Question

Let $S=\{x\in R:x\ge 0$ and $2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0\}$. Then $S$:

• A. contains exactly two elements.
• B. contains exactly four elements.
• C. is an empty set.
• D. contains exactly one element.

Answer 1

The correct option is A

contains exactly two elements.

Explanation for the correct option:

Determine $S$

On solving, we have:

$2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0$

$$\begin{gathered} \mathrm{let},\sqrt{x}=t \\ \Rightarrow 2|t-3|+t(t-6)+6=0 \end{gathered}$$

Case (i)

$$\begin{gathered} t<3 \\ \therefore -2t+6+t^2-6t+6=0 \\ \Rightarrow t^2-8t+12=0 \\ \Rightarrow t=2,6 \end{gathered}$$

Since, $t<3$

Hence $t=2$

$$\begin{gathered} \sqrt{x}=2 \\ \Rightarrow x=4 \end{gathered}$$

Case (ii)

$t\ge 3$

$$\begin{gathered} \therefore 2t-6+t^2-6t+6=0 \\ \Rightarrow t^2-4t=0 \\ \Rightarrow t=4,0 \end{gathered}$$

Since, $t\ge 3$

hence $t=4$

$$\begin{gathered} \therefore \sqrt{x}=4 \\ \Rightarrow x=16 \end{gathered}$$

Therefore, $S$ contains exactly two elements.

Hence option A is the correct answer.