Complex numbers practice · Medium

Option C (interior and boundary of an ellipse) is correct because the given inequality defines the set of points whose sum of distances from two fixed points is less than or equal to a constant, which is the definition of an ellipse's interior and boundary.

Option C ($x^2+y^2$) is correct because multiplying the given equation by its conjugate results in the product of magnitudes, which equals $x^2 + y^2$.

Option B ($\left| {z - 3i} \right| = 6,\;\mathrm{Im\;}\left( z \right) > 0$) is correct because the locus represents a circle with center (0, 3) and radius 6, but only the part above the x-axis is considered due to the argument condition.

Option C ($1+i \cot (\pi / 2 n)$) is correct because it represents the sum of the geometric series with complex terms, simplified using trigonometric identities.

Option A ($\frac{{12}}{5} + \frac{{16}}{5}i$) is correct because it satisfies both the positive argument condition and the inequality |2 - 3i| < |z|.

Option B (${2^{2n}}$) is correct because each pair of factors simplifies to 4, and there are n such pairs.

Option C ($1+i \cot (\pi / 2 n)$) is correct because after simplifying the geometric series and using trigonometric identities, the expression equals $1+i \cot (\pi / 2 n)$.

Option B is correct because the given equation represents a circle in the complex plane, and a circle has an infinite number of points.

Option C is correct. Given the modulus of each complex number is 1, the expression $\left| {{z_1} + {z_2} + ... + {z_n}} \right|$ is equal to $\left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + ... + \frac{1}{{{z_n}}}} \right|$ because the conjugate of each $z_i$ is its reciprocal.

Option A ($\frac{{3 - 5i}}{2}$) is correct because it is one of the roots of the quadratic equation and has a greater modulus than the other root.

Option A (0) is correct because the given condition leads to a contradiction, implying there are no such complex numbers z.

Option B ($\frac{1}{2}$) is correct because by recognizing the geometric series and using the properties of complex numbers, the real part of z can be found to have an absolute value of 1/2.

Option B (1) is correct because by equating the real parts of the given equation involving complex numbers, we directly obtain the desired expression.

Option D ($\frac{1}{2}$) is correct because solving the equation $e^{\log 2^{\frac{|z|^2-|z|+4}{|z|^2+1}}}=\log _{\sqrt{2}} (16)$ leads to $|z| = \frac{1}{2}$.

Option B ($- \frac{{2n}}{{1 - \alpha }}$) is correct because it is derived by multiplying the original series by alpha, subtracting, and using the properties of the nth root of unity.

The correct answer is C ($\frac{{4m\pi }}{{n\left( {n + 1} \right)}}$) because using De Moivre's theorem and equating to 1 leads to this solution for $\theta$.

Option C (${\alpha ^3} + {\beta ^3}$) is correct because substituting the given expressions for $x, y, z$ and using the properties of cube roots of unity simplifies the product $xyz$ to $\alpha^3 + \beta^3$.

Option C ($\frac{{7^n-1}}{{6}}$) is correct because by using the factorization of $x^n - 1$ and substituting $x=7$, we can find the value of the given expression.

Option C (a line) is correct because the condition |w| = 1 leads to an equation that represents a straight line in the complex plane.

Option C (87) is correct because the maximum value of the expression is obtained by minimizing the negative terms, which leads to the result 87.

Option B (Re($\frac{{Z-\alpha}}{{\beta-\gamma}}$)=0) is correct because the perpendicularity condition implies that the argument of the complex number (Z-α)/(β-γ) is π/2, making its real part zero.

Option B (whose centre is (0,-1) and radius =3) is correct because it correctly identifies the center and radius of the circle on which the transformed complex number lies.

Option C (Both are correct) is correct because both statements correctly calculate the roots of the given quadratic equations using the quadratic formula.

Option C $\frac{{4m\pi }}{{n\left( {n + 1} \right)}}$ is correct because it is derived by using De Moivre's theorem and equating the imaginary part to zero to find the general solution for $\theta$.

Option B (1) is correct because by substituting $p = \cot \theta$ and simplifying using trigonometric identities and Euler's formula, the expression equals 1.

Option A (1) is correct because the magnitude of a complex number divided by its conjugate is always 1.

Option B (13) is correct because solving the equation cos(25θ) = -1 within the given interval (0, π] yields 13 possible values for θ.

Option B ($6,0$) is correct because these are the greatest and least values of $|z+1|$ based on the given inequality and geometric interpretation.

Option B (1) is correct because the expression simplifies to 1 using Euler's formula and trigonometric identities.

Option B ($a = 2 - \sqrt 3 ,\;b = 2 - \sqrt 3$) is correct because by using the properties of equilateral triangles and complex numbers, we can derive the values of a and b.