`A_(1)` and `A_(2)` are two vectors such that `|A_(1)| = 3 , |A_(2)| = 5` and `|A_(1)+A_(2)| = 5` the value of `(2A_(1)+3A_(2)).(2A_(1)-2A_(2))` is

Options

A. `(237)/(2)`
B. `-123`
C. `(-337)/(2)`
D. `(337)/(2)`

Answer

To solve the problem, we need to find the value of the expression \((2A_1 + 3A_2) \cdot (2A_1 - 2A_2)\).

### Step-by-step Solution:

1. **Understanding the Dot Product**:
The dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is given by:
\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) \]
where \( \theta \) is the angle between the two vectors.

2. **Expanding the Expression**:
We can expand the expression using the distributive property of the dot product:
\[ (2A_1 + 3A_2) \cdot (2A_1 - 2A_2) = 2A_1 \cdot 2A_1 + 2A_1 \cdot (-2A_2) + 3A_2 \cdot 2A_1 + 3A_2 \cdot (-2A_2) \]
This simplifies to:
\[ 4A_1 \cdot A_1 - 4A_1 \cdot A_2 + 6A_2 \cdot A_1 - 6A_2 \cdot A_2 \]

3. **Combining Like Terms**:
We can combine the terms:
\[ 4|A_1|^2 + (6 - 4)A_1 \cdot A_2 - 6|A_2|^2 \]
This simplifies to:
\[ 4|A_1|^2 + 2A_1 \cdot A_2 - 6|A_2|^2 \]

4. **Substituting Known Values**:
We know:
- \( |A_1| = 3 \) so \( |A_1|^2 = 9 \)
- \( |A_2| = 5 \) so \( |A_2|^2 = 25 \)
- We need to find \( A_1 \cdot A_2 \).

5. **Finding \( A_1 \cdot A_2 \)**:
From the given information, we know:
\[ |A_1 + A_2| = 5 \]
Using the formula for the magnitude of the sum of two vectors:
\[ |A_1 + A_2|^2 = |A_1|^2 + |A_2|^2 + 2A_1 \cdot A_2 \]
Substituting the known values:
\[ 5^2 = 9 + 25 + 2A_1 \cdot A_2 \]
This gives:
\[ 25 = 34 + 2A_1 \cdot A_2 \]
Rearranging gives:
\[ 2A_1 \cdot A_2 = 25 - 34 = -9 \quad \Rightarrow \quad A_1 \cdot A_2 = -\frac{9}{2} \]

6. **Substituting Back**:
Now substituting back into our expression:
\[ 4(9) + 2\left(-\frac{9}{2}\right) - 6(25) \]
This simplifies to:
\[ 36 - 9 - 150 = 36 - 159 = -123 \]

### Final Answer:
The value of \((2A_1 + 3A_2) \cdot (2A_1 - 2A_2)\) is \(-123\).