Topic: Rigid Bodies
Concepts: Moments of inertia

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Solution:

Recall the moment of inertia of a rod about its center is

We are given $I=md^2$ which means

so the answer is D.

Topic: Oscillatory Motion
Concepts: Parallel-axis theorem, Physical pendulum

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Solution:

Recall the angular frequency of a physical pendulum is given by

where $s$ is the distance between the CM and the pivot point and $I$ is the moment of inertia around the pivot point. In our case, we have

using $I_{\text{rod}}=md^2$ (by definition of $d$ in the problem) and the parallel-axis theorem. Then

Since we are given

we can identify

so the answer is E.

Topic: Oscillatory Motion
Concepts: Physical pendulum

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Solution:

From the previous problem, we have

To maximize $\beta$, we can equivalently minimize $1/\beta^2$:

which occurs at $k=1$. Thus,

so the answer is C.