Complex Numbers and Quadratic Equations

If \( z = 3 + 4i \), what is the modulus of \( z \)?

The roots of the quadratic equation \( x^2 - 2x + 5 = 0 \) are:

If \( z = 1 + i \), what is \( z \times \overline{z} \) where \( \overline{z} \) is the conjugate of \( z \)?

If \( \alpha \) and \( \beta \) are roots of \( x^2 - 4x + 8 = 0 \), then \( \alpha^2 + \beta^2 \) equals:

Find the principal argument (in radians) of the complex number \( -1 - i \).

If \( z = \cos 60^\circ + i \sin 60^\circ \), compute \( z^{3} \).

The quadratic equation whose roots are \( \frac{1}{3} \) and \( -\frac{2}{5} \) is:

If \( z_1 = 2 + 3i \) and \( z_2 = 1 - 4i \), find \( \frac{z_1}{z_2} \).

If the quadratic equation \( x^2 + px + q = 0 \) has roots \( 2 + \sqrt{3}i \) and \( 2 - \sqrt{3}i \), find the values of \( p \) and \( q \).

Find the value of \( k \) such that the equation \( x^2 + 2(k - 1)x + k + 3 = 0 \) has equal roots.