Question Type 2: Working with conjugate of complex numbers Exercises

Let $z=x+yi$ satisfy $z+\bar{z}=10$. Find the value of $x$.

Simplify the expression $\frac{3 + 2i}{1 - i}$ by rationalizing the denominator.

Compute the modulus of $5 - 12i$ by using its conjugate.

Simplify the expression $\frac{1+2i}{3-4i} + \frac{5 - i}{2 + i}$ and express the result in the form $a + bi$.

Multiply the complex number $4 - 3i$ by its conjugate and simplify the result.

Let $z=x+yi$. Show that $z\bar{z}=x^2+y^2$.

Solve for $z$ if $\bar{z}=2+3i$.

Find the conjugate of $-2 + 5i$ and then compute the product of the number with its conjugate.

Express $\frac{1}{1 + i}$ in the form $a + bi$.

Find the conjugate of the complex number $4 - 3i$.

Simplify $\frac{2 - 3i}{4 + i}$ and express your answer in the form $a + bi$.

Simplify $\frac{5 + 3i}{2 + 3i}$ and write your answer in the form $a + bi$.