Question

Let the relation R be defined in N by aRb if 3a + 2b = 27 then R is

• A. {(1, 12), (3, 9), (5,6), (7, 3), (9, 0)}
• B. {(1, 12), (3, 9), (5, 6), (7, 3)}
• C. {(2, 1), (9, 3), (6, 5), (3, 7)}
• D. \(\left\{(0,\frac{27}{2}),(1, 12), (3, 9), (5,6),(7,3)\right\}\)

Hint:

To solve a relation given by an equation, substitute different values for \( a \) or \( b \) and solve for the corresponding variable. In this case, solving \( 3a + 2b = 27 \) for various values of \( a \) gives us the pairs that form the relation. This method is useful for finding valid pairs in mathematical relations defined by equations.

Answer 1

The Correct Option is B

The correct answer is (B) : {(1, 12), (3, 9), (5, 6), (7, 3)}.

Answer 2

The correct answer is: (B) {(1, 12), (3, 9), (5, 6), (7, 3)}.

We are given the relation \( R \) defined on the natural numbers \( \mathbb{N} \) by the condition:

\(aRb \) if and only if \( 3a + 2b = 27\)

We need to find all pairs \( (a, b) \) in \( \mathbb{N} \) that satisfy the equation \( 3a + 2b = 27 \).

Let's solve this equation for various values of \( a \):

• For \( a = 1 \): \(3(1) + 2b = 27 \Rightarrow 2b = 24 \Rightarrow b = 12\)
• For \( a = 3 \): \(3(3) + 2b = 27 \Rightarrow 2b = 18 \Rightarrow b = 9\)
• For \( a = 5 \): \(3(5) + 2b = 27 \Rightarrow 2b = 12 \Rightarrow b = 6\)
• For \( a = 7 \): \(3(7) + 2b = 27 \Rightarrow 2b = 6 \Rightarrow b = 3\)

Therefore, the relation \( R \) contains the pairs \( (1, 12), (3, 9), (5, 6), (7, 3) \). 
Thus, the correct answer is (B) {(1, 12), (3, 9), (5, 6), (7, 3)}.