250+ TOP MCQs on Binary Operations | Class 12 Maths

Mathematics Multiple Choice Questions 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

Answer: a
Clarification: 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=6a⁴-9b⁴ be a binary operation on R, then * is commutative.
a) True
b) False

Answer: b
Clarification: The given statement is false. The binary operation ‘*’ is commutative if a*b=b*a
Here, a*b=6a⁴-9b⁴ and b*a=6b⁴-9a⁴
⇒a*b≠b*a
Hence, the ‘*’ is not commutative.

3. Let ‘*’ be a binary operation on N defined by a*b=a-b+ab², then find 4*5.
a) 9
b) 88
c) 98
d) 99

Answer: d
Clarification: The binary operation is defined by a*b=a-b+ab².
∴4*5=4-5+4(5²)=-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=ab²
d) a*b=a^b

Answer: a
Clarification: 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

Answer: c
Clarification: 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) 4a² b
b) 16a² b
c) 16ab²
d) 4ab²

Answer: b
Clarification: Given that, a*b=4ab.
Then, (a*b)*a=(4ab)*a
=4(4ab)(a)=16a² b.

7. Let ‘*’ and ‘^’ be two binary operations such that a*b=a² b and a ^ b = 2a+b. Find (2*3) ^ (6*7).
a) 256
b) 286
c) 276
d) 275

Answer: c
Clarification: Given that, a*b=a² b and a ^ b = 2a+b.
∴(2*3)^(6*7)=(2²×3)^(6²×7)
=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

Answer: a
Clarification: 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

Answer: d
Clarification: It is given that a*b=3a^b+5.
Then, 8*3=3(8³)+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

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