Find the coefficient of
(iv) `x^4` in the expansion of `((x)/(2) - (3)/(x^2) )^(10)`.

Answer

To find the coefficient of \( x^4 \) in the expansion of \( \left( \frac{x}{2} - \frac{3}{x^2} \right)^{10} \), we will use the Binomial Theorem.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = \frac{x}{2} \), \( b = -\frac{3}{x^2} \), and \( n = 10 \).

2. **Write the General Term**:
Substituting the values into the formula, we get:
\[ T_{r+1} = \binom{10}{r} \left( \frac{x}{2} \right)^{10-r} \left( -\frac{3}{x^2} \right)^r \]

3. **Simplify the General Term**:
Simplifying \( T_{r+1} \):
\[ T_{r+1} = \binom{10}{r} \left( \frac{x^{10-r}}{2^{10-r}} \right) \left( -\frac{3^r}{x^{2r}} \right) \]
This can be rewritten as:
\[ T_{r+1} = \binom{10}{r} \cdot (-3)^r \cdot \frac{x^{10-r-2r}}{2^{10-r}} = \binom{10}{r} \cdot (-3)^r \cdot \frac{x^{10-3r}}{2^{10-r}} \]

4. **Find the Power of \( x \)**:
We need the term where the power of \( x \) is 4:
\[ 10 - 3r = 4 \]
Solving for \( r \):
\[ 3r = 10 - 4 \implies 3r = 6 \implies r = 2 \]

5. **Substitute \( r \) Back into the General Term**:
Now substitute \( r = 2 \) into the general term:
\[ T_{3} = \binom{10}{2} \cdot (-3)^2 \cdot \frac{x^{10-3 \cdot 2}}{2^{10-2}} \]
This simplifies to:
\[ T_{3} = \binom{10}{2} \cdot 9 \cdot \frac{x^4}{2^8} \]

6. **Calculate the Coefficient**:
Calculate \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]
Therefore, the coefficient becomes:
\[ 45 \cdot 9 \cdot \frac{1}{256} = \frac{405}{256} \]

### Final Answer:
The coefficient of \( x^4 \) in the expansion is \( \frac{405}{256} \).