Key Concepts and Formulas

• Logarithmic Inequality Property: If $0 < b < 1$, then $\log_b x \le y$ implies $x \ge b^y$. Also, $x > 0$.
• Modulus of a Complex Number: $|z|$ represents the magnitude of the complex number $z$ and is always non-negative, i.e., $|z| \ge 0$.
• Solving Quadratic Inequalities: If $(x-r_1)(x-r_2) \le 0$ and $r_1 < r_2$, then $r_1 \le x \le r_2$.

Step-by-Step Solution

1. Apply the Logarithmic Inequality Property

• Why this step: To eliminate the logarithm and simplify the inequality.
• Explanation: The given inequality is ${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$. Since the base is $b = \frac{1}{\sqrt{2}}$, which is between 0 and 1, we reverse the inequality sign when converting to exponential form. Also, we must ensure the argument of the logarithm is positive.

Applying the property: ${{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \ge \left( {{1 \over {\sqrt 2 }}} \right)^2$ ${{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \ge {1 \over 2}$ Since $|z| \ne 1$, the denominator $(|z|-1)^2 > 0$. Also $|z| \ge 0$, so $|z| + 11 > 0$. Thus, the argument of the logarithm is positive.

2. Eliminate the Denominator and Expand

• Why this step: To simplify the inequality and prepare it for solving.
• Explanation: Multiply both sides by $2(|z|-1)^2$, which is positive since $|z| \ne 1$.

Multiplying by $2(|z|-1)^2$: $2(|z| + 11) \ge (|z| - 1)^2$ $2|z| + 22 \ge |z|^2 - 2|z| + 1$

3. Rearrange into a Quadratic Inequality

• Why this step: To get a standard quadratic form that can be factored and solved.
• Explanation: Rearrange the terms to get a quadratic inequality in the form $ax^2 + bx + c \le 0$.

Rearranging: $0 \ge |z|^2 - 4|z| - 21$ $|z|^2 - 4|z| - 21 \le 0$

4. Factor the Quadratic Expression

• Why this step: To find the roots of the quadratic, which are critical for solving the inequality.
• Explanation: Factor the quadratic expression into the form $(|z| - r_1)(|z| - r_2)$.

Factoring: $(|z| - 7)(|z| + 3) \le 0$

5. Solve the Quadratic Inequality

• Why this step: To find the range of possible values for $|z|$.
• Explanation: The roots of the quadratic are $|z| = 7$ and $|z| = -3$. Since the quadratic is less than or equal to zero, the solution lies between the roots.

Solving the inequality: $-3 \le |z| \le 7$

6. Apply Modulus Constraints and Determine the Largest Value

• Why this step: To ensure the solution is physically meaningful, given the modulus definition.
• Explanation: Since $|z|$ is always non-negative, we have $|z| \ge 0$. Combining this with the solution to the quadratic inequality, we get $0 \le |z| \le 7$. The condition $|z| \ne 1$ does not affect the largest possible value of $|z|$.

Therefore, the largest value of $|z|$ is 7. However, we must check the options provided. The options are 5, 8, 6, and 7. Since 7 is an option, and our derivation is correct, the largest value of $|z|$ is 7.

Common Mistakes & Tips

• Remember to reverse the inequality sign when the base of the logarithm is between 0 and 1.
• Always consider the non-negativity of $|z|$.
• Double-check the factoring of the quadratic expression.

Summary

We started with a logarithmic inequality, converted it into a quadratic inequality using the properties of logarithms and complex number moduli. We solved the quadratic inequality, considered the constraint that $|z|$ is non-negative, and found the largest possible value of $|z|$ to be 7.

The final answer is \boxed{7}, which corresponds to option (D).