Complex numbers practice · Mixed difficulty

Option B ([3,13]) is correct because it represents the range of distances from a point on the ellipse to the point (7,0).

Option A ($\frac{{3\sqrt 3 }}{4}$) is correct because solving the equation $\overline z = i{z^2}$ gives the vertices of a polygon, and calculating the area of the polygon results in $\frac{{3\sqrt 3 }}{4}$.

The correct answer is D (None of these) because the smallest positive integer n for which the given expression equals 1 is 4, not any of the provided options.

Option B (72) is correct because simplifying the given expression and equating the powers leads to m=48 and n=24, thus m+n=72.

Option D (3) is correct because the expression simplifies to 3 using the change of base formula for logarithms.

Option C ($\sqrt{2}\left[\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right]$) is correct because it represents the polar form of the given complex number after simplification.

Option B (-tan^(-1)(sqrt(2)/3)) is correct because after simplifying x, y, and z, the argument of the resulting complex number is -tan^(-1)(sqrt(2)/3).

Option B ($\sqrt 3 + 1$) is correct because it represents the maximum possible value of the modulus of z that satisfies the given condition.

Option C ($\frac{\pi}{2}$) is correct because the complex number z + iz is obtained by rotating z by 90 degrees counterclockwise in the Argand plane, making the angle OPQ a right angle.

Option B (1) is correct because if $\frac{2z_1}{3z_2}$ is purely imaginary, then by substituting $\frac{z_1}{z_2} = ki$ and simplifying the expression $\left| \frac{z_1 - z_2}{z_1 + z_2} \right|^4$, we get 1.

Option B ($\pm 2 \sqrt{2}$) is correct because by equating the magnitudes and using the given condition x=ky and y being a natural number, we find that k can be $\pm 2 \sqrt{2}$.

Option B (${\log _b}a = {\log _c}b$) is correct because given that x, y, z are in GP and ${a^x} = {b^y} = {c^z}$, by taking logarithms and using the properties of logarithms, we can derive the relationship ${\log _b}a = {\log _c}b$.

Option A (0) is correct because the argument of the absolute value of any complex number is always 0, and the greatest integer function of 0 is 0.

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 D ( - 1) is correct because the infinite product of the given complex numbers simplifies to -1 using trigonometric identities.

Option D (0) is correct because using the property 1 + ω = -ω² and 1 + ω² = -ω, the expression simplifies to 0.

Option D (32) is correct because solving the logarithmic equation leads to $x = 32$.

Option C ($2 \cos \left(10 \cos ^{-1} \frac{2}{3}\right)$) is correct because by converting to polar form and using De Moivre's theorem, the expression simplifies to this result.

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 A (2) is correct because the roots of the equation are $\pm i \sqrt{2}$, therefore, n = 2.

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

Option C (${x^2} + x + 1 = 0$) is correct because the new roots ${\alpha ^{31}}$ and ${\alpha ^{62}}$ satisfy the same quadratic equation as $\alpha$ and ${\alpha ^2}$.

Option B ($\frac{{2\;ab}}{{{{a^2} - {b^2}}}}$) is correct because it represents the simplified form of the given expression after applying logarithmic and trigonometric identities.

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 A (1) is correct because when the given complex expression is simplified and the modulus is calculated, the result is 1.

The value of 1/λ is 3/2. This is found by using the properties of equilateral triangles and the relationship between the roots and coefficients of the quadratic equation.

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