Correct Answer  Option 2 : MLT
^{3} and MLT
^{4 }
CONCEPT:
Principle of homogeneity of dimensions:

According to this principle, a physical equation will be dimensionally correct if the dimensions of all the terms occurring on both sides of the equation are the same.
 This principle is based on the fact that only the physical quantities of the same kind can be added, subtracted, or compared.
 Thus, velocity can be added to velocity but not to force.
EXPLANATION:
 The dimension of the force is given as,
⇒ [F] = [M L T^{2}] (1)
Let the dimensional formula of a and b is given as,
⇒ [a] = [M^{x} L^{y} T^{z}]
⇒ [b] = [M^{p} L^{q} T^{r}]
The dimensional formula of at and bt^{2} should be equal to the dimensional formula of F.
⇒ [a][t] = [F]
⇒ [M^{x} L^{y} T^{z}][T] = [MLT^{2}]
⇒ [M^{x} L^{y} T^{z+1}] = [MLT^{2}] (2)
By eqauation 2,
⇒ x = 1
⇒ y = 1
⇒ z +1 = 2
⇒ z = 3
Similarly,
⇒ [b][t^{2}] = [F]
⇒ [M^{p} L^{q} T^{r}][T^{2}] = [MLT^{2}]
⇒ [M^{p} L^{q} T^{r+2}] = [MLT^{2}] (3)
By equation 3,
⇒ p = 1
⇒ q = 1
⇒ r + 2 = 2
⇒ r = 4
⇒ [a] = [M^{x} L^{y} T^{z}] = [M L T^{3}]
⇒ [b] = [M^{p} L^{q} T^{r}] = [M L T^{4}]
 Hence, option 2 is correct.