Complex numbers practice · Easy

Option C (0) is correct because the sum of any four consecutive powers of i (the imaginary unit) is always zero.

Option D ($\frac{1}{2}$) is correct because, after simplifying the expression using properties of complex cube roots of unity, the sine of the resulting angle is $\frac{1}{2}$.

Option B ($- \frac{{3\pi }}{{4}}$) is correct because after simplification and rationalization, the argument of the complex number is found to be $- \frac{{3\pi }}{{4}}$.

Option D (0) is correct because if $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then $z_1$ and $z_2$ must have the same argument.

Option C ($\frac{1}{4}(1 \pm \sqrt{3} i)$) is correct because it satisfies the condition for the points to form an equilateral triangle.

Option C ($x^2+y^2+z^2-y z-z x-x y$) is the correct answer because it results from expanding and simplifying the given expression using the properties of $\omega$.

Option B (4) is correct because by substituting $z = x + iy$ into the equation and solving the resulting system of equations, we find four distinct solutions.

Option D ($p^2-3 q$) is correct because, given $\alpha, \beta$ are roots of $x^2+p x+q=0$, and $\omega$ is a cube root of unity, the expression $(\omega \alpha+\omega^2 \beta)(\omega^2 \alpha+\omega \beta)$ simplifies to $p^2 - 3q$ using Vieta's formulas.

Option A ($- {\left( { - 2} \right)^n}$) is correct because the expression simplifies to $-(-2)^n$ using properties of complex numbers and the given condition on n.

Option D (1) is correct because the sum of the cube roots of unity is 0, and the product simplifies to 1, resulting in 0 - 1 = -1. However, the question asks for z = sum - product, so z = 0 - (-1) = 1.

Option D ($\{z:|z|=1, z \neq 1\}$) is correct because the condition leads to $|z|=1$ and $z \neq 1$ due to the purely real constraint.

Option B (1 and 0) is correct because the given complex number simplifies to 1, which has a modulus of 1 and an argument of 0.

Option B ($3 \sqrt{2}$) is correct because the locus of z is a circle, and $|z-7-9i|$ represents the distance from the center of the circle to the point (7, 9).

Option A (6) is correct because solving the given equations for the complex number $\alpha = x + iy$ yields a real part of 6.

Option D (lie in the interval (1,2)) is correct because the minimum value of |z + 1/2| lies between | |z| - 1/2 | and |z| + 1/2, and since |z| >= 2, the minimum value is greater than 3/2 and less than 5/2, thus in the interval (1,2).

Option B is correct. The product simplifies to a closed form expression involving z_0 and its powers, ultimately leading to the given result.

Option D (1) is correct because solving the equations $f(1)=2$ and $f(i^2)=0$ gives $f(x) = x^2 + x$, and solving $f(z+z^{-1})=0$ leads to $|z|=1$.

Option A ($2i$) is correct because simplifying the summation using geometric series properties results in $2i$.

Option C ($a = b = c$) is correct because the quadratic equation must have roots that are also roots of the cubic equation, leading to the conclusion that the quadratic is a scaled version of a factor of the cubic.

Option A ($\frac{{\pi}}{{2}}$) is correct because the argument of the complex number is $\pm \frac{{\pi}}{{2}}$, and we take the positive value.

Option C ($\frac{{1}}{{2}}$) is correct because after simplifying the expression using properties of cube roots of unity, the sine of the resulting angle is 1/2.

Option D (None of these) is correct because after simplifying using the properties of cube roots of unity, the expression evaluates to 4, which is not among the given options.

Option D (2) is correct because when (z - alpha)/(z + alpha) is purely imaginary and |z| = 2, solving for alpha gives alpha = 2.

Option A (0) is correct because the given condition implies that $z_1$ and $z_2$ are orthogonal, leading to the expression simplifying to 0.

Option D ($\left| {{z_k}} \right| = \left| {{z_{k + 1}}} \right|$) is correct because all nth roots of unity lie on the unit circle, hence their magnitudes are equal to 1.

Option D ($e^{-r \sin \theta}$) is correct because it is the result of calculating the magnitude of the complex exponential function.

Option C (The perpendicular bisector of $AB$) is correct because the equation $\left| {z - 3 + i} \right| = \left| {z - 2 - i} \right|$ represents the locus of a point $P$ such that its distance from $A$ and $B$ are equal, which is the perpendicular bisector of $AB$.

The correct answer is D ($\frac{{ - 1 - \sqrt 3 i}}{2}$) because it is one of the complex cube roots of unity.

Option B ($1,1$) is correct because by using the property $1 + \omega = -\omega^2$ and simplifying, we find that $A = 1$ and $B = 1$.

Option A (Circle of radius $\;\frac{{\sqrt 5 }}{2}$) is correct because when $\frac{{z + i}}{{z + 2}}$ is purely imaginary, the locus of z is a circle with the specified radius.