Complex numbers practice · Easy

Option B ($- {110^0}$) is correct because it is the principal amplitude of the given complex number after simplification using trigonometric identities and properties of complex numbers.

Option A (1) is correct because the expression is simplified using the properties of cube roots of unity.

Option D ($\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)$) is correct because it correctly identifies the modulus and argument of the complex number after simplification.

Option A ($x$-axis) is correct because the given condition implies that the distance from z to 2i is equal to the distance from z to -2i, which means z lies on the perpendicular bisector of the line segment joining 2i and -2i, which is the x-axis.

Options B and C are correct because they represent the possible scenarios where the product of the imaginary parts of the two complex numbers is negative, satisfying the given conditions.

Option C (4) is correct because by cubing $Z^{\frac{{1}}{{3}}}=a-ib$ and equating real and imaginary parts, we find that $\frac{{x}}{{a}}-\frac{{y}}{{b}} = 4(a^2 - b^2)$.

The polar form of the complex number $1+\sqrt{3} i$ is $2\left(\cos \frac{{\pi}}{{3}}+i \sin \frac{{\pi}}{{3}}\right)$ because the modulus is 2 and the argument is $\frac{{\pi}}{{3}}$.

Option A (5) is correct because simplifying the fraction z1/z2 results in (1/5)(9 + 12i), so the value of 'a' is 5.

Option A ($\pi$) is correct because when $\arg(z) < 0$, adding $\pi$ to the argument of $z$ gives the argument of $-z$, and the difference is $\pi$.

Option C (a straight line) is correct because the given condition translates to the perpendicular bisector of the line segment joining 0 and 5 in the complex plane.

Option C (1) is correct because using the given conditions |a|=|b|=|c|=1 and abc=a+b+c, we can manipulate the equations to show that bc+ca+ab = 1.

Option B (-1) is correct because the value of the given expression simplifies to $\omega + \omega^2$, which equals -1.

Option D (None of these) is correct because the correct formula leads to $\frac{{n(2^{n-1})}}{{2^n - 1}}$, which is not among the options.

Option C (2$\cos(m\theta + n\phi)$) is correct because by using Euler's formula and simplifying the expression, we arrive at 2$\cos(m\theta + n\phi)$.

Option A (-1) is correct because the expression simplifies to $\omega + \omega^2$, which equals -1 due to the properties of complex roots of unity.

Option C (form an equilateral triangle) is correct because the given condition leads to $|z_1| = |z_2|$ and the angle between $z_1$ and $z_2$ being 60 degrees, thus forming an equilateral triangle with the origin.

Option A (0) is correct because simplifying the expression i⁹ + i¹⁹ results in 0 + 0i, so the value of 'a' is 0.

Option B is correct because the given equation represents a circle in the complex plane with center (0, 1) and radius $\sqrt{2}$, derived by converting to Cartesian coordinates.

Option A ($C + i\;S$) is correct because it is the simplified form of the given expression using trigonometric identities.

Option D (${240^0}$) is correct because the argument of the complex number $\frac{{1 - i\sqrt 3 }}{{1 + i\sqrt 3 }}$ is calculated to be ${240^0}$ after rationalizing and finding the angle in the correct quadrant.

Option B (II quadrant) is correct because the simplified complex number has a negative real part and a positive imaginary part.

Option B ($\quad\left|z_1\right| \geq 1$) is correct because if $|z_1| < 1$, the inequality leads to a contradiction.

Option D (0) is correct because the sum of any four consecutive powers of $i$ is always 0.

Option B (A circle) is correct because the given equation simplifies to the standard form of a circle's equation, ${x^2} + {y^2} = constant$.

Option A (${p^2} + {q^2} + {r^2} = 1$) is correct because it is the condition derived from the given equation using componendo and dividendo.

Option B (3/2 - 2i) is correct because substituting z = x + iy into the equation and solving for x and y yields x = 3/2 and y = -2.

Option D (1) is correct because when $\frac{Z_{2}}{Z_{1}}$ is purely imaginary, the magnitude of the expression simplifies to 1.

Option C, $\frac{\left(a^2+1\right)^2}{4 a^2-1}$, is the correct answer because it results from multiplying the complex conjugate of the given expression.

Option B ($4 / \sqrt{3}$) is correct. By using the given conditions and properties of complex numbers, we can find the imaginary part of the difference between the two complex numbers.

Option B (20) is correct because the area of the triangle formed by -z, iz, and z-iz is (3/2)|z|^2, which equals 600, leading to |z| = 20.