Question

Let S be the set of all real roots of the equation, 3^x(3^x – 1) + 2 = |3^x – 1| + |3^x – 2|. Then S ______.

Options

• contains exactly two elements.

• is a singleton.

• is an empty set.

• contains at least four elements.

Answer 1

Let S be the set of all real roots of the equation, 3^x(3^x – 1) + 2 = |3^x – 1| + |3^x – 2|. Then S is a singleton.

Explanation:

Let 3^x = y

∴ y(y – 1) + 2 = |y – 1| + |y – 2|

Case I: When y > 2

y² – y + 2 = y – 1 + y – 2

⇒ y² – 3y + 5 = 0

∵ D < 0  ...[∵ Equation not satisfy.]

Case II: When 1 ≤ y ≤ 2

y² – y + 2 = y – 1 + y – 2; y² – y + 1 = 0

∵ D < 0  ...[∵ Equation not satisfy.]

Case III: When y ≤ 1

y² – y + 2 = –y + 1 – y + 2

⇒ y² + y – 1 = 0

∴ y = `(-1 + sqrt(5))/2` = `(-1 - sqrt(5))/2`  ...[∵ Equation not satisfy]

∴ Only one `-1 + sqrt(5)/2`satisfy equation