Complex numbers practice · Mixed difficulty

Option D (None of these) is correct because the value of $\arg(z) + \arg(\bar{z})$ depends on whether z is a negative real number or not. If z is a negative real number, the sum is $2\pi$, otherwise it is 0.

Option D ($-\frac{2}{25}-\frac{11}{25} i$) is correct because simplifying the expression and then finding the conjugate yields this result.

The value of $\theta$ is $\frac{{2 \pi}}{{3}}$ because the complex number lies in the second quadrant and the argument is calculated accordingly.

Option C ($i$) is correct because it is one of the values obtained by converting the complex number to polar form and applying De Moivre's theorem.

Option B ($0 \leq a \leq 4$) is correct because it correctly determines the range of possible values for 'a' based on the given conditions and properties of complex numbers.

Option C ($18\sin \left( {\alpha + \beta + \gamma } \right)$) is correct because it is derived from the complex number manipulation and equating imaginary parts.

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 B ($2 \pm 2 \sqrt{2} i$) is correct because by substituting z = x + iy into the given equation and solving for x and y, we find that x = 2 and y = $\pm 2\sqrt{2}$.

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.

Option B ($\left( {1/3,\;5} \right)$) is correct because it represents the solution set for the given inequality after considering different cases based on the absolute value expressions.

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

The correct answer is None of the above because the magnitude of z is $\sqrt{\frac{{17}}{2}}$, and z is neither purely real nor purely imaginary.

Option A ($|\operatorname{Re}(z)| \leq 2$) is correct because $z$ can be expressed as $2\operatorname{Re}(\omega)$, and since $|\omega|=1$, the real part of $z$ is bounded by 2.

Option C ($- 2 < x < - 1$) is correct because it represents the interval where the inequality holds true after simplifying and analyzing the sign changes.

Option B ($3 \frac{\sqrt{26}}{7}$) is correct because it represents the perpendicular distance calculated using vector cross products and area of the triangle.

Option D (n1 > 0, n2 > 0) is correct because the expression simplifies to a real number only if n1 and n2 are positive integers.

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 simplifying the expression inside the modulus leads to $|2i| = 2$.

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 ($-1365 \sqrt{3} i$) is correct because by finding the value of 'a' such that $z^3$ is real and then summing the geometric series, we get $-1365 \sqrt{3} i$.

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.