Interpret the system

The standard form for time period of SHM is $T = 2\pi\sqrt{\frac{M}{k_{\text{eq}}}}.$

We are told that $T = \pi\sqrt{\frac{\alpha M}{5k}}.$

So we need to find the equivalent spring constant $k_{\text{eq}}$ of the arrangement shown in the figure, and then compare both expressions.

Use the spring combination from the figure

From the given arrangement, the effective spring constant comes out to be $k_{\text{eq}} = \frac{5k}{12}.$

Write the time period using this equivalent spring constant

Compare with the given form

Given, $T = \pi\sqrt{\frac{\alpha M}{5k}}.$

Therefore, $\alpha = 48.$

Compare with stored answer

Stored correct answer: $12$

But from the SHM formula, $T = 2\pi\sqrt{\frac{M}{k_{\text{eq}}}}$ and with $k_{\text{eq}}=\frac{5k}{12},$ the result is clearly $T = \pi\sqrt{\frac{48M}{5k}},$ so $\alpha=48$, not $12$.

The stored answer appears to have missed the factor of $4$ arising from the leading $2\pi$ in the time-period formula.