Why is it necessary to check that n modifyingabove p with caret greater than or equals 5np≥5 and n modifyingabove p with caret greater than or equals 5nq≥5 ?

It is important to check both conditions

np ≥ 5

nq = n (1 - p) ≥ 5

when you try to approximate a binomial distribution with a normal distribution.

If these two conditions are verified, it means that the sample is "big enough" to use the approximation, otherwise, the normal distribution does not fit well enough the binomial distribution and the values of mean, etc, could be different.

Therefore if those conditions are not verified, the normal distribution is not suitable to represent the binomial distribution.