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To solve this problem, let's first analyze the given information and find the required value step by step.

### Given:
- \(\cos \theta = -\frac{5}{13}\) in Quadrant II.
- We need to find \(\cos 2\theta\).

### Step-by-Step Solution:

1. **Understand the Quadrant and Significance:**

   Since \(\theta\) is in Quadrant II:
   - \(\cos \theta < 0\) (which is given as \(-\frac{5}{13}\))
   - \(\sin \theta > 0\)

2. **Find \(\sin \theta\) using Pythagorean Identity:**

   The Pythagorean identity states:
   \[
   \sin^2 \theta + \cos^2 \theta = 1
   \]
   
   Substituting \(\cos \theta = -\frac{5}{13}\):
   \[
   \sin^2 \theta + \left(-\frac{5}{13}\right)^2 = 1
   \]
   \[
   \sin^2 \theta + \frac{25}{169} = 1
   \]
   \[
   \sin^2 \theta = 1 - \frac{25}{169}
   \]
   \[
   \sin^2 \theta = \frac{169}{169} - \frac{25}{169}
   \]
   \[
   \sin^2 \theta = \frac{144}{169}
   \]
   \[
   \sin \theta = \sqrt{\frac{144}{169}} = \frac{12}{13}
   \]

   Since \(\sin \theta > 0\) in Quadrant II, we have:
   \[
   \sin \theta = \frac{12}{13}
   \]

3. **Calculate \(\cos 2\theta\):**

   The double angle formula for cosine is:
   \[
   \cos 2\theta = \cos^2 \theta - \sin^2 \theta
   \]

   Substitute \(\cos \theta = -\frac{5}{13}\) and \(\sin \theta = \frac{12}{13}\):

   \[
   \cos 2\theta = \left(-\frac{5}{13}\right)^2 - \left(\frac{12}{13}\right)^2
   \]
   \[
   \cos 2\theta = \frac{25}{169} - \frac{144}{169}
   \]
   \[
   \cos 2\theta = \frac{25 - 144}{169}
   \]
   \[
   \cos 2\theta = \frac{-119}{169}
   \]

4. **Match the Answer:**

   The value of \(\cos 2\theta = -\frac{119}{169}\), which corresponds to option (A).

### Conclusion:

The correct answer is **A. \(-\frac{119}{169}\)**.

Would you like further details or have any questions?

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Here are five related questions to deepen your understanding:

1. How do you determine the sign of trigonometric functions in different quadrants?
2. What are the double angle formulas for sine and tangent?
3. How can we use trigonometric identities to solve equations involving trigonometric functions?
4. How would you solve for \(\sin 2\theta\) given the same initial conditions?
5. What are the Pythagorean identities and how are they derived?

**Tip:** Memorize the signs of trigonometric functions in all four quadrants; this helps in solving many trigonometric problems quickly.

Learn how to calculate the cosine of double angle using the given trigonometric values. This problem involves trigonometric identities and the double angle formula for cosine, with step-by-step explanations. Suitable for high school students studying trigonometry.

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