Complex numbers practice · Mixed difficulty

Option A ($6ab$) is correct because after substituting the given values of x, y, and z and simplifying using the properties of cube roots of unity, the expression ${x^2} + {y^2} + {z^2}$ reduces to $6ab$.

Option A (6) is correct because the magnitude of the product of two complex numbers is the product of their magnitudes, resulting in $\sqrt{6}$.

Option A (1) is correct because there is only one complex number that satisfies the given conditions, which is z=0.

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

Option A (32) is correct because solving the logarithmic equation step-by-step leads to x = 32.

Option C (512 & -512√3) is correct. By converting the complex number to polar form and using De Moivre's theorem, the result is 512 - 512√3i.

Option D (-7/2) is correct because the sum of the real parts of the roots is -7/2 based on the properties of complex numbers and the given equation.

Option B ($2^n-1$) is correct because by substituting x=2 into the polynomial equation and using the properties of complex numbers, we can find the product of the distances.

Option A (5) is correct because by substituting z = 2ω into the expression and using the properties of cube roots of unity, the expression simplifies to 5.

Option C ($\omega$, an imaginary cube root of unity) is correct because if a root remains unchanged when a, b, and c are decreased by one, then that root must satisfy the equation $x^2 + x + 1 = 0$, which has roots $\omega$ and $\omega^2$.

Option C (1) is correct because the roots are complex cube roots of unity, and substituting these values into the expression ${\alpha ^{2009}} + {\beta ^{2009}}$ simplifies to 1.

Option B (0) is correct because using the given condition $z_1 + z_2 + z_3 = 0$ and the fact that $|z_i| = 1$, the expression simplifies to zero.

Option C is correct because both statements are correct. The modulus and argument of the given complex number are correctly calculated.

Option B is correct. Statement 1 and Statement 2 are both true due to the triangle inequality properties of complex numbers, but Statement 2 does not explain Statement 1.

Option A ($2 \pi / 3$) is correct because the complex number lies in the second quadrant, and the argument is calculated accordingly.

Option C (1) is correct because the sum simplifies to 1 after using the properties of complex numbers and geometric series.

The correct answer is B (the line through the origin with slope 1) because the equation |z-i|=|z-1| represents the perpendicular bisector of the line segment joining the points (1,0) and (0,1) in the complex plane, which is a line through the origin with slope 1.

Option A ($1+i$) is correct because $(1+i)^2 = 2i$, so the square root of $2i$ is $1+i$.

Option D (128) is correct because using the property of cube roots of unity, the expression simplifies to 64 + 64 = 128.

Option A ($i\cot \frac{\theta }{2}$) is correct because it is the simplified form of the given expression using trigonometric identities.

Option B is correct. By expressing z in polar form and using trigonometric identities, we can derive that z = (1 + i tan(arg z / 2)) / (1 - i tan(arg z / 2)).

Option B (2) is correct because the equation holds for three distinct values of x only when a = 2 and r = 0, making a + r = 2.

The correct answer is C (f is neither injective nor surjective) because the function f(z) = |z| maps multiple complex numbers to the same real number (not injective), and its range is only non-negative real numbers, not all real numbers (not surjective).

Option B ($\frac{{{{\left( { - 1} \right)}^n}}}{{{{2^{3n}}}}}$) is correct because it is the simplified form of the given expression using trigonometric identities and De Moivre's theorem.

Option B (1) is correct because the expression simplifies to sin(π/2) + cos(π/2), which equals 1.

Option D ($\pm \left( {b - ia} \right)$) is correct because by squaring the given equation and comparing real and imaginary parts, we can find the expression for $\sqrt { - x - iy}$.

Option A (Real axis) is correct because the given condition implies that the imaginary part of z is zero, which means z lies on the real axis.

Option B ($\frac{{1}}{{2}}(e+\frac{{1}}{{e}})\cos 1, \frac{{1}}{{2}}(e-\frac{{1}}{{e}})\sin 1$) is correct because by using the definition of cosine in terms of complex exponentials and simplifying, we find the real and imaginary parts of cos(1-i).

Option C (both g(x) and h(x) are divisible by (x-1)) is correct because if f(x) is divisible by x²⁺x+1, then both g(1) and h(1) must be zero, implying divisibility by (x-1).

Option D (They form an isosceles triangle) is correct because two of the sides formed by the complex numbers have equal lengths.