Complex numbers practice · Mixed difficulty

The correct answer is C (On the $y$-axis (imaginary axis)) because after simplifying the complex number, the real part becomes zero, leaving only the imaginary part.

The expression $\frac{{1+a}}{{1-a}}$ is equal to $i \cot \frac{{\theta}}{{2}}$ because it simplifies to this form using trigonometric identities and Euler's formula.

Option A ($\pi / 4$) is correct because by simplifying the expression for z, we find that one possible value of arg(z) is $\pi/4$.

Option A (I and II are correct) is correct because the modulus of (1+i)/(1-i) is 1 and its argument is π/2, while statements III and IV are incorrect.

Option C ($\frac{{\sqrt 5 }}{5}$ unit) is correct because it is the modulus of the complex number obtained after simplifying the given expression.

Option C (A line through the origin) is correct because the equation |z-1| = |z+i| represents the perpendicular bisector of the line segment joining 1 and -i, which is the line x + y = 0, passing through the origin.

Option A (2) is correct because by using the properties of roots of unity and trigonometric identities, the value simplifies to 2.

Option B (17) is correct because by substituting $z = x + iy$ into the given equation and solving the resulting system of equations, we find $|z|^2 = x^2 + y^2 = 17$.

Option C ($\frac{3}{2}$) is correct because it represents the minimum distance between the complex number z (with |z| >= 1) and the given complex number, found using the triangle inequality.

Option A (1) is correct because if (z-1)/(z+1) is purely imaginary, then its conjugate is its negative, leading to |z| = 1.

Option C (2(1+i)) is incorrect. Option B (2(1-i)) is correct because by converting to polar form and using De Moivre's theorem, the product of the four roots simplifies to 2(1-i).

Option D ($- \theta$) is correct because the argument of the conjugate of a complex number is the negative of the argument of the original complex number.

Option B ($2\sqrt 5$) is correct because the radius of the circle is calculated using the formula $\sqrt {{{\left| a \right|}^2} - b}$.

Option C (Both are correct) is correct because both statements are true after evaluating the polynomial at x=1+2i and solving the cubic equation.

Option A (a circle) is correct because the given equation, after simplification, represents the general form of a circle in the complex plane.

Option C (0) is correct because the sum of any four consecutive powers of i (the imaginary unit) is always zero.

Option D ($\frac{1}{2}$) is correct because, after simplifying the expression using properties of complex cube roots of unity, the sine of the resulting angle is $\frac{1}{2}$.

Option A (1) is correct because the radius of the circle representing the locus of z is 1, derived from the given condition.

Option B ($- \frac{{3\pi }}{{4}}$) is correct because after simplification and rationalization, the argument of the complex number is found to be $- \frac{{3\pi }}{{4}}$.

Option D (0) is correct because if $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then $z_1$ and $z_2$ must have the same argument.

Option A ($\frac{{10}}{{\sqrt{{2}}}}(1+i)$) is correct because after simplifying using trigonometric identities, the expression equals $\frac{{10}}{{\sqrt{{2}}}}(1+i)$.

Option B ($2\sqrt{13}$) is correct because the difference between the greatest and least values of $|z|$ is found by using the triangle inequality on $|(z-3+2i)+(3-2i)|$.

Option B ($i \tan n \theta$) is correct because after applying De Moivre's theorem and trigonometric identities, the expression simplifies to $i \tan n \theta$.

Option C ($\frac{1}{4}(1 \pm \sqrt{3} i)$) is correct because it satisfies the condition for the points to form an equilateral triangle.

Option B (2) is correct because by using the properties of complex numbers and inequalities, we find the difference between the maximum and minimum values of |z| is 2.

Option C ($x^2+y^2+z^2-y z-z x-x y$) is the correct answer because it results from expanding and simplifying the given expression using the properties of $\omega$.

Option B (4) is correct because by substituting $z = x + iy$ into the equation and solving the resulting system of equations, we find four distinct solutions.

Option D ($p^2-3 q$) is correct because, given $\alpha, \beta$ are roots of $x^2+p x+q=0$, and $\omega$ is a cube root of unity, the expression $(\omega \alpha+\omega^2 \beta)(\omega^2 \alpha+\omega \beta)$ simplifies to $p^2 - 3q$ using Vieta's formulas.

Option A ($- {\left( { - 2} \right)^n}$) is correct because the expression simplifies to $-(-2)^n$ using properties of complex numbers and the given condition on n.

Option D (1) is correct because the sum of the cube roots of unity is 0, and the product simplifies to 1, resulting in 0 - 1 = -1. However, the question asks for z = sum - product, so z = 0 - (-1) = 1.