Complex numbers practice · Mixed difficulty

Option C (infinite number of points) is correct because the minimum value is attained for all points on the line segment joining 0 and i, which represents an infinite number of points.

The correct answer is 3 because when $\alpha$ and $\beta$ are complex cube roots of unity, substituting $\omega$ and $\omega^2$ simplifies the expression to 3.

Option B ($i - 1$) is correct because the sum telescopes, leaving only the first two terms.

Option D (2) is correct because by recognizing omega and omega squared as roots of an equation, the third root can be found and used to evaluate the expression.

Option C (${\alpha ^3} + {\beta ^3}$) is correct because substituting the given expressions for $x, y, z$ and using the properties of cube roots of unity simplifies the product $xyz$ to $\alpha^3 + \beta^3$.

Option C ($\frac{{7^n-1}}{{6}}$) is correct because by using the factorization of $x^n - 1$ and substituting $x=7$, we can find the value of the given expression.

Option C (A straight line) is correct because the equation represents the locus of a point where the sum of its distances from two fixed points is constant, which degenerates into a straight line when the constant equals the distance between the fixed points.

Option B (Im(z)=0) is correct because the expression is the sum of a complex number raised to the 7th power and its conjugate raised to the 7th power, resulting in a real number, hence the imaginary part is zero.

Option C (26) is correct because by solving the quadratic equation and using properties of cube root of unity, and last digit properties, the value of the expression can be found.

Option C (a line) is correct because the condition |w| = 1 leads to an equation that represents a straight line in the complex plane.

Option A ($3 \sqrt{2} \text{}$) is correct because it represents the magnitude of the complex number |z - 7 - 9i| after simplifying the given amplitude condition.

Option A ($\frac{{4{{\log }_e}3}}{{{{\log }_e}7 - {{\log }_e}6}}$) is correct because it is obtained by taking the logarithm of both sides of the equation and solving for x.

Option A ($z(\frac{{1}}{{a}}-\frac{{i}}{{b}})+\bar{{z}}(\frac{{1}}{{a}}+\frac{{i}}{{b}})=2$) is correct because it represents the equation of a line passing through the complex numbers $a$ and $ib$.

Option A ($9 \sqrt{{3}} / 4$) is correct because the triangle formed by $z$, $\omega z$, and $z + \omega z$ is equilateral with side length $|z| = 3$, and the area of an equilateral triangle is $\frac{{\sqrt{{3}}}}{{4}} \times side^2$.

Option A is correct because the expression simplifies to $\frac{{(n - 1)n(n^2 + 3n + 4)}}{{4}}$ after expanding and summing the terms using the properties of cube roots of unity.

The correct answer is B ($3{x^2} + 3{y^2} + 10y + 3 = 0$) because substituting $z = x + iy$ and simplifying the given equation leads to this equation of a circle.

Option C (87) is correct because the maximum value of the expression is obtained by minimizing the negative terms, which leads to the result 87.

Option B (Re($\frac{{Z-\alpha}}{{\beta-\gamma}}$)=0) is correct because the perpendicularity condition implies that the argument of the complex number (Z-α)/(β-γ) is π/2, making its real part zero.

Option A ($y+x \omega$) is correct because by taking the conjugate and manipulating the expression using the properties of cube roots of unity, we find that $(b + a\omega)^7 = y + x\omega$.

Option B (whose centre is (0,-1) and radius =3) is correct because it correctly identifies the center and radius of the circle on which the transformed complex number lies.

Option C (Both are correct) is correct because both statements correctly calculate the roots of the given quadratic equations using the quadratic formula.

Option C ($\left| z \right| > 2$) is correct because solving the logarithmic inequality leads to the condition that the magnitude of z is greater than 2.

Option A ($\frac{a^2+b^2}{c^2+d^2}$) is correct because multiplying the complex number by its conjugate and squaring the result gives the correct expression.

Option B ($i - 1$) is correct because the sum simplifies to $i^{13} + i^{14}$ which equals $i - 1$ due to the cyclic nature of powers of i.

Option C $\frac{{4m\pi }}{{n\left( {n + 1} \right)}}$ is correct because it is derived by using De Moivre's theorem and equating the imaginary part to zero to find the general solution for $\theta$.

Option B (1) is correct because by substituting $p = \cot \theta$ and simplifying using trigonometric identities and Euler's formula, the expression equals 1.

Option A (25) is correct because the maximum distance from the origin to the point z is found by solving the inequality derived from the given condition using properties of complex numbers.

Option D (a hyperbola) is correct because the equation $\operatorname{Re}(\bar{z}^2) = k^2$ simplifies to $x^2 - y^2 = k^2$, which is the equation of a hyperbola.

Option C ($\operatorname{Im}(z) > 0$) is correct because the inequality $|z-i| < |z+i|$ represents the set of all points in the complex plane that are closer to $i$ than to $-i$, which is the upper half-plane.

Option D (Both (2) and (3)) is correct because the product simplifies to $\omega^{\frac{{n(n+1)}}{{2}}}$, which can be either 1 or $\omega$ depending on the value of n.