Complex numbers practice · Easy

Option B (-1 + i) is correct because it is the additive inverse that, when added to the original number, results in zero.

Option B (1, 1) is correct because simplifying the expression using properties of cube roots of unity leads to A = 1 and B = 1.

Option D ($x = 0, y = 0$) is correct because the determinant is zero after applying column operations.

Option A ($\bar zw = - i$) is correct because, given the conditions on the magnitudes and arguments of z and w, the expression simplifies to -i.

The correct answer is C ($\frac{3}{2}{\left| z \right|^2}$) because the area of the triangle formed by the given points in the Argand plane is $\frac{3}{2}$ times the square of the modulus of z.

Option B (${y^2} \le 2(x + \frac{1}{2})$) is correct because it represents the inequality derived from the given condition $|z| - Re(z) \le 1$ after substituting $z = x + iy$ and simplifying.

Option B (The line passing through the origin with slope 1) is correct because the equation simplifies to y = x, which represents a line through the origin with a slope of 1.

Option B (2r) is correct because when the roots are equal in magnitude but opposite in sign, the coefficient of x in the quadratic equation derived from the given equation must be zero, leading to p + q = 2r.

Option D ($\pi /3$) is correct because the argument of the complex number ${z^2}{e^{z - i}}$ simplifies to $\frac{\pi}{3}$ when $z = \sqrt{3} + i$.

Option A (3) is correct because for a quadratic equation to have equal roots, the discriminant must be zero, and solving for 'a' gives a = 3.

Option C is correct because the real part of z can be either 3/2 or 7/2, satisfying the given condition based on the minimum distance.

Option D ($\frac{{4 n \pi }}{{r(r + 1)}}, n \in I$) is correct because it correctly sums the geometric series in the exponent and solves for $\theta$ using the properties of complex exponentials.

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 (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 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 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 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 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.