Complex numbers practice · Easy

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

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 (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 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.

Option C is correct because A Δ B represents the symmetric difference, which includes elements in B but not in A.

Option D (12) is correct because the sum can be simplified using the properties of complex roots of unity.

Option D (1) is correct because by using the change of base formula for logarithms and simplifying the sum, the result is 1.

Option D ($\frac{3}{4}$) is correct because if $a = \frac{3}{4}$, then z would be real, contradicting the condition that the imaginary part of z is non-zero.

Option B (9) is correct because, after simplifying using the properties of cube roots of unity, the expression evaluates to 9.

Option C (1) is correct because the expression simplifies to 1 using the property that if $|z|=1$, then $\bar{z} = \frac{1}{z}$.

The correct answer is D (${\sin ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)\,$) because the real part of the complex number must be zero for it to be purely imaginary.

Option A ($x = 4n$, where $n$ is any positive integer) is correct because it represents the general solution for x that satisfies the equation, derived by simplifying the complex fraction and using the properties of imaginary units.

Option C ($-\frac{1}{2}(\sqrt{3}-i)$) is correct because after simplifying the given expression using trigonometric identities and De Moivre's theorem, the result matches this option.

Option B ($\cos \frac{\pi }{8} + i\sin \frac{\pi }{8}$) is correct because it is derived from solving the equation $i{z^4} + 1 = 0$ using De-Moivre’s theorem.

Option C (25) is correct because it uses the logarithmic property to simplify the expression to 5 squared, which equals 25.

Option D ($1 / 2-\sqrt{3}$) is the correct answer. By equating the given complex numbers and solving for 'a', we arrive at the solution after rationalizing the denominator.

Option B ($z$ is purely real) is correct because the conjugate of the determinant is equal to the determinant itself, implying that $z$ is a real number.

Option B is correct because by cubing the given expression and equating real and imaginary parts, we can find the value of k.

Option B (0) is correct because substituting $\alpha = \omega$ and using the properties of cube roots of unity ($\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$) simplifies the expression to 0.

Option B (-z) is correct because simplifying the determinant using properties of $\omega$ leads to $k = -z$.