Complex numbers practice · Mixed difficulty

Option B (A hyperbola) is correct because the difference in distances from the center of the variable circle to the centers of the two fixed circles is constant, which is the definition of a hyperbola.

Option B ($a d-b c=0$) is correct because if three complex numbers are collinear, the imaginary part of the ratio of the differences between them is zero, which leads to the given condition.

Option B ($- 2\sqrt 6$) is correct because it is the imaginary part of the simplified expression after taking the square root of the complex numbers.

Option A (${x^2} - \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0$) is correct because it is the quadratic equation whose roots are the real and imaginary parts of the complex number.

Option C (3) is correct because using the triangle inequality and the given condition $|z| < 1/2$, the expression is bounded by 3.

Option C (230) is correct because the maximum value of the sum of the complex numbers is found by summing the maximum possible magnitudes of each complex number, which forms an arithmetic progression.

Option B is correct because the expression simplifies to $2(\left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2)$ using the properties of complex numbers and their conjugates.

Option A (0) is correct because substituting z = 2 - 3i into the expression z² - 4z + 13 results in 0.

The value of $p^2+q^2$ is $\frac{{\left(a^2+1\right)^2}}{{4 a^2-1}}$ because it is obtained by multiplying the complex number with its conjugate.

Option B is correct because the inequality $|x|+|y| \leq \sqrt{2}|z|$ holds true, where $k = \sqrt{2}$.

The correct answer is A (1) because the minimum value is achieved when z lies on the line segment connecting 0 and 1.

Option D (7) is correct because it is the largest value of |z| that satisfies the given logarithmic inequality.

Option C (${z_1},{z_2}$ and the origin from an equilateral triangle) is correct because the given condition implies the complex numbers form an equilateral triangle with the origin.

Option A (${p^2} = 3q$) is correct because it satisfies the condition for the points represented by the roots and the origin to form an equilateral triangle.

Option C ($x+y=a, y>0$) is correct because it represents the locus of the complex number satisfying the given argument condition.

Option C ($\frac{\pi }{{10}}$) is correct because it accurately calculates the argument (amplitude) of the given complex number using trigonometric identities.

Option B (-y - ix) is correct because by using the properties of complex conjugates and manipulating the expression, we find that $(b + ai)^{11} = -y - ix$.

Option B (-2) is correct because the given condition implies that z is a complex cube root of unity (specifically $-\omega$ or $-\omega^2$), and substituting this into the expression yields -2.

Option C (0) is correct because when α is expressed in terms of ω, the expression simplifies to 0 since n is not a multiple of 3.

Option B ($- 1,\;1 - 2\omega ,\;1 - 2{\omega ^2}$) is correct because it represents the roots of the given equation after simplification and using the properties of cube roots of unity.

Option A ($2 \sqrt{3}$) is correct because by using properties of cube root of unity and modulus, the minimum value of the expression can be found.

Option B is correct because |z1+z2|^2 + |z1-z2|^2 simplifies to 2(|z1|^2 + |z2|^2) using the properties of complex numbers.

Option B ($z$ lies on the real axis) is correct because $|w|=1$ implies $|z+i|=|z-i|$, which means $z$ is equidistant from $i$ and $-i$, thus lying on the real axis.

Option A is correct because the real part of $\tan(\alpha+i \beta)$ is derived using trigonometric identities and Euler's formula.

Option D ($4k$) is correct because the argument difference between two nth roots of unity subtending a right angle at the origin must be $\pm \frac{\pi}{2}$, leading to $n = 4k$.

Option A (\left[\begin{array}{cc}1 & -2023 i \\ 0 & 1\end{array}\right]) is correct because after simplifying the expression and using the properties of matrix multiplication, the result is a matrix with -2023i in the top right corner.

The correct answer is D ($\pi / 2$) because the odd functions integrate to zero over the symmetric interval, leaving only the integral of the constant function 1.

Option B (A circle) is correct because the equation derived from the given condition $\arg \left( {\frac{{z - 1}}{{z + 1}}} \right) = \frac{\pi }{3}$ represents a circle.

Option B (0) is correct because using the properties of cube roots of unity, the expression simplifies to zero.

Option C (0) is correct because when n is not a multiple of 3, the expression $1 + {\omega ^n} + {\omega ^{2n}}$ simplifies to 0, where $\omega$ is a complex cube root of unity.