Question 35

b = (-3 + √5)/2

When the complex number (√5+3i)/(1+i) is expressed in the form a + bi, the value of b is found to be (-3 + √5)/2.

A) -1.5

This option is incorrect because it suggests a specific numerical value for b that does not match the result obtained from simplifying the expression. The value of b must derive from the correct simplification of the complex number, which does not yield -1.5.

B) 2

This option is incorrect as well; while it represents a potential value for b, it does not correspond to the outcome of the expression (√5 + 3i) / (1 + i). The calculation of b must align with the final result after performing complex division.

C) (3-√5)/2

This option is incorrect because, although it is a fraction, it does not correctly represent the imaginary part of the expression after simplification. The calculations must reflect the proper operations on the complex number to yield the accurate value for b.

D) (- 3 + √(5))/2

This option is correct as it accurately reflects the value of b after simplifying the expression (√5 + 3i) / (1 + i). The calculation involves multiplying by the conjugate to eliminate the imaginary part in the denominator, leading to this result.

E) (- 3 - √(5))/2

This option is incorrect since it suggests a different value for b that does not arise from the simplification process. The correct imaginary part of the expression does not yield this negative result.

Conclusion

The correct answer is definitively D) (-3 + √5)/2 because it accurately represents the imaginary part of the complex number after proper simplification. All other options fail to reflect the correct calculations derived from the original expression, highlighting the importance of careful manipulation in complex numbers.