Question

Given below are four statements: \(M\geq X\), \(P<B\), \(B=M\), \(X>Q\). Conclusions: I. \(M\geq Q\), II. \(B>Q\), III. \(P<Q\). Which conclusions follow?

• A. III only
• B. II only
• C. I and II only
• D. II and III only

Hint:

In inequality questions, a conclusion follows only when it is definitely true in all possible cases.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question is based on mathematical inequalities. We need to combine the given inequality statements into a single chain and verify which conclusions logically follow.

Step 2: Key Formula or Approach:
Combine the statements by linking common variables: \[ B = M, \quad M \ge X, \quad X > Q \implies B = M \ge X > Q \] And we also have $P < B$.

Step 3: Detailed Explanation:
Let us analyze each conclusion based on our linked inequality chain:

• Conclusion (I): $M \ge Q$
From the chain, we have $M \ge X$ and $X > Q$. Since one of the inequalities is a strict inequality ($>$), the relation between $M$ and $Q$ is strictly greater than: \[ M > Q \] Therefore, saying $M \ge Q$ (greater than or equal to) as a definite conclusion is mathematically incorrect, as $M$ can never be equal to $Q$.

• Conclusion (II): $B > Q$
From our chain, we have: \[ B = M \ge X > Q \implies B > Q \] Since $B$ is strictly greater than $Q$, this conclusion is definitely true.

• Conclusion (III): $P = Q$
We know $P < B$ and $B > Q$. There is no direct relationship or path linking $P$ and $Q$ since the inequality signs are opposing ($P<B > Q$). Thus, we cannot conclude that $P = Q$.
Therefore, only Conclusion (II) is definitely true.

Step 4: Final Answer:
Only Conclusion (II) is true, which corresponds to option (B).