Question

Assertion (A): The relation \( R = \{(x, y) : (x + y) { is a prime number and } x, y \in \mathbb{N}\} \) is not a reflexive relation.
Reason (R): The number \( 2n \) is composite for all natural numbers \( n \).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.
• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Answer 1

The Correct Option is C

Step 1: Analyze Assertion (A)
For \( R \) to be reflexive, \( (x, x) \) must belong to \( R \) for all \( x \in \mathbb{N} \). This means \( x + x = 2x \) must be a prime number. 
However, for \( x>1 \), \( 2x \) is not a prime number as it is divisible by \( 2 \). 
Therefore, \( R \) is not reflexive, and Assertion (A) is true.
 Step 2: Analyze Reason (R)
The Reason states that \( 2n \) is composite for all \( n \). 
This is false because when \( n = 1 \), \( 2n = 2 \), which is a prime number. 
Therefore, Reason (R) is false. 
Step 3: Conclusion
Since Assertion (A) is true and Reason (R) is false, the correct answer is option (C). 
 

Answer 2

Step 1: Analyze Assertion (A)
From the graph, the line \( Z = x + 2y \) passes through two corner points \( (60, 0) \) and \( (120, 60) \), providing the same maximum value. This indicates that the maximum value occurs at infinite points along this segment. 
Thus, Assertion (A) is true. 
Step 2: Analyze Reason (R)
In general, the optimal solution of an LPP occurs at corner points of the feasible region. This is true; however, in this case, the solution lies along a line segment connecting two corner points. 
Thus, Reason (R) is not the correct explanation of Assertion (A). 
Step 3: Conclusion
Both Assertion (A) and Reason (R) are true, but Reason (R) does not explain Assertion (A). 
Hence, the correct answer is option (B).