Complex numbers: แบบฝึกหัด

Given the complex number \(z = 5 - 2i\), find the value of \(z \bar{z}\), where \(\bar{z}\) is the complex conjugate of \(z\).

Given that \( 1 - 3i \) is a root of the cubic equation \( z^3 + az^2 + bz + 20 = 0 \), where \( a \) and \( b \) are real constants, determine the real root of the equation.

The quartic equation \( z^4 - 2z^3 + 3z^2 - 2z + 2 = 0 \) has roots \( z_1, z_2, z_3 \) and \( z_4 \). Given that \( z_1 = i \) is a root, find the other three roots of the equation.

Find the modulus of the complex number \(z = 3 - 4i\).

The cubic equation \(z^3 + az^2 + bz + c = 0\), where \(a, b,\) and \(c\) are real constants, has roots \(z_1 = 2\) and \(z_2 = 1 + i\). Determine the value of \(a\).

Given the complex number \(z = 7 - 4i\), calculate the value of \(\text{Re}(z) - \text{Im}(z)\).

Find the complex number \(z\) that satisfies the equation \(2z + \bar{z} = 9 - 2i\), where \(\bar{z}\) is the complex conjugate of \(z\).

In an Argand diagram, the locus of points \(z\) satisfying the equation \(\text{arg}(z - 2) = \frac{\pi}{3}\) is a half-line. Describe the position and direction of this locus in the complex plane.

Given that \(z = 5 - 12i\), find:
(a) the modulus of \(z\), \(|z|\),
(b) the argument of \(z\), \(\arg z\), in radians to 3 decimal places.
(c) Represent \(z\) and its conjugate \(z^*\) on a single Argand diagram.

The complex number \(z_1\) is a root of the quadratic equation \(z^2 - 6z + 25 = 0\).
(a) Solve the equation to find the two roots, \(z_1\) and \(z_2\), giving your answers in the form \(a {+} bi\).
(b) Represent these roots as points \(P_1\) and \(P_2\) on an Argand diagram, labeling the coordinates clearly.