Question

Given $L \ge N < M = Q \le R$, then which of the following are true?
A. L = R
B. Q > N
C. R > N
D. L > Q
E. R $\ge$ M
Choose the correct answer from the options given below :

• A. B and E only
• B. A, C and E only
• C. C and D only
• D. B, C and E only

Hint:

In inequality chains, whenever you encounter two symbols pointing in opposite directions between two variables (e.g., $<$ and $>$), no definite conclusion can be drawn between those variables.
This immediately identifies statements comparing $L$ with $M, Q,$ or $R$ as false, allowing quick elimination of options.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question involves mathematical inequalities and transitive relations.
We are given a chain of inequalities relating five variables: $L, N, M, Q,$ and $R$.
We must analyze each proposed statement (A to E) to determine which inequalities are always mathematically true based on the given relationship.

Step 2: Detailed Explanation:

• Analyze the Given Inequality Chain:
The given relationship is:
\[ L \ge N < M = Q \le R \]

• Evaluate Statement A ($L = R$):
The relationship between $L$ and $R$ spans across a change in inequality direction ($L \ge N < M$).
Since the inequalities point in opposite directions, there is no defined relationship between $L$ and $R$.
Therefore, $L = R$ is not necessarily true. (Statement A is False).

• Evaluate Statement B ($Q > N$):
We have the path: $N < M = Q$.
This simplifies to $N < Q$, which can be rewritten as $Q > N$.
Therefore, $Q > N$ is always true. (Statement B is True).

• Evaluate Statement C ($R > N$):
We have the path: $N < M = Q \le R$.
This simplifies to $N < R$, which can be rewritten as $R > N$.
Therefore, $R > N$ is always true. (Statement C is True).

• Evaluate Statement D ($L > Q$):
The path between $L$ and $Q$ contains a change in inequality direction ($L \ge N < M = Q$).
Because of this sign change, no definite comparison can be made between $L$ and $Q$.
Therefore, $L > Q$ is not necessarily true. (Statement D is False).

• Evaluate Statement E ($R \ge M$):
We have the path: $M = Q \le R$.
This simplifies to $M \le R$, which can be rewritten as $R \ge M$.
Therefore, $R \ge M$ is always true. (Statement E is True).

Step 3: Final Answer:
Statements B, C, and E are true, while statements A and D are not necessarily true.
Hence, the correct option is (D).