This set of Class 12 Maths Chapter 1 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=6a4-9b4 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=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
View Answer
Explanation: 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=ab
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) 4a2 b
b) 16a2 b
c) 16ab2
d) 4ab2
View Answer
Explanation: 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
View Answer
Explanation: 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
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=3ab+5. Find 8*3.
a) 1547
b) 1458
c) 1448
d) 1541
View Answer
Explanation: It is given that a*b=3ab+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
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.
Sanfoundry Global Education & Learning Series – Mathematics – Class 12.
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