If $x = 3 - 2\sqrt{3}i$, then evaluate $x^4 - 12x^3 + 54x^2 - 108x - 54 = $
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- 0
- 6
- -6
- 9
The Correct Option is D
Solution and Explanation
Top Questions on Complex numbers
Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
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If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]
Then the value of \( \overline{z\omega} \) is:
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If \( x^a y^b = e^m, \)
and
\[ x^c y^d = e^n, \]
and
\[ \Delta_1 = \begin{vmatrix} m & b \\ n & d \\ \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} a & m \\ c & n \\ \end{vmatrix}, \quad \Delta_3 = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} \]
Then the values of \( x \) and \( y \) respectively (where \( e \) is the base of the natural logarithm) are:
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\( z_1, z_2, z_3 \) represent the vertices A, B, C of a triangle ABC respectively in the Argand plane. If
\[ |z_1 - z_2| = \sqrt{25 - 12 \sqrt{3}}, \] \[ \left|\frac{z_1 - z_3}{z_2 - z_3}\right| = \frac{3}{4}, \] \[ \text{and } \angle ACB = 30^\circ, \]
Then the area (in sq. units) of that triangle is:
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- If \( \alpha \) is the common root of the quadratic equations \( x^2 - 5x + 4a = 0 \) and \( x^2 - 2ax - 8 = 0 \), where \( a \in \mathbb{R} \), then the value of \( \alpha^4 - \alpha^3 + 68 \) is:
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