Observe the following statements :
Statement -I : In the expansion of `(1+x)^50,` the sum of the coefficients of odd powers of x is `2^50` .
Statement -II : The coefficient of `x^4` in the expansion of `(x/2 - (3)/(x^2))^10` is equal to `504/259` .
Then the true statements are :

Options

A. a) only I
B. b) only II
C. c) both I and II
D. d) neither I nor II

Answer

To determine the validity of the two statements provided, we will analyze each statement step by step.

### Statement I:
**In the expansion of \( (1+x)^{50} \), the sum of the coefficients of odd powers of \( x \) is \( 2^{50} \).**

1. **Understanding the Binomial Expansion**:
The binomial expansion of \( (1+x)^{50} \) is given by:
\[ (1+x)^{50} = \sum_{k=0}^{50} \binom{50}{k} x^k \]
where \( \binom{50}{k} \) are the binomial coefficients.

2. **Sum of Coefficients**:
The sum of all coefficients in the expansion is obtained by substituting \( x = 1 \):
\[ (1+1)^{50} = 2^{50} \]

3. **Separating Odd and Even Coefficients**:
The coefficients of odd powers of \( x \) can be derived from the expansion. The sum of the coefficients of odd powers can be calculated as:
\[ \text{Sum of odd coefficients} = \frac{(1+1)^{50} - (1-1)^{50}}{2} = \frac{2^{50} - 0}{2} = 2^{49} \]
This indicates that the sum of the coefficients of odd powers is \( 2^{49} \), not \( 2^{50} \).

4. **Conclusion for Statement I**:
Therefore, **Statement I is false**.

### Statement II:
**The coefficient of \( x^4 \) in the expansion of \( \left( \frac{x}{2} - \frac{3}{x^2} \right)^{10} \) is equal to \( \frac{504}{259} \).**

1. **Using the Binomial Theorem**:
The general term in the expansion of \( \left( a + b \right)^n \) 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. **Finding the Term for \( x^4 \)**:
We need to find \( r \) such that the power of \( x \) in the term equals 4:
\[ T_{r+1} = \binom{10}{r} \left( \frac{x}{2} \right)^{10-r} \left( -\frac{3}{x^2} \right)^r \]
The power of \( x \) in this term is:
\[ x^{10-r} \cdot x^{-2r} = x^{10 - 3r} \]
Setting \( 10 - 3r = 4 \) gives:
\[ 3r = 6 \implies r = 2 \]

3. **Calculating the Coefficient**:
Substitute \( r = 2 \) into the term:
\[ T_{3} = \binom{10}{2} \left( \frac{x}{2} \right)^{8} \left( -\frac{3}{x^2} \right)^{2} \]
This simplifies to:
\[ T_{3} = \binom{10}{2} \cdot \frac{x^8}{2^8} \cdot \frac{9}{x^4} \]
Thus, the coefficient of \( x^4 \) is:
\[ \binom{10}{2} \cdot \frac{9}{256} \]
Calculating \( \binom{10}{2} = \frac{10 \times 9}{2} = 45 \):
\[ \text{Coefficient} = 45 \cdot \frac{9}{256} = \frac{405}{256} \]

4. **Conclusion for Statement II**:
Since \( \frac{405}{256} \neq \frac{504}{259} \), **Statement II is also false**.

### Final Conclusion:
Both statements are false.