Question

If \( \mathrm{Re}(1+iy)^3 = -26 \), where \(y\) is a real number, then the value of \( |y| \) is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(6\)
• E. \(9\)

Hint:

When dealing with powers of complex numbers, expand step-by-step and separate real and imaginary parts clearly.

Answer 1

The Correct Option is B

Concept: To find the real part of a complex expression, expand it and collect real terms. Use: \[ (a+ib)^2 = a^2 - b^2 + 2iab \]

Step 1: Expand the expression
\[ (1+iy)^2 = 1 - y^2 + 2iy \] Now multiply by \( (1+iy) \): \[ (1+iy)^3 = (1 - y^2 + 2iy)(1+iy) \]

Step 2: Multiply carefully
\[ = (1 - y^2)(1+iy) + 2iy(1+iy) \] \[ = (1 - y^2) + i y(1 - y^2) + 2iy + 2i^2 y^2 \] \[ = (1 - y^2) + iy(1 - y^2 + 2) - 2y^2 \] \[ = (1 - y^2 - 2y^2) + i y(3 - y^2) \] \[ = (1 - 3y^2) + i y(3 - y^2) \]

Step 3: Extract real part
\[ \mathrm{Re}(1+iy)^3 = 1 - 3y^2 \]

Step 4: Use given condition
\[ 1 - 3y^2 = -26 \] \[ 3y^2 = 27 \] \[ y^2 = 9 \]

Step 5: Final value
\[ |y| = 3 \] \[ \boxed{3} \]