SSC Selection Post 2022 PYQ — Question 55

The correct answer is Option 1 i.e. $​​-1\over5$.

We know that,

a³ + b³ + c³ = 3abc

when, a + b + c = 0

⇒ (x + $1\over2$)³ + (3x + $2\over3$)³ + (x - $1\over6$)² = $1\over12$ × (9x + 2)(2x + 1)(6x - 1)

⇒ (x + $1\over2$)³ + (3x + $2\over3$)³ + (x - $1\over6$)² = $1\over12$ × 3(3x + $2\over3$) × 2(x + $1\over2$) × 6(x - $1\over6$)

⇒ (x + $1\over2$)³ + (3x + $2\over3$)³ + (x - $1\over6$)² = 3 × (3x + $2\over3$) × (x + $1\over2$) × (x - $1\over6$) 

So,

⇒ x + $1\over2$ + 3x + $2\over3$ + x - $1\over6$ = 0

⇒ 5x + $3+4-1\over6$ = 0

⇒ 5x + $6\over6$ = 0

⇒ 5x = -1

⇒ x = $​​-1\over5$