Question 7

The complete conjugate of z is 5 - 3i.

The complete conjugate of a complex number is obtained by changing the sign of the imaginary part. Therefore, for the complex number z = 5 + 3i, the complete conjugate is 5 - 3i.

A) -5 - 3i

This option is incorrect because it not only changes the sign of the imaginary part but also alters the real part of the complex number. The correct form of the conjugate retains the real part as is.

B) -5 + 3i

This option is also incorrect as it incorrectly changes the sign of the real part while keeping the imaginary part's sign positive. The complete conjugate should only change the sign of the imaginary component.

C) -3 + 5i

This option is incorrect because it completely alters both the real and imaginary parts of the original complex number, which does not align with the definition of a conjugate.

D) 3 + 5i

This option is incorrect as it changes the sign of the real part while incorrectly keeping the imaginary part positive. The complete conjugate should keep the real part unchanged.

E) 5 - 3i

This is the correct option as it accurately reflects the definition of the complete conjugate, where only the sign of the imaginary part is inverted while the real part remains the same.

Conclusion

The correct answer, 5 - 3i, effectively represents the complete conjugate of the complex number z = 5 + 3i, adhering strictly to the rule of sign inversion for the imaginary part. All other options fail to meet the criteria for a conjugate, either changing the real part or misrepresenting the imaginary part entirely.