If `Z` is a non-real complex number, then find the minimum value of |`(Imz^5)/(Im^5z)`|

Options

A. `-1`
B. `-2`
C. `-4`
D. `-5`

Explanation

To find the minimum value of \(\left| \frac{\text{Im}(z^5)}{\text{Im}(z)^5} \right|\) where \(z\) is a non-real complex number, we can follow these steps:

### Step 1: Represent the Complex Number
Let \(z = x + iy\), where \(x\) and \(y\) are real numbers and \(y \neq 0\) since \(z\) is non-real.

### Step 2: Calculate \(z^5\)
Using the binomial theorem, we can expand \(z^5\):
\[ z^5 = (x + iy)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (iy)^k \]
This expansion gives us:
\[ z^5 = x^5 + 5x^4(iy) + 10x^3(iy)^2 + 10x^2(iy)^3 + 5x(iy)^4 + (iy)^5 \]
Simplifying the powers of \(i\):
\[ = x^5 + 5x^4(iy) - 10x^3y^2 - 10x^2y^3i + 5xy^4 + iy^5 \]
Combining real and imaginary parts:
\[ = (x^5 - 10x^3y^2 + 5xy^4) + i(5x^4y - 10x^2y^3 + y^5) \]

### Step 3: Identify the Imaginary Part
The imaginary part of \(z^5\) is:
\[ \text{Im}(z^5) = 5x^4y - 10x^2y^3 + y^5 \]

### Step 4: Calculate \(\text{Im}(z)^5\)
The imaginary part of \(z\) is simply \(y\), so:
\[ \text{Im}(z)^5 = y^5 \]

### Step 5: Set Up the Expression
Now we can set up the expression we need to minimize:
\[ \left| \frac{\text{Im}(z^5)}{\text{Im}(z)^5} \right| = \left| \frac{5x^4y - 10x^2y^3 + y^5}{y^5} \right| = \left| \frac{5x^4}{y^4} - \frac{10x^2}{y^2} + 1 \right| \]

### Step 6: Let \(t = \frac{x}{y}\)
Let \(t = \frac{x}{y}\). Then, we can rewrite the expression as:
\[ \left| 5t^4 - 10t^2 + 1 \right| \]

### Step 7: Find the Minimum Value
To find the minimum value of \(5t^4 - 10t^2 + 1\), we can treat it as a function of \(u = t^2\):
\[ f(u) = 5u^2 - 10u + 1 \]
This is a quadratic function, and its minimum occurs at:
\[ u = -\frac{b}{2a} = -\frac{-10}{2 \times 5} = 1 \]
Substituting \(u = 1\) back into the function:
\[ f(1) = 5(1)^2 - 10(1) + 1 = 5 - 10 + 1 = -4 \]

### Step 8: Conclusion
Thus, the minimum value of \(\left| \frac{\text{Im}(z^5)}{\text{Im}(z)^5} \right|\) is:
\[ \left| -4 \right| = 4 \]

### Final Answer
The minimum value is \(\boxed{4}\).

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