Suppose p, q, r are positive rational numbers | vedclass

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(C) The correct option is $(C)$.Given that $p, q, r \in \mathbb{Q}^{+}$ and $x = \sqrt{p}+\sqrt{q}+\sqrt{r} \in \mathbb{Q}$.If any of $\sqrt{p}, \sqrt

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$\sqrt{p}, \sqrt{q}, \sqrt{r}$ are rational

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(C) The correct option is $(C)$.Given that $p, q, r \in \mathbb{Q}^{+}$ and $x = \sqrt{p}+\sqrt{q}+\sqrt{r} \in \mathbb{Q}$.If any of $\sqrt{p}, \sqrt{q}, \sqrt{r}$ is irrational,say $\sqrt{p}$,then we can write $\sqrt{p} = x - (\sqrt{q}+\sqrt{r})$.Squaring both sides,$p = x^2 + q + r + 2\sqrt{qr} - 2x(\sqrt{q}+\sqrt{r})$.This implies $\sqrt{qr}$ must be of the form $a + b\sqrt{q} + c\sqrt{r}$ for some rational $a, b, c$.By analyzing the cases where one,two,or three of these square roots are irrational,we find that the only way for the sum to be rational when $p, q, r$ are rational is if each individual term $\sqrt{p}, \sqrt{q}, \sqrt{r}$ is itself a rational number.For example,if $\sqrt{p}, \sqrt{q}, \sqrt{r}$ were irrational,their sum could only be rational if they were of the form $k_1\sqrt{n}, k_2\sqrt{n}, k_3\sqrt{n}$ for some non-square integer $n$,which would imply $\sqrt{p}, \sqrt{q}, \sqrt{r}$ are rational multiples of the same irrational number,but the sum condition forces them to be rational.

Suppose $p, q, r$ are positive rational numbers such that $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is also rational. Then

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