Complex numbers practice · Medium

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

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 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 $i \tan(\arg(z-1))$.

A straight line. Option B is correct because the condition |w| = 1 leads to a linear equation in terms of the real and imaginary parts of z.

Option B ($|\operatorname{Re}(z)|=|\operatorname{Im}(z)|$) is correct because if $\operatorname{Re}(z^2)=0$, then the real and imaginary parts of z must have equal magnitudes.

Option B ($-15$) is correct because by substituting x = 3 + i into the given expression and simplifying, we find that the result is -15.

Option B (a circle) is correct because the condition for the expression to be purely imaginary leads to the equation x² + y² = 1, which represents a circle.

Option B ($4(a^2-b^2)$) is correct because cubing both sides of the given equation and equating real and imaginary parts allows us to express x/a + y/b in terms of $a^2$ and $b^2$.

Option C ($\frac{9^n-1}{8}$) is correct because it is obtained by substituting x=9 into the factored form of $x^n - 1$.

The curves $C_1$ and $C_2$ intersect at 4 points because $C_1$ is a circle and $C_2$ is an ellipse where the radius of the circle is smaller than the major axis of the ellipse.

The correct answer is C ((0, 3)) because the condition for the equation to represent an ellipse requires the distance between the foci to be less than the length of the major axis, leading to the inequality $|a^2 - 2a| < 3$.

Option D (1) is correct because the expression simplifies to 1 using the property that |β| = 1.

Option B ($\frac{{1 \pm \sqrt{{3}} i}}{{4}}$) is correct because it is the solution for z when the points A(z), B(-z), and C(z-1) form an equilateral triangle.

Option A ($C+i S$) is correct because the expression simplifies to $C+iS$ using trigonometric identities.

Option C ($n\pi \pm \frac{\pi }{3}$) is correct because setting the real part of the expression to zero yields this general solution for $\theta$.

Option C (1) is correct because the modulus of a product is the product of the moduli, and the modulus of a quotient is the quotient of the moduli, leading to the answer 1.

Option D (None of these) is correct because the value of $\arg(z) + \arg(\bar{z})$ depends on whether z is a negative real number or not. If z is a negative real number, the sum is $2\pi$, otherwise it is 0.

Option D ($-\frac{2}{25}-\frac{11}{25} i$) is correct because simplifying the expression and then finding the conjugate yields this result.

The value of $\theta$ is $\frac{{2 \pi}}{{3}}$ because the complex number lies in the second quadrant and the argument is calculated accordingly.

Option B ($0 \leq a \leq 4$) is correct because it correctly determines the range of possible values for 'a' based on the given conditions and properties of complex numbers.

Option B ($2 \pm 2 \sqrt{2} i$) is correct because by substituting z = x + iy into the given equation and solving for x and y, we find that x = 2 and y = $\pm 2\sqrt{2}$.

The correct answer is None of the above because the magnitude of z is $\sqrt{\frac{{17}}{2}}$, and z is neither purely real nor purely imaginary.