Q55 / 150 · MHT CET 2026 (20 April Shift 1) · Mathematics
If $w$ is a complex cube root of unity , then the value of $w^{10} - w^7 + w^5 - w^2 + 1$ is.
- A$0$
- B$-1$
- Correct answerC$1$
- D$w$
Solution
Since $w$ is a complex cube root of unity, $w^3 = 1$ and $1 + w + w^2 = 0$.
Simplifying each term using $w^3 = 1$:
$w^{10} = (w^3)^3 \cdot w = 1 \cdot w = w$
$w^7 = (w^3)^2 \cdot w = 1 \cdot w = w$
$w^5 = w^3 \cdot w^2 = 1 \cdot w^2 = w^2$
Substituting these values into the given expression:
$w^{10} - w^7 + w^5 - w^2 + 1 = w - w + w^2 - w^2 + 1$
$= 0 + 0 + 1 = 1$
Answer: $1$
Simplifying each term using $w^3 = 1$:
$w^{10} = (w^3)^3 \cdot w = 1 \cdot w = w$
$w^7 = (w^3)^2 \cdot w = 1 \cdot w = w$
$w^5 = w^3 \cdot w^2 = 1 \cdot w^2 = w^2$
Substituting these values into the given expression:
$w^{10} - w^7 + w^5 - w^2 + 1 = w - w + w^2 - w^2 + 1$
$= 0 + 0 + 1 = 1$
Answer: $1$
Hence, the correct answer is option: 'C'.