Complex numbers practice · Mixed difficulty

Option B ($z$ lies on real axis) is correct because the condition $|w| = 1$ implies that the imaginary part of $z$ must be zero, meaning $z$ is real.

Option D (-2) is correct because the expression simplifies to -2.

Option D is correct because $\left|\frac{{z-1}}{{z+1}}\right|=1$ represents a straight line, while the other options represent circles.

Option D (0) is correct because after solving the logarithmic equation, the solution x=2 is extraneous.

Option D (${x^2} + x + 1 = 0$) is correct because the roots of the new equation are ${\alpha ^{19}}$ and ${\beta ^7}$, which simplify to $\omega$ and ${\omega ^2}$, respectively, leading to the original equation.

Option C (4) is correct because it is the smallest positive integer $n$ for which ${\left( {\frac{{1 + i}}{{1 - i}}} \right)^n} = 1$.

Option B ($2 \cos (\alpha+\beta+\gamma+\delta)$) is correct because by applying De Moivre's theorem and simplifying, the expression equals $2 \cos (\alpha+\beta+\gamma+\delta)$.

Option B ($\sqrt{5}$) is correct because it is derived from the parallelogram law relating the sum and difference of two complex numbers to their magnitudes.

Option C (a hyperbola) is correct because the locus of a point where the difference of its distances from two fixed points is constant defines a hyperbola.

Option D ($\frac{1}{{5 + 3\cos \theta }}$) is correct because it is the real part of the given complex expression after rationalizing the denominator.

Option A (are vertices of an equilateral triangle) is correct because the complex cube roots of unity, $1, \omega, \omega^2$, are equally spaced on the unit circle in the complex plane, forming an equilateral triangle.

Option A (10) is correct because by substituting z = x + iy into the given equation and solving for x and y, we find that $z\bar{z} = x^2 + y^2 = 10$.

Option A ($-1$) is correct because it is the simplified form of the given expression using properties of complex cube roots of unity.

Option D ($-128\omega^2$) is correct because the expression simplifies to $(-2\omega^2)^7$, which equals $-128\omega^{14}$ and further simplifies to $-128\omega^2$.

Option D (None of these) is correct because the given conditions lead to contradictory requirements for x and y, indicating no solution exists for z.

The correct answer is B (12) because the value of $3(\alpha+\beta+\gamma)$ is calculated correctly based on the center and radius of the given circle.

Option B (ellipse) is correct because the set of points can be parameterized as $x = \frac{3}{2} \cos \theta$ and $y = \frac{5}{2} \sin \theta$, which represents an ellipse.

Option A, 7, is correct because it uses the properties of complex numbers and the triangle inequality to find the maximum value of the given expression.

Option B (An equilateral triangle) is correct because if the magnitudes of three complex numbers are equal and their sum is zero, they form the vertices of an equilateral triangle.

Option A (-88) is correct because substituting x = 4 + 3i into the expression and simplifying using polynomial division results in -88.

The correct answer is B ($\mathrm{IM\;}\left( z \right) = 0$) because the imaginary part of the simplified complex number $z$ is zero, indicating that $z$ is a real number.

Option D (6) is correct because the number of common roots is equal to the greatest common divisor (GCD) of 12 and 18, which is 6.

Option A ($f(z_1, z_2, z_3)=f(z_1+a, z_2+a, z_3+a)$) is correct because adding a constant to each variable in the determinant does not change its value due to properties of determinants.

Option A (2) is correct because the minimum value is obtained when any two are zero and the third is a minimum magnitude integer i.e. 1.

Option C (A circle) is correct because the given condition leads to the equation of a circle when $z$ is expressed as $x+iy$.

Option A (1) is correct because there is only one quadratic equation that can be formed with the given roots, regardless of the combination of alpha, beta, and gamma.

Option C ($-\frac{5 \pi}{12}$) is correct because calculating the argument of each complex number and using the properties of arguments leads to the principal argument being $-\frac{5 \pi}{12}$.

Option B ($|z-7-5 i|=\sqrt{2}$) is correct because it represents the locus of the complex number $z$ satisfying the given argument condition.

Option B (a straight line parallel to $y$-axis) is correct because the condition $|z+1|=|z|$ implies that z lies on the perpendicular bisector of the line segment joining -1 and 0 in the complex plane, which is a vertical line.

Option A (An equilateral triangle) is correct because the given condition implies that two sides are equal and the angle between them is 60 degrees.