Step 1: Read the data.
Two coils sit close together with a mutual inductance of $M = 5\ H$. The current in the first coil changes from $0$ to $12\ A$. We want the change in flux linkage in the second coil.
Step 2: Recall what mutual inductance does.
Mutual inductance tells how much flux links the second coil for each ampere of current in the first coil. The link is $N\phi = M I$.
Step 3: Note the change form.
Since $M$ is fixed, the change in flux linkage is $\Delta(N\phi) = M\,\Delta I$. The time given is extra information and is not needed for the flux change itself.
Step 4: Find the current change.
The current goes from $0$ to $12\ A$, so $\Delta I = 12 - 0 = 12\ A$.
Step 5: Multiply the values.
$\Delta(N\phi) = M\,\Delta I = 5 \times 12 = 60$.
Step 6: State the answer with units.
Flux linkage is measured in webers, so the change is $60\ Wb$. \[ \boxed{60~\text{Wb}} \]