Let the lines (2 – i) z = (2 + i) $\overline{z}and\left(2+i\right)z+\left(i-2\right)\overline{z}-4i=0,$ (here i² = -1) be normal to a circle C. If the line iz + $\overline{z}+1+i=0$  is tangent to this circle C, then its radius is:

A quadratic function f(x) satisfies f(x) ≥ 0 for all real x. If f(2) = 0 and f(4) = 12, then the value of f(6) is :
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The area of the triangle vertices A(z), B(iz) and C(z + iz) is:
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Let z be those complex numbers which satisfy

$\left|z+5\right|\le 4andz\left(1+i\right)+\overline{z}\left(1-i\right)\ge -10,i=\sqrt[]{-1}.$

If the maximum value of ${\left|z+1\right|}^{2}is\alpha +\beta \sqrt[]{2},$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then the value of (α + β) is…….

Let $z=\frac{1-i\sqrt[]{3}}{2},i=\sqrt[]{-1}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> Then the value of

$21+{\left(z+\frac{1}{z}\right)}^3+{\left(z^2+\frac{1}{z^2}\right)}^3+{\left(z^3+\frac{1}{z^3}\right)}^3+.....+{\left(z^{21}+\frac{1}{z^{21}}\right)}^{3}is\_\_\_\_\_\_\_\_\_.$

If a, b, $\gamma$ <!-- [if gte mso 9]><xml> </xml><![endif]-->, $\delta$ <!-- [if gte mso 9]><xml> </xml><![endif]-->, are the roots of the equation x⁴ + x³ + x² + x + 1 = 0, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is equal to: