Complex numbers practice · Easy

Option B ($\left| z \right| < 5$) is correct because solving the inequality results in the condition that the modulus of $z$ must be less than 5.

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

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 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.

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 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)).

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 (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.

Option B (17) is correct because the position of the first significant figure is determined by the characteristic of the logarithm of the number.

Option A ($\bar{a} \bar{z}+a z=0$) is correct because reflecting a complex number in the real axis means taking its conjugate, which transforms the equation accordingly.

Option B is correct because the complex number $i+\sqrt{3}$ can be expressed in polar form as $2(\cos{\frac{\pi}{6}}+i \sin{\frac{\pi}{6}})$.

Option A (a circle) is correct because the locus of the point representing z is a circle, derived from the given condition on the real part of the complex expression.

Option C (1) is correct because if z_2/z_1 is purely real, the magnitude of the given expression simplifies to 1.

Option A ($z\bar z - z{\bar z_0} - \bar z{z_0} + {z_0}{\bar z_0} = {r^2}$) is correct because it represents the equation of a circle in the complex plane with center z0 and radius r.

Option B ($\infty$) is correct because the intersection of the two circles is empty for an infinite number of values of n.