Let's go through the questions one by one:

A1. Given that $\ln z = xy$, which of the following is equivalent to $\frac{\partial z}{\partial y}$?

The equation given is $\ln z = xy$. To differentiate with respect to $y$, we will apply implicit differentiation.

1. Differentiate both sides with respect to $y$: $\frac{d}{dy}(\ln z) = \frac{d}{dy}(xy)$

2. Recall the chain rule for $\ln z$: $\frac{1}{z} \cdot \frac{\partial z}{\partial y} = x$

3. Therefore: $\frac{\partial z}{\partial y} = xz$

Thus, the correct answer is (b) $xz$.

A2. How many arbitrary constants are there in the particular solution of a second-order linear differential equation?

For a second-order linear differential equation, the general solution will typically contain two arbitrary constants (since it is of order 2).

Thus, the correct answer is (c) 2.

A3. Using Simpson's rule with only the values in the table below, what is the maximum number of strips we can take to estimate $\int_0^{10} y \, dx$?

The table gives values for $x$ and $y$, and you are to apply Simpson's rule. Simpson's rule requires an even number of strips to make use of the method correctly.

The values provided range from $x = 0$ to $x = 10$, so the total number of strips will be $10 - 0 = 10$ divided by the step size. Since the maximum number of strips must be an even number, the correct number is 8.

Thus, the correct answer is (d) 8.

A4. Consider the following two types of differential equations:

• Type I: (Variable) Separable
• Type II: Linear

Which of these types describe the equation $(x^2 + 2) \frac{dy}{dx} + xy \ln |x| = y$?

We need to determine whether this is separable or linear. Let's analyze:

• The equation is linear in $y$, because $y$ and its derivative $\frac{dy}{dx}$ appear in a linear form.
• The equation does not separate into a product of functions involving only $y$ and only $x$.

Therefore, this is a linear differential equation.

Thus, the correct answer is (b) Type II only.

A5. The following is the Fourier Series of a function $f(t)$:

$f(t) = \sum_{n=1}^{\infty} \frac{1}{n^2} (\cos(nt) + \sin(nt))$

Which of the following describes the function $f(t)$?

To check the symmetry of the function, notice that the Fourier series contains both $\cos(nt)$ and $\sin(nt)$, which are typically even and odd functions, respectively.

Thus, the function described by this series has components of both odd and even functions.

Therefore, the correct answer is (d) Both Odd and Even.

If you'd like further details on any of these steps or need clarification, feel free to ask! Here are some follow-up questions you may find useful:

1. Can you explain why implicit differentiation was used for A1?
2. What is the reasoning behind Simpson's rule requiring an even number of strips?
3. How would you apply this to a more complex differential equation in A4?
4. Can you explain why Fourier series involve both sine and cosine terms?
5. What are the properties of linear differential equations?

Tip: Always check for symmetry when working with Fourier series. It can help determine if the function is odd, even, or neither.