Verified Answer :

  • Evaluate each expression by summing the real and imaginary components separately.
  • Option (a): \(7+1 + (2-1)i = 8 + i\).
  • Option (b): \(9+1 + (2-1)i = 10 + i\).
  • Option (c): \(9-1 + (4-7)i = 8 - 3i\).
  • Option (d): \(7-1 + (4-7)i = 6 - 3i\).
  • The correct expression is option (c), which results in \(\boxed{c}\).

Explanation

  1. Analysis of the problem

To determine which of the provided addition expressions results in the sum \(8 - 3i\), we must perform the addition of complex numbers for each option. The general rule for adding two complex numbers \((a + bi)\) and \((c + di)\) is to add their real parts and their imaginary parts separately: \((a + bi) + (c + di) = (a + c) + (b + d)i\). We will apply this to each option.

  1. Performing the addition for each option

We perform the addition for each option:

A\((7+2i)+(1-i) = (7+1) + (2-1)i = 8 + i\)
B\((9+2i)+(1-i) = (9+1) + (2-1)i = 10 + i\)
C\((9+4i)+(-1-7i) = (9-1) + (4-7)i = 8 - 3i\)
D\((7+4i)+(-1-7i) = (7-1) + (4-7)i = 6 - 3i\)
  1. Conclusion

Comparing the results obtained in the previous step to the target sum \(8 - 3i\):

AResults in \(8 + i eq 8 - 3i\)
BResults in \(10 + i eq 8 - 3i\)
CResults in \(8 - 3i\), which matches the target sum.
DResults in \(6 - 3i eq 8 - 3i\)

Thus, option (c) is the correct expression. The final answer is \(\boxed{c}\).

Examples

In electrical engineering, complex numbers are frequently used to represent impedance in AC circuits. When adding impedances in series, you sum the real parts (resistance) and imaginary parts (reactance) in exactly the same way as shown in this problem. For example, if one component has an impedance of \((9 + 4i)\) Ohms and another has \((-1 - 7i)\) Ohms, their series combination is calculated by adding these values, helping engineers design circuits that behave as expected at different frequencies.

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