Complex numbers practice · Mixed difficulty

Option D ($-{2^7}i$) is correct because, after simplifying z and raising it to the power of 14 using De Moivre's theorem, the result is $-{2^7}i$.

Option C is correct because it accurately determines the least value of n for which the given condition is satisfied, using complex number properties.

Option C (-1) is correct because after substituting $z = \cos \theta \pm i \sin \theta$ into the expression and simplifying, the result is -1.

Option D (None of these) is correct because the only integral solution to the equation is x=0, which is not a non-zero solution.

Option B (4) is correct because, by equating the imaginary parts of the given equation, we directly find that Im(z) = 4.

Option C ($\beta \in(1, \infty)$) is correct because the product of the roots, which is $\beta$, must be greater than 1 due to the roots being distinct and lying on Re(z) = 1.

Option A (1) is correct because substituting the root into the polynomial and equating the real part to zero leads to the given expression being equal to -1, and taking the absolute value gives 1.

Option C ($\theta$) is correct because using the property $\bar{z} = \frac{1}{z}$ when $|z|=1$, we can simplify the expression to $\arg(z) = \theta$.

Option C ($1+c \cos \theta$) is the correct answer because after simplification and using the given equation, the expression reduces to $1+c \cos \theta$.

Option A ($\frac{{3\pi}}{{4}}$) is correct because the given conditions lead to $\arg(z) = \frac{{3\pi}}{{4}}$ after substituting and simplifying the arguments.

Option B (either on the real axis or on a circle not passing through the origin) is correct because the given condition leads to two possibilities: z lies on the real axis or on a circle centered at (1, 0) with radius 1, which does not pass through the origin.

Option B (6) is correct because it is the smallest integer value of n for which both z^n = 1 and (1+z)^n = 1 are satisfied, given the constraints on z.

Option B ($(a^2 + b^2)(|z_1|^2 + |z_2|^2)$) is correct because expanding and simplifying the given expression using properties of complex numbers leads to this result.

Option B (I is true, II is false) is correct because statement I is based on the property of complex number magnitudes, while statement II is only true when z1 and z2 have the same argument.

The correct answer is B ($x=4n$, where $n$ is any positive integer) because it satisfies the condition $\left(\frac{1+i}{1-i}\right)^{x}=1$ by simplifying to $i^x = 1$, which holds true for multiples of 4.

Option B (y² + 4x - 4 = 0) is correct because substituting z = 1 + iα into z² = x + iy and eliminating α results in the equation y² + 4x - 4 = 0.

Option B (7) is correct because the maximum value is found using the triangle inequality and the given condition |z-i| <= 2.

Option D (Straight line) is correct because the argument being constant implies a straight line emanating from the point 'a'.

Option A (5) is correct because simplifying the given equation leads to |z|^2 = 25, thus |z| = 5.

Option B ($-1, 1-2 \omega, 1-2 \omega^2$) is correct because these are the roots obtained by solving the given equation using the cube roots of unity.

Option A (55) is correct because the expression simplifies to the square root of 3025, which equals 55.

Option C (16) is correct because by simplifying the expression using the properties of complex cube roots of unity, the result is 16.

Option A (1000) is correct because by finding the minimum value of v and the corresponding z_0, we can substitute these values into the given expression and evaluate it to 1000.

Option B ($\mathrm{Im\\;}\left( {{{{z}}^2}} \right) = 0$) is correct because if the real part of z is 0, then z is purely imaginary, and the square of a purely imaginary number is a real number, thus having an imaginary part of 0.

The correct answer is B (on a straight line parallel to imaginary-axis) because the equation |w| = |w-1| represents the perpendicular bisector of the line joining 0 and 1 in the complex plane, which is a vertical line.

Option C (3) is correct because the expression simplifies to (a² + b²)(|z_1|^2 + |z_2|^2), and since (a, b) lies on the unit circle, a² + b² = 1, resulting in 1 * 3 = 3.

Option C ((a²⁺b²)(|z1|^2+|z2|^2)) is correct because expanding the given expression and simplifying using the properties of complex numbers leads to the result (a²⁺b²)(|z1|^2+|z2|^2).

Option B (32) is correct because substituting x=i into the binomial expansion and taking the magnitude squared results in 32.

Option C ($x$-axis) is correct because the locus of the complex number z satisfying the given condition is the x-axis.

Option B ($\frac{5}{3}$) is correct because the center of the circle is found by converting the complex equation to Cartesian form and comparing it with the standard circle equation.