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If ω is a non-real cube root of unity, then what is a root of the following equation?

 

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NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)
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  1. x=0
  2. x=1
  3. x=ω
  4. x=ω2

Answer (Detailed Solution Below)

Option 1 : x=0
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Detailed Solution

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Calculation:

Given,

Let ω be a non-real cube root of unity, so and .

Consider the determinant

Step 1 — Column operation:  Replace the first column by :

Step 2 — Expansion along the third row:

which simplifies to

Step 3 — Equate to zero:

∴ The root of the equation is  .

Hence, the correct answer is Option 1.

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