This questions tests the conceptual understanding of definition of mean and sd.
An alternative solution is to calculate the new mean if one cannot explain why.
Original mean = 137 Apples = 6 Total = 137 x 6 = 822
Increase in each apple = 25 => Total increase = 25 x 6 = 150
New total = 822 + 150 = 972 Hence New mean = 972/6 = 162 g
Since the increase in mass for each apple is a constant, this implies the difference in mass between each apple is the same. By definition, the standard deviation is the spread from the mean, hence it remains the same.
Standard deviation has no change = 5.48g