Suppose p,q,r are positive rational numbers such that `sqrtp+sqrtq+sqrtr` are irrational. Then

Options

A. `sqrtp, sqrtq,sqrtr` are irrational
B. `sqrt(pq)sqrt(pr),sqrt(qr)` are rational, but `sqrtp,sqrtq,sqrtr` are irrational
C. `sqrtp,sqrtq,sqrtr` are rational
D. `sqrt(pq)sqrt(pr)sqrt(qr)` are irrational

Answer

To solve the problem, we need to analyze the conditions given and derive the conclusion step by step.

### Step 1: Understand the Given Information
We are given that \( p, q, r \) are positive rational numbers and that \( \sqrt{p} + \sqrt{q} + \sqrt{r} \) is irrational.

### Step 2: Analyze the Implications
Since \( p, q, r \) are rational, their square roots \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) can either be rational or irrational. However, the sum of three rational numbers is always rational. Therefore, at least one of \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) must be irrational for the sum to be irrational.

### Step 3: Case Analysis
We will analyze three cases based on how many of \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) are irrational.

#### Case 1: Exactly One of \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) is Irrational
Assume \( \sqrt{p} \) is irrational and \( \sqrt{q}, \sqrt{r} \) are rational. Then:
\[ \sqrt{p} + \sqrt{q} + \sqrt{r} = \text{(irrational)} + \text{(rational)} + \text{(rational)} = \text{irrational} \]
This case holds true.

#### Case 2: Exactly Two of \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) are Irrational
Assume \( \sqrt{p} \) and \( \sqrt{q} \) are irrational, and \( \sqrt{r} \) is rational. Then:
\[ \sqrt{p} + \sqrt{q} + \sqrt{r} = \text{(irrational)} + \text{(irrational)} + \text{(rational)} = \text{irrational} \]
This case also holds true.

#### Case 3: All Three \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) are Irrational
If all three are irrational, then:
\[ \sqrt{p} + \sqrt{q} + \sqrt{r} = \text{(irrational)} + \text{(irrational)} + \text{(irrational)} = \text{irrational} \]
This case also holds true.

### Step 4: Conclusion
From the analysis, we see that the sum \( \sqrt{p} + \sqrt{q} + \sqrt{r} \) can be irrational in all three cases. However, if we consider the implications of rationality and irrationality, we conclude that:

- If all three square roots were irrational, their sum would not necessarily be irrational unless they are specifically chosen to be conjugates.

Thus, the only consistent conclusion is that at least one of the square roots must be rational, leading to the conclusion that at least one of \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) must be rational.

### Final Answer
The correct statement is that at least one of \( \sqrt{p}, \sqrt{q}, \sqrt{r} \) must be rational.

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