if the coefficients of `x^(5)" and "x^(15)` in the expansion of `(x^(2)+(a)/(x^(3)))^(10)` are equal then then the positive value of 'a' is:

Options

A. `2sqrt(3)`
B. `1`
C. `(1)/(sqrt(3))`
D. `(1)/(2sqrt(3))`

Explanation

To solve the problem, we need to find the positive value of 'a' such that the coefficients of \( x^5 \) and \( x^{15} \) in the expansion of \( (x^2 + \frac{a}{x^3})^{10} \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_r \) in the binomial expansion of \( (x^2 + \frac{a}{x^3})^{10} \) is given by:
\[ T_r = \binom{10}{r} (x^2)^{10-r} \left(\frac{a}{x^3}\right)^r \]
Simplifying this, we have:
\[ T_r = \binom{10}{r} a^r x^{20 - 2r - 3r} = \binom{10}{r} a^r x^{20 - 5r} \]

2. **Find the Coefficient of \( x^5 \)**:
For \( x^5 \), we set the exponent equal to 5:
\[ 20 - 5r = 5 \implies 5r = 15 \implies r = 3 \]
The coefficient of \( x^5 \) is:
\[ \text{Coefficient of } x^5 = \binom{10}{3} a^3 \]

3. **Find the Coefficient of \( x^{15} \)**:
For \( x^{15} \), we set the exponent equal to 15:
\[ 20 - 5r = 15 \implies 5r = 5 \implies r = 1 \]
The coefficient of \( x^{15} \) is:
\[ \text{Coefficient of } x^{15} = \binom{10}{1} a^1 \]

4. **Set the Coefficients Equal**:
According to the problem, the coefficients of \( x^5 \) and \( x^{15} \) are equal:
\[ \binom{10}{3} a^3 = \binom{10}{1} a \]

5. **Calculate the Binomial Coefficients**:
We know:
\[ \binom{10}{3} = \frac{10!}{3! \cdot 7!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120 \]
\[ \binom{10}{1} = 10 \]

6. **Substitute the Values**:
Substitute the values of the binomial coefficients into the equation:
\[ 120 a^3 = 10 a \]

7. **Rearrange the Equation**:
Rearranging gives:
\[ 120 a^3 - 10 a = 0 \]
Factor out \( a \):
\[ a(120 a^2 - 10) = 0 \]

8. **Solve for \( a \)**:
This gives us two cases:
- \( a = 0 \) (not positive)
- \( 120 a^2 - 10 = 0 \)
\[ 120 a^2 = 10 \implies a^2 = \frac{10}{120} = \frac{1}{12} \]
Taking the positive root:
\[ a = \sqrt{\frac{1}{12}} = \frac{1}{2\sqrt{3}} \]

### Final Answer:
The positive value of \( a \) is:
\[ a = \frac{1}{2\sqrt{3}} \]