Complex numbers practice · Mixed difficulty

Option C, $\frac{9^n-1}{8}$, is correct because it is derived by substituting $x=9$ into the factored form of $x^n - 1$ divided by $x-1$, where the roots of unity are used.

Option A (-$\frac{{1}}{{5}}$) is correct because the real part of the simplified expression is -$\frac{{1}}{{5}}$.

Option D ($x - y + 1 = 0$) is correct because it represents the locus of z, derived from the given amplitude condition by equating the tangent of the angle to 1.

Option A (91) is correct because expanding $\left(-2-\frac{1}{3} i\right)^{3}$, equating it to $\frac{x+i y}{27}$, and then calculating $y-x$ yields 91.

Option B ($\operatorname{Im}(z)=0$) is correct because the imaginary part of the complex number $z$ is zero, making it a real number.

Option D (14) is correct because it represents the maximum possible value of the sum of the magnitudes of the complex numbers, which occurs when each complex number lies on the circle's circumference farthest from the origin.

Option C (-120) is correct because cubing (2+3i) gives x+iy = -46 + 9i, and then 3x + 2y = -120.

Option B (0) is correct because the sum of the fifth powers of the 100th roots of unity is zero.

Option A (0) is correct because after simplification, the expression equals 0.

Option D (-$\frac{1}{2}(\sqrt{3}-i)$) is correct because it is the simplified form of the given complex expression after applying Euler's formula.

Option D is correct because both statements are correct. Statement I defines the polar form, and Statement II describes the argument of a quotient of complex numbers.

Option C (parabola) is correct because the given equation represents a parabola when converted to Cartesian coordinates.

Option B (${x^2} - 2x\cos n\emptyset + 1 = 0$) is correct because the roots ${\alpha ^n}$ and ${\beta ^n}$ satisfy this equation, derived using De Moivre's theorem.

Option D (${z}=\frac{\sqrt{{3}}}{{3}}+{{i}}$) is correct because it is the solution obtained by substituting $z = x + iy$ into the given equation and solving for x and y.

The correct answer is D (lies in the interval (1,2)) because the minimum value of $|z + (1/2)|$ occurs when $z$ is on the circle $|z| = 2$ and closest to $-1/2$, resulting in a value of 3/2, which lies in the interval (1,2).

Option D ($a^{2}=3 b$) is correct because it is the relationship derived from the equilateral triangle condition and Vieta's formulas.

Option B ($\frac{2 b}{(1+a)^2+b^2}$) is correct because rationalizing the denominator and equating real and imaginary parts allows us to express x in terms of a and b.

Option B (8) is correct because it represents the maximum possible value of |z| derived from the given equation by using the triangle inequality.

Option C (${225^0}$) is correct because the argument of the simplified complex number -1-i lies in the third quadrant.

Option B is correct because by substituting $z = \omega$ or $z = \omega^2$, the expression simplifies to 12, where $\omega$ is a complex cube root of unity.

Option D (purely imaginary) is correct because the expression simplifies to a purely imaginary number using Euler's formula and trigonometric identities.

Option B ($24 / 5$) is correct because the least value of the expression is found by using the triangle inequality and substituting the minimum value of |z|.

Option C ($\frac{\pi}{4}$) is correct because the principal argument of the limit of the product of complex numbers $Z_n$ is $\frac{\pi}{4}$.

Option B ([5,19]) is correct because by using the triangle inequality and the given conditions, we can find the range of values for $|z_1 - z_2|$.

Option D ($- \left( {4/5} \right)$) is correct because it is the imaginary part of the simplified complex number.

Option A ($2i$) is correct because by squaring the first equation and using the second equation, the value of the expression can be found.

Option C is correct because simplifying the expression using properties of complex numbers, specifically omega ($\omega$), leads to the result $i \sqrt{3}$.

Option A ($4(a^2-b^2)$) is correct because it results from equating real and imaginary parts of the complex equation and simplifying the expression.

Option A (-2cosα\leftcos ( π + α ) + isin ( π + α ) $-2\cos\mathrm{\alpha}\left{\cos \left( {\pi + \alpha } \right) + i\sin \left( {\pi + \alpha } \right)} \right}$) is correct because it correctly represents the modulus and argument of the given complex number in the specified range for $\alpha$.

Option B ($3 z_0^2$) is correct because in an equilateral triangle, the circumcenter coincides with the centroid, and using the properties of equilateral triangles, $z_1^2+z_2^2+z_3^2 = 3z_0^2$.