Question

Let S = {x ∈ R : x ≥ 0 and `2|sqrt(x) - 3| + sqrt(x)(sqrt(x) - 6) + 6 = 0}`. Then S ______.

पर्याय

• contains exactly one element.

• contains exactly two elements.

• contains exactly four elements.

• is an empty set.

Answer 1

Let S = {x ∈ R : x ≥ 0 and `2|sqrt(x) - 3| + sqrt(x)(sqrt(x) - 6) + 6 = 0}`. Then S contains exactly two elements.

Explanation:

∵ `|sqrt(x) - 3| = {{:(sqrt(x) - 3;, x ≥ 9),(3 - sqrt(x), x < 9):}`

Case-I: x ∈ [0, 9)

`2(3 - sqrt(x)) + x - 6sqrt(x) + 6` = 0

⇒ `x - 8sqrt(x) + 12` = 0

⇒ `sqrt(x)` = 6, 2

⇒ x = 36, 4

Since x ∈ [0, 9) ; ∴ x = 4

Case-II: x ∈ [9, ∞)

`2(sqrt(x) - 3) + x - 6sqrt(x) + 6` = 0

⇒ `x - 4sqrt(x)` = 0

⇒ x = 16, 0

Since x ∈ [9, ∞) ; ∴ x = 16

Hence, x = 4 and 16