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.

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.