Set of 12 advanced complex number problems from a mathematics worksheet.

We’ll solve Problem 12 from your list:

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12. If $ |z + i| = z + 2(1 + i) $, then find $ z $.

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Step 1: Understand the equation

We are given:

$$
|z + i| = z + 2(1 + i)
$$

Note that the left-hand side (LHS) is a modulus, which is always a non-negative real number.

The right-hand side (RHS) is a complex expression: $ z + 2(1 + i) $. But since LHS is real and non-negative, the RHS must also be real and non-negative.

So, we can deduce:

> The complex number $ z + 2(1 + i) $ must be real and ≥ 0.

Let’s write $ z = x + iy $, where $ x, y \in \mathbb{R} $.

Then,

- LHS: $ |z + i| = |x + i(y + 1)| = \sqrt{x^2 + (y + 1)^2} $
- RHS: $ z + 2(1 + i) = (x + 2) + i(y + 2) $

Since RHS must be real, its imaginary part must be zero:

$$
\text{Im}(RHS) = y + 2 = 0 \Rightarrow y = -2
$$

Also, since RHS must be ≥ 0 (because modulus is ≥ 0), we require:

$$
\text{Re}(RHS) = x + 2 \geq 0 \Rightarrow x \geq -2
$$

Now plug $ y = -2 $ into the LHS:

$$
|z + i| = \sqrt{x^2 + (-2 + 1)^2} = \sqrt{x^2 + 1}
$$

And RHS becomes:

$$
x + 2 + i(-2 + 2) = x + 2
$$

So the equation becomes:

$$
\sqrt{x^2 + 1} = x + 2
$$

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Step 2: Solve the equation $ \sqrt{x^2 + 1} = x + 2 $

Since square root is non-negative, we require $ x + 2 \geq 0 $, which we already have ($ x \geq -2 $).

Now square both sides:

$$
x^2 + 1 = (x + 2)^2 = x^2 + 4x + 4
$$

Subtract $ x^2 $ from both sides:

$$
1 = 4x + 4
\Rightarrow 4x = -3
\Rightarrow x = -\frac{3}{4}
$$

Check if this satisfies $ x \geq -2 $: yes, $ -\frac{3}{4} > -2 $.

Also check in original equation:

LHS: $ \sqrt{(-3/4)^2 + 1} = \sqrt{\frac{9}{16} + 1} = \sqrt{\frac{25}{16}} = \frac{5}{4} $

RHS: $ x + 2 = -\frac{3}{4} + 2 = \frac{5}{4} $

✔ Matches.

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Step 3: Write final answer

Recall $ z = x + iy $, with $ x = -\frac{3}{4} $, $ y = -2 $

So,

$$
\boxed{z = -\frac{3}{4} - 2i}
$$

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✔ Final Answer: $ \boxed{z = -\dfrac{3}{4} - 2i} $

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*Note: You uploaded an image with multiple problems — I solved #12 as requested. If you’d like solutions to others, please specify which one(s).*