Let a*b=6a^{4}-9b^{4} be a binary operation on R, then * is commutative.

a) True

b) False

View Answer

Explanation: The given statement is false. The binary operation ‘*’ is commutative if a*b=b*a

Here, a*b=6a

^{4}-9b

^{4}and b*a=6b

^{4}-9a

^{4}

⇒a*b≠b*a

Hence, the ‘*’ is not commutative.

3. Let ‘*’ be a binary operation on N defined by a*b=a-b+ab^{2}, then find 4*5.

a) 9

b) 88

c) 98

d) 99

View Answer

Explanation: The binary operation is defined by a*b=a-b+ab

^{2}.

∴4*5=4-5+4(5

^{2})=-1+100=99.

4. Let ‘*’ be defined on the set N. Which of the following are both commutative and associative?

a) a*b=a+b

b) a*b=a-b

c) a*b=ab^{2}

d) a*b=a^{b}

View Answer

Explanation: The binary operation ‘*’ is both commutative and associative for a*b=a+b.

The operation is commutative on a*b=a+b because a+b=b+a.

The operation is associative on a*b=a+b because (a+b)+c=a+(b+c).

5. Let ‘&’ be a binary operation defined on the set N. Which of the following definitions is commutative but not associative?

a) a & b=a-b

b) a & b=a+b

c) a & b=ab – 8

d) a & b=ab

View Answer

Explanation: The binary operation ‘&’ is commutative but not associative for a*b=ab-8.

For Commutative: a & b=ab-8 and b & a=ba-8

ab-8=ba-8. Hence, a & b=ab-8 is commutative.

For Associative: (a &b)& c=(ab-8)& c=(ab-8)c-8=abc-8c-8=abc-8c-8.

a& (b &c)=a&(bc-8)=a(bc-8)-8=abc-8a-8.

⇒(a&b) & c≠a& (b& c). Hence, the function is not associative.

6. Let ‘*’ be a binary operation defined by a*b=4ab. Find (a*b)*a.

a) 4a^{2} b

b) 16a^{2} b

c) 16ab^{2}

d) 4ab^{2}

View Answer

Explanation: Given that, a*b=4ab.

Then, (a*b)*a=(4ab)*a

=4(4ab)(a)=16a

^{2}b.

7. Let ‘*’ and ‘^’ be two binary operations such that a*b=a^{2} b and a ^ b = 2a+b. Find (2*3) ^ (6*7).

a) 256

b) 286

c) 276

d) 275

View Answer

Explanation: Given that, a*b=a

^{2}b and a ^ b = 2a+b.

∴(2*3)^(6*7)=(2

^{2}×3)^(6

^{2}×7)

=12^252=2(12)+252=276.

8. An element is said to be invertible only if there is an identity element in that binary operation.

a) True

b) False

View Answer

Explanation: The given statement is true. If there is a binary operation *:M×M → M with an identity element a∈ M is said to be invertible with respect to the binary operation * if there exists an element b ∈ M such that a*b = e = b*a, b is called inverse of a.

9. Let ‘*’ be a binary operation defined by a*b=3a^{b}+5. Find 8*3.

a) 1547

b) 1458

c) 1448

d) 1541

View Answer

Explanation: It is given that a*b=3a

^{b}+5.

Then, 8*3=3(8

^{3})+5=3(512)+5=1536+5=1541.

10. Which of the following is not a type of binary operation?

a) Transitive

b) Commutative

c) Associative

d) Distributive

View Answer

Explanation: Transitive is not a type of binary operation. It is a type of relation. Distributive, associative, commutative are different types of binary operations.

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