A suitable financial calculator can give you the answer more or less directly. Here, we find that the answer choices presented are the result of improper calculation. The multiplier was rounded to 5 decimal digits before it was used to compute the amount. As a consequence the numbers shown in your problem statement are incorrect. The only number that should be rounded is the final answer.
1. The correct amount is $14,141.98. The best of the available choices is
... $14,142.03
2. The correct amount is $10,754.65. The best of the available choices is
... $10,754.61
_____
In each case, the multiplier is (1 + r/n)^(nt), where r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. Since you want results good to 7 significant digits, your value of multiplier must be good to at least 8 significant digits.
For problem 1: (1 + .06/4)^(4·5) = 1.015^20 ≈ 1.3468550
For problem 2: (1 + .12/12)^(12*2) = 1.01^24 ≈ 1.2697346
The final account balance is the initial balance multiplied by the corresponding multiplier value.
