$$Working\ simul\tan eously;\ \frac{1}{K}+\frac{1}{M}+\frac{1}{P}=\frac{1}{24}\min utes$$
Target question => How long does it take machine k working alone at its constant rate to complete the task?

Statement 1 =>
Machines M and P working simultaneously and independently at their respective constant rates can complete the task in 36 minutes
$$\ \frac{1}{M}+\frac{1}{P}=\frac{1}{36}$$
$$Given\ that\ \ \frac{1}{K}+\frac{1}{M}+\frac{1}{P}=\frac{1}{24}$$
$$\ \ \frac{1}{K}+\frac{1}{36}=\frac{1}{24}$$
$$\ \ \frac{1}{K}=\frac{1}{24}-\frac{1}{36}=\frac{3-2}{12}=\frac{1}{12}$$
K= 12 minutes. Statement 1 is SUFFICIENT

Statement 2 =>
Machines K and P working simultaneously and independently at their respective constant rates can complete the task in 48 minutes
$$\ \frac{1}{K}+\frac{1}{P}=\frac{1}{48}$$
$$\ \frac{1}{K}=\frac{1}{48}-\frac{1}{P}=\frac{P-48}{48P}$$
$$K=\frac{48P}{P-48}$$
The value of P is unknown and K's minute cannot be estimated. Hence, statement 2 is NOT SUFFICIENT

Since statement 1 alone is Sufficient, answer = A