Complex numbers practice · Easy

Option C ($-2^{62}$) is correct because by expressing the complex numbers in terms of $\omega$ and $\omega^2$ (cube roots of unity), the expression simplifies to $-2^{62}$.

Option C (1) is correct because the expression simplifies to 1 using the property that |β| = 1 and |z| = |conjugate(z)|.

Option D (None of these) is correct because the correct value of the sum is (n * 3^(n-1))/(3^n - 1) - 1/2, which is not among the given options.

Option D (collinear) is correct because the given relation $a z_1+b z_2+c z_3=0$ with $a+b+c=0$ implies that the points $z_1, z_2, z_3$ are collinear.

The correct answer is A (12) because when n=12, the given expression simplifies to 1 using De Moivre's theorem.

Option A (0) is correct because by substituting $w=u+iv$ and using $|z|=1$, we find that the real part of w, Re(w), is 0.

Option C ($\frac{{-1}}{{i+1}}$) is correct because the conjugate of a complex number is obtained by changing the sign of the imaginary part.

Option D ($z_1^3 + z_2^3 + z_3^3 + 3\;{z_1}{z_2}{z_3} = 0$) is not correct because the correct relation for an equilateral triangle in the Argand plane is $z_1^2 + z_2^2 + z_3^2 = {z_1}{z_2} + {z_2}{z_3} + {z_3}{z_1}$.

Option C (2) is correct because the minimum value is found using the triangle inequality property of complex numbers.

Option A, 2, is correct because it uses the triangle inequality to find the minimum distance between two complex numbers given their individual magnitudes and the magnitude of their difference with another complex number.

Option B (z is purely imaginary) is correct because if z = -conjugate(z), then the real part of z must be zero.

Option B ($x^2+y^2=1$) is correct because simplifying the given equation leads to this equation of a circle centered at the origin with radius 1.

Option D (0) is correct because the given condition implies that point A lies on the line segment BC, hence the distance of A from the line BC is zero.

Option B ($a=\frac{1}{2}\left(e+\frac{1}{e}\right) \cos 1, b=\frac{1}{2}\left(e-\frac{1}{e}\right) \sin 1$) is correct because it uses the correct hyperbolic trigonometric identities to expand $\cos(1-i)$ and separate the real and imaginary parts.

Option C ($\sqrt{{5}}+1$) is correct because it is the maximum value of $|z|$ that satisfies the given inequality.

Option A ($\sqrt{5}$) is correct because the minimum value is found using the triangle inequality property of complex numbers.

Option C (a+|a|) is correct because the sum of the squares of the fourth roots of a negative number is 0, which is equivalent to a + |a| when a < 0.

Option C (${\alpha ^2} + {\beta ^2} = 1$) is correct because by taking the modulus of both sides of the given equation and simplifying, we find that ${\alpha ^2} + {\beta ^2} = 1$.

Option A ($a = 1 + i$) is correct because solving $\sqrt{2i}$ yields $\pm(1+i)$, and $a = 1 + i$ is one of the possible solutions.

Option D $\frac{4}{{1 + i}}$ is correct because the conjugate of a complex number involves changing the sign of the imaginary part.

Option D ($\left|z_1\right| < \left|z_3\right| < \left|z_2\right|$) is correct because $z_3$ is the midpoint of $z_1$ and $z_2$, and $x_1 < x_2$ which implies $|z_1| < |z_3| < |z_2|$.

Option D (A hyperbola) is correct because the equation Im(${z^2}$) = k, where k is a non-zero real number, simplifies to xy = k/2, which represents a hyperbola.

Option A (244) is correct because expanding z⁵ and (z_conjugate)⁵ and adding them results in 244.

Option D ($(0,3)$) is correct because the given condition represents an ellipse only if the distance between the foci is less than the length of the major axis.

Option C ($\pm 3$) is correct because it represents the real part of the square root of the complex number -7 + 24i, calculated using the appropriate formula.

Option D ($\frac{1}{{4\sin {2^0}}}$) is correct because the summation of the imaginary parts of the complex number series simplifies to this expression using trigonometric identities.

The correct answer is A ($x = 4,\;9$) because these are the only values that satisfy the given absolute value equation.

Option B (1,1) is correct because $(1+\omega)^{2017}$ simplifies to $-\omega^2$, which can be rewritten as $1 + \omega$, thus A=1 and B=1.

Option C ($18\sin \left( {\alpha + \beta + \gamma } \right)$) is correct because it is derived from complex number manipulation and trigonometric identities given the initial conditions.

Option A (1) is correct because using the given conditions, we can simplify the expression to |z_2|^2 which is equal to 1.