Complex Numbers Flashcards

If z = (3 + 4i)/(1 - 2i), what is the imaginary part of z?

Answer: 2

Multiply numerator and denominator by the conjugate of the denominator: (3 + 4i)(1 + 2i) / ((1 - 2i)(1 + 2i)). Numerator: 3 + 6i + 4i + 8i² = 3 + 10i - 8 = -5 + 10i. Denominator: 1 + 4 = 5. So z = (-5 + 10i)/5 = -1 + 2i. The imaginary part is 2.

For which value of k does the equation x² + kx + (k + 3) = 0 have exactly one real solution (a repeated root)?

Answer: k = 2 or k = 6

A quadratic has exactly one real (repeated) root when the discriminant equals zero: b² - 4ac = 0. Here a=1, b=k, c=k+3, so k² - 4(k+3) = 0 → k² - 4k - 12 = 0 → (k-6)(k+2) = 0 → k = 6 or k = -2.

What is the modulus of the complex number z = (1 + i)⁸?

Answer: 16

The modulus of (1 + i) is √(1² + 1²) = √2. Using the property |z^n| = |z|^n, we get |(1 + i)⁸| = (√2)⁸ = 2⁴ = 16.

If z is a complex number such that z + 1/z = √3, what is z³ + 1/z³?

Answer: 0

Let s = z + 1/z = √3. Then z² + 1/z² = s² - 2 = 3 - 2 = 1. Then z³ + 1/z³ = (z + 1/z)(z² - 1 + 1/z²) = √3 · (1 - 1) = √3 · 0 = 0.

In the complex plane, the complex number w = -√3 + i has an argument (principal angle) of:

Answer: 5π/6

The point -√3 + i is in the second quadrant (negative real part, positive imaginary part). The reference angle is arctan(1/√3) = π/6. Since it's in the second quadrant, the principal argument is π - π/6 = 5π/6.

If the polynomial p(x) = x⁴ - 2x³ + 6x² - 2x + 5 has (x - i) as a factor, which of the following must also be a factor?

Answer: (x² + 1)

For a polynomial with real coefficients, complex roots always come in conjugate pairs. If (x - i) is a factor, then (x + i) must also be a factor. Together, (x - i)(x + i) = x² + 1 must be a factor. Therefore (x² + 1) is a factor.