In a triangle `ABC,` if `angle A = 30^@, b = 10 and a = x,` then the values of `x` for which there are 2 possible triangles is given by(All symbols used have usual meaning in a triangle.)

Options

A. (a)` 5 lt x lt 10`
B. (b)`x lt 5/2`
C. (c)`5/3 lt x lt 10`
D. (d)`5/2 lt x lt 10`

Answer

To solve the problem, we need to find the values of \( x \) for which there are two possible triangles given \( \angle A = 30^\circ \), \( b = 10 \), and \( a = x \).

### Step-by-Step Solution

1. **Understanding the Triangle**:
We have a triangle \( ABC \) where \( \angle A = 30^\circ \), side \( b = 10 \), and side \( a = x \). We need to find the conditions under which two triangles can be formed.

2. **Using the Law of Sines**:
According to the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Rearranging gives:
\[ \sin B = \frac{b \cdot \sin A}{a} = \frac{10 \cdot \sin 30^\circ}{x} = \frac{10 \cdot \frac{1}{2}}{x} = \frac{5}{x} \]

3. **Finding Conditions for Two Triangles**:
For two triangles to exist, the sine of angle \( B \) must satisfy:
\[ 0 < \sin B < 1 \]
This leads to two inequalities:
- \( \frac{5}{x} > 0 \) (which is always true for \( x > 0 \))
- \( \frac{5}{x} < 1 \)

4. **Solving the Inequality**:
From \( \frac{5}{x} < 1 \):
\[ 5 < x \quad \Rightarrow \quad x > 5 \]

5. **Considering the Triangle Inequality**:
We also need to ensure that the triangle inequality holds. The triangle inequality states:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

In our case, we need to find the upper limit for \( x \):
- Since \( b = 10 \), we need \( x < 10 + c \).

6. **Finding the Maximum Value of \( x \)**:
To ensure that \( x < 10 + c \), we can analyze the situation where \( c \) approaches its minimum value. The minimum value of \( c \) occurs when \( x \) is maximized. Therefore, we need to find the maximum \( x \) such that:
\[ x < 10 \]

7. **Final Range for \( x \)**:
Combining the inequalities, we have:
\[ 5 < x < 10 \]

### Conclusion
Thus, the values of \( x \) for which there are 2 possible triangles is:
\[ \boxed{(5, 10)} \]