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Let's break down the problem step by step.

### Given Information:

We are analyzing the expression for the **shear modulus** \( \mu_0 \), which is a function of various parameters, including the temperature \( T \), Boltzmann constant \( K_B \), and the membrane spring length \( l_0 \). The key equation is:

\[
\mu_0 = \frac{\sqrt{3 K_B T}}{4 p l_m x_0} \left( \frac{x_0}{2(1-x_0)^3} - \frac{1}{4(1-x_0)} + \frac{1}{4} \right) + \frac{\sqrt{3 k_p (n+1)}}{(4/m + 1)}
\]

We are asked to evaluate the **truth** of three statements regarding \( \mu_0 \) as a function of temperature \( T \):

1. \( \mu_0 \) is proportional to \( T \).
2. The graph of \( \mu_0 \) as a function of \( T \) is linear.
3. The slope of the graph of \( \mu_0 \) as a function of \( T \) is \( \frac{\sqrt{3 K_B}}{2 p l_m} \).

### Key assumptions:

- \( x_0 = \frac{1}{2} \)
- \( k_p = 0 \)

This simplifies the equation significantly.

### Simplification for \( x_0 = \frac{1}{2} \):

When we substitute \( x_0 = \frac{1}{2} \) into the expression, we can simplify terms like \( (1 - x_0) \) and other constants.

### Checking the statements:

#### 1. Proportionality to \( T \):
The term \( \sqrt{3 K_B T} \) appears in \( \mu_0 \), which grows as \( \sqrt{T} \), not linearly with \( T \). Hence, **\( \mu_0 \) is not proportional to \( T \)**.

#### 2. Linearity of \( \mu_0 \) in \( T \):
Since \( \mu_0 \) contains \( \sqrt{T} \), the relationship is **not linear** but a square root function of \( T \).

#### 3. Slope of the graph of \( \mu_0 \) as a function of \( T \):
If we differentiate \( \mu_0 \) with respect to \( T \), the derivative will yield a term involving \( \frac{1}{2 \sqrt{T}} \), and the constant multiplier for \( T \)-dependence does match the form \( \frac{\sqrt{3 K_B}}{2 p l_m} \). Hence, **this slope expression is correct**.

### Conclusion:
- **Only statement (iii) is correct**, so the correct answer is **(E) (iii) only**.

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Would you like a more detailed explanation of the simplifications and derivatives? Here are some follow-up questions that might interest you:

1. What happens to \( \mu_0 \) if \( k_p \) is not zero?
2. How does the temperature dependence of \( \mu_0 \) affect membrane mechanics?
3. Can the shear modulus \( \mu_0 \) exhibit other forms of nonlinearity?
4. What is the physical significance of the slope in statement (iii)?
5. How would the model change if \( x_0 \neq 1/2 \)?

**Tip:** For proportionality questions, always check if the function is directly dependent on a variable or involves square roots or higher powers.

Analyze the expression for the shear modulus μ₀ given by the formula and assess the truth of three statements about its relationship with the temperature T.

Explore how the shear modulus μ₀, as given in thermodynamic models of hexagonal networks, relates to the temperature T. Assess the truth of statements about proportionality, linearity, and the slope of μ₀ as a function of T. Suitable for advanced physics students.

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