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Question

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

This question was previously asked in
NDA 2 2025 GAT Question Paper (14-Sep-2025)
The correct answer is

x = ω

In this case, x = ω is a root of the equation because ω is the cube root of unity.

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Similar Questions

  1. If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
  2. Let \(z_1\) and \(z_2\) be two complex numbers such that \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), then what is \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equal to ?
  3. If \(\omega \ne 1\) is a cube root of unity, then what are the solutions of \((z-100)^3 + 1000 = 0\) ?
  4. What is \((1+i)^4 + (1-i)^4\) equal to, where \(i = \sqrt{-1}\)?
  5. If \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?
  6. What is the value of the sum
    \(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
    where \(i = \sqrt{-1}\)?
  7. If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
  8. If \(\alpha\), \(\beta\), \(\gamma\) are cube roots of -8, then what is \(\frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2}\) equal to ?
  9. What is the value of \(\left|\frac{Z_1}{Z_2}\right|\)?
  10. What is the value of \(\frac{1}{2} + \text{Re}\left(\frac{Z_1}{Z_2}\right)\)?

Important Questions from Complex Numbers

  1. If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -

  2. If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:

  3. If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:

  4. If iz3 + z2 - z + i = 0, then the value of |z| is:

  5. Nature of the triangle formed by the points representing the complex numbers 3 + 4i, 8 - 6i and 13 + 9i is:

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