Dimensional analysis is a method used to check the correctness of equations by comparing the dimensions of the physical quantities involved in the equation. The dimensions of a physical quantity refer to the powers to which the fundamental units of mass, length, and time are raised to represent that quantity.

Given that 10(at 3) is a quantity, we need to determine its dimensions. Here, 'at' denotes acceleration due to gravity, which has dimensions of length per unit time squared.

So, we can write 10(at 3) as 10 multiplied by (length/time^2)^3.

Breaking it down further, we get:

10 * (length^3 / time^6)

Now, we can compare the dimensions of this quantity with the given options:

a) M0L0T0 - This option implies that the quantity has no dimensions, which is not true.

b) M0L0T1 - This option includes a time dimension of 1, which is not sufficient to represent the given quantity.

c) M0L0T-1 - This option includes a time dimension of -1, which is correct as it cancels out the time dimension in the quantity.

d) None of these - This option is not correct as option c) is the correct answer.

Therefore, the dimension of 10(at 3) is M0L0T-1.