Complex numbers practice · Mixed difficulty

Option C (-1) is correct because the expression simplifies to $\omega + {\omega ^2}$ which equals -1, where $\omega$ is a complex cube root of unity.

Option B (1) is correct because the modulus of the given expression simplifies to 1 using the property that |z1| = 1.

Option C (1/3) is correct because by substituting a = 0 and b = 1, we can solve for x and y and then find the value of 1/x - 1/y.

Option B ($2n\pi$) is the correct answer because it satisfies the condition for the given expression to be real by making the imaginary part zero.

Option C ($(1,0)$) is correct because it is the simplified form of the given complex number expression using De Moivre's theorem.

Option D (None of these) is correct because the complex number z has a non-zero imaginary part and Re(z) = 0, which is purely imaginary.

Option A ($- 12 + 9i$) is correct because it represents the complex number obtained by tripling the magnitude of $4 - 3i$ and rotating it by $\pi$ radians counterclockwise.

Option A ($|z-\bar{z}| \leq |z||\arg (z)-\arg (\bar{z})|$) is correct because it follows from the geometric interpretation of complex numbers and their arguments.

Option C (Collinear) is correct because the determinant of the matrix formed by the coordinates of the points is zero, indicating they lie on the same line.

Option C ($y < x < 0$) is correct because for $\frac{{\bar{{z}}}}{{z}}$ to lie in the third quadrant, both its real and imaginary parts must be negative, leading to the condition $y < x < 0$.

Option A ($\omega, \omega^2$) is correct because these are the common roots of both equations, where $\omega$ is a complex cube root of unity.

Option B ($\left(\frac{{n(n+1)}}{{2}}\right)^2-n$) is correct because the given expression can be simplified to a sum of cubes minus a linear term, which results in the given expression.

Option C ($\frac{3}{4}$) is correct because by substituting $x = -y^2$ into the equation derived from the purely imaginary condition, we can solve for $y^4 + y^2 - y$.

Option B (a rectangular hyperbola) is correct because the condition Im($z^2$) = 4 translates to $2xy = 4$, which is the equation of a rectangular hyperbola.

Option D (0) is correct because since the coefficients are real, the complex conjugate is also a root, leading to the given expression equaling 0.

Option C (1 or $- 1$) is correct because by substituting $z = x_1 + iy_1$ and $w = x_2 + iy_2$ into the given equations, we find that $z$ must be either 1 or $- 1$.

The correct answer is C (Straight line with the sum of its intercepts on the coordinate axes equals 14) because the equation simplifies to a linear equation x + y = 7, representing a straight line with intercepts at (7,0) and (0,7).

Option A ($x=4, 0 \leq y \leq 2$) is correct because it satisfies the given condition $0 \leq \sin^{-1}(\frac{{z-4}}{{2i}}) \leq \frac{{\pi}}{{2}}$.

Option C ($1-z$) is correct because the remainder is found by using the polynomial remainder theorem and solving a system of equations to determine the coefficients of the remainder polynomial.

Option A (Non-negative) is correct because the expression can be rewritten as a sum of squares, which is always non-negative.

Option A ($a^2+b^2$) is correct because it follows from taking the modulus of both sides of the equation and simplifying.

Option B (${{{2^5}}}$) is correct because it is the result of simplifying the expression using De Moivre's theorem and trigonometric identities.

Option D ($\frac{{\sin n \theta}}{{\sin ^n \theta}}$) is correct because it is obtained by substituting the roots and simplifying using trigonometric identities.

The correct answer is A (13) because the series ${a_1} + {a_3} + {a_5} + ... + {a_{25}}$ simplifies to a sum of 13 ones.

Option B (2) is correct because when $\frac{{z+2i}}{{z-2i}}$ is purely imaginary, the magnitude of z, $|z|$, equals 2.

Option B is correct because Rolle's Theorem guarantees the existence of a point 'c' between two roots 'a' and 'b' where the derivative of the function is zero.

Option A (1) is correct because the inverse of the conjugate multiplied by the conjugate equals 1.

Option A ($\sqrt{2}(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12})$) is correct because it involves converting the complex number to polar form and simplifying using Euler's formula.

Option D ($\sqrt{5}+1$) is correct because it represents the greatest possible value of |z| derived from the given condition and triangle inequality.

Option C ($i \tan (\arg (z-1))$) is correct because, by letting $z-1 = \cos \theta + i \sin \theta$, we can express $\frac{{z-2}}{{z}}$ as $i \tan(\frac{{\theta}}{{2}})$, which is equal to $i \tan(\arg(z-1))$.