Complex numbers practice · Mixed difficulty

Option D (a rectangular hyperbola) is correct because the condition Re($z^2$) = 4 translates to $x^2 - y^2 = 4$, which is the equation of a rectangular hyperbola.

Option B ($\frac{13}{2}$) is correct because it's the perpendicular distance calculated using the formula for the distance of a point from a line in the complex plane.

Option C ($-\frac{i}{2}$) is correct because $z^2 = (1+i)^2 = 2i$, and the multiplicative inverse of $2i$ is $\frac{1}{2i} = -\frac{i}{2}$.

Option C ($\frac{{5 \pi}}{{6}}$) is correct because by using the properties of equilateral triangles and complex numbers, we can find the argument of z_0.

Option B is correct because, given $z=a+ib$ where $a > 0, b > 0$, then $|z| \geq \frac{1}{\sqrt{2}}(a+b)$.

Option D (4) is correct because there are four complex numbers that satisfy the given equation: one where z=0 and three others derived from z³ = -i.

Option A (${\mathop\mathrm{Re}\nolimits} \left( {{z^2}} \right) = 0$) is correct because if the real and imaginary parts of z are equal, then the real part of z² is zero.

Option B (-$\frac{1}{2}$) is correct because it is the real part of the sum of complex numbers on the unit circle.

Option D (0) is correct because the expression simplifies to zero using the properties of cube roots of unity.

Option B ($\frac{1}{4}(n - 1)n(n^2 + 3n + 4)$) is correct because it is the result of summing the series after simplifying the general term using properties of complex cube roots of unity.

Option B ($\frac{1}{2}{\left| z \right|^2}\sin \mathrm{\alpha }$) is correct because the area of the triangle can be found using the determinant formula with the given vertices, which simplifies to the expression involving the sine of the angle.

Option A ($\operatorname{Im}(z)=0$) is correct because the relation implies that $z$ is equidistant from $i$ and $-i$ when normalized, thus lying on the real axis.

Option D ($\frac{{a+i b}}{{1+c}}$) is correct because, after substituting $z = \frac{{b+ic}}{{1+a}}$ and simplifying, the expression $\frac{{1+iz}}{{1-iz}}$ reduces to $\frac{{a+ib}}{{1+c}}$.

Option C (-i) is correct because the infinite product simplifies to -i using trigonometric identities and limits.

Option C ($a^2+b^2=1$) is correct because by taking the modulus of both sides of the equation and simplifying, we arrive at the relationship $a^2 + b^2 = 1$.

Option D (None of these) is correct because the order relation (>, <) is not defined for complex numbers.

Option A (12) is correct because the maximum value of $|iz + 6 - 8i|$ is found using the triangle inequality, resulting in $|iz| + |6-8i| = 2 + 10 = 12$.

Option B ($2\sqrt 2 ,\frac{{3\pi }}{4}$) is correct because it represents the magnitude and amplitude of the given complex number after simplification.

Option D ($x=0, y=0$) is correct because after applying the column operation, the determinant becomes 0.

Option B ($z_1 \cdot z_2$ is unimodular) is correct because the given condition implies that $|z_1 z_2| = 1$, which means $z_1 z_2$ is unimodular.

Option B ($\frac{1}{2}(|\alpha+\beta|^2+|\alpha-\beta|^2)$) is correct because expanding the magnitudes and simplifying leads to the given equality.

Option C ($-\sqrt{3}$) is correct because it is found by simplifying the expression inside the tangent function using properties of cube roots of unity.

Option C is correct because the triangle inequality states that the magnitude of the sum of two complex numbers is less than or equal to the sum of their magnitudes.

Option C (-1) is the correct answer because after simplification using trigonometric identities and De Moivre's theorem, the expression evaluates to -1.

Option D (0) is correct because (z + conjugate(z))/(1 + z*conjugate(z)) is a real number, and the argument of any positive real number is 0.

Option C ($-\frac{\pi}{2}$) is correct because if arg(z) < 0, then (z - conjugate(z))/2 is a purely imaginary number with a negative imaginary part, implying an argument of -$\frac{\pi}{2}$.

Option A (${e^{ - r\sin \mathrm{\theta }}}$) is correct because it is derived from Euler's formula and properties of complex exponentials.

Option D (either on the real axis or on a circle passing through the origin) is correct because the imaginary part of the expression being real leads to the equation of a circle passing through the origin or the real axis.

Option C (Four solutions) is correct because the equation ${z^2} = \bar z$ has four solutions: three from ${z^3} = 1$ and one at z = 0.

Option B ($\frac{{1}}{{2}}(i z+1)+i$) is correct because it is the remainder obtained after dividing f(z) by $z^2 + 1$.