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12 problems match “complex conjugate”. Clear

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1. Dividing complex numbers ●○○○○ 10 ptsCompute $\dfrac{20+10i}{1-3i}$. Write the result in the form $a+bi$ and find $a$ and $b$.
2. Conjugate product of complex numbers ●○○○○ 10 ptsThe complex numbers $a$ and $b$ satisfy $a \overline{b} = -1 + 5i$. Find the real part and the imaginary part of $\overline{a}\, b$.
3. Product of a complex number and its conjugate ●○○○○ 10 ptsGiven a complex number $z$ satisfying $z(1+i)=2i$, find $z\cdot\overline{z}$.
4. Product of a complex root and its conjugate ●○○○○ 10 ptsLet $z$ be a complex number that satisfies $z^2 + 3 = 0$. Find the value of $z \cdot \overline{z}$, where $\overline{z}$ denotes the complex conjugate of $z$.
5. Inverse of a complex number ●○○○○ 10 ptsFind the real part and the imaginary part of $(7+9i)^{-1}$. Round each to two decimal places.
6. Real part of reciprocal on unit circle ●●○○○ 20 ptsLet $z$ be a complex number with $|z| = 1$ and $z \neq -1$. Show that $\operatorname{Re}\left(\frac{1}{z+1}\right)$ is a constant, and find its value.
7. Modulus of a symmetric complex ratio ●●○○○ 20 ptsLet $z_1, z_2, z_3$ be complex numbers with $|z_1|=|z_2|=|z_3|=1$, and assume $z_1+z_2+z_3\neq 0$. Find $\left|\frac{z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}}{z_{1}+z_{2}+z_{3}}\right|$.
8. Minimise |z|² under a conjugate constraint ●●○○○ 20 ptsOver complex numbers $z$, minimize $z^*z$ subject to $z = z^* + i$, where $z^*$ is the complex conjugate of $z$. Write the minimizing $z$ as $z = a + bi$ with real $a$ and $b$, and find $a$ and $b$.
9. Complex solutions of a conjugate equation ●●○○○ 20 ptsFind all complex numbers $z$ satisfying $z\bar z+z^2=\overline{\bar z}-z+2$. List the solutions in increasing order.
10. Complex equation with a conjugate ●●○○○ 20 ptsLet $z$ be a complex number and let $z'$ denote its complex conjugate. Find $z$ if $\frac{z}{2}+3=\frac{z'}{3}-2$.

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