Consider a parallelogram constructed on the vectors `vecA=5vecp+2vecq and vecB=vecp-3vecq`. If `|vecp|=2, |vecq|=5`, the angle between `vecp` and `vecq` is `(pi)/(3)` and the length of the smallest diagonal of the parallelogram is k units, then the value of `k^(2)` is equal to

Answer

To solve the problem, we need to find the value of \( k^2 \), where \( k \) is the length of the smallest diagonal of the parallelogram formed by the vectors \( \vec{A} \) and \( \vec{B} \).

### Step 1: Define the vectors
Given:
\[ \vec{A} = 5\vec{p} + 2\vec{q} \]
\[ \vec{B} = \vec{p} - 3\vec{q} \]

### Step 2: Calculate the diagonals
The diagonals of the parallelogram can be represented as:
1. \( \vec{D_1} = \vec{A} + \vec{B} \)
2. \( \vec{D_2} = \vec{A} - \vec{B} \)

Calculating \( \vec{D_1} \):
\[ \vec{D_1} = (5\vec{p} + 2\vec{q}) + (\vec{p} - 3\vec{q}) = (5\vec{p} + \vec{p}) + (2\vec{q} - 3\vec{q}) = 6\vec{p} - \vec{q} \]

Calculating \( \vec{D_2} \):
\[ \vec{D_2} = (5\vec{p} + 2\vec{q}) - (\vec{p} - 3\vec{q}) = (5\vec{p} - \vec{p}) + (2\vec{q} + 3\vec{q}) = 4\vec{p} + 5\vec{q} \]

### Step 3: Calculate the magnitudes of the diagonals
To find the lengths of the diagonals, we need to compute the magnitudes \( |\vec{D_1}| \) and \( |\vec{D_2}| \).

Calculating \( |\vec{D_1}|^2 \):
\[ |\vec{D_1}|^2 = |6\vec{p} - \vec{q}|^2 = (6\vec{p} - \vec{q}) \cdot (6\vec{p} - \vec{q}) = 36|\vec{p}|^2 - 12(\vec{p} \cdot \vec{q}) + |\vec{q}|^2 \]

Calculating \( |\vec{D_2}|^2 \):
\[ |\vec{D_2}|^2 = |4\vec{p} + 5\vec{q}|^2 = (4\vec{p} + 5\vec{q}) \cdot (4\vec{p} + 5\vec{q}) = 16|\vec{p}|^2 + 40(\vec{p} \cdot \vec{q}) + 25|\vec{q}|^2 \]

### Step 4: Substitute the known values
Given:
- \( |\vec{p}| = 2 \)
- \( |\vec{q}| = 5 \)
- The angle \( \theta \) between \( \vec{p} \) and \( \vec{q} \) is \( \frac{\pi}{3} \), thus \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

Now, calculate \( \vec{p} \cdot \vec{q} \):
\[ \vec{p} \cdot \vec{q} = |\vec{p}||\vec{q}|\cos\theta = 2 \cdot 5 \cdot \frac{1}{2} = 5 \]

### Step 5: Calculate \( |\vec{D_1}|^2 \)
Substituting the values into \( |\vec{D_1}|^2 \):
\[ |\vec{D_1}|^2 = 36(2^2) - 12(5) + (5^2) = 36(4) - 60 + 25 = 144 - 60 + 25 = 109 \]

### Step 6: Calculate \( |\vec{D_2}|^2 \)
Substituting the values into \( |\vec{D_2}|^2 \):
\[ |\vec{D_2}|^2 = 16(2^2) + 40(5) + 25(5^2) = 16(4) + 200 + 25(25) = 64 + 200 + 625 = 889 \]

### Step 7: Determine the smallest diagonal
The smallest diagonal is \( |\vec{D_1}| \) since \( 109 < 889 \).

### Step 8: Find \( k^2 \)
Thus, we have:
\[ k^2 = 109 \]

### Final Answer
The value of \( k^2 \) is \( \boxed{109} \).