Complex numbers practice · Easy

Option B (3) is correct because after simplifying the expression and substituting, the value reduces to 3.

Option C ($\sin^{-1}(\frac{1}{\sqrt{3}})$) is correct because when the given expression is purely imaginary, its real part must be zero, leading to the equation $\sin^2 \theta = \frac{1}{3}$, which gives $\theta = \sin^{-1}(\frac{1}{\sqrt{3}})$.

Option A (a straight line) is correct because the given condition implies that the distance from z to the origin is equal to the distance from z to the point i/3, which defines a straight line.

Option B (1) is correct because when the given ratios are equal to k, taking the logarithm of xyz results in 0, which means xyz equals 1.

Option C ($1+i \cot (\pi / 2 n)$) is correct because it is derived by simplifying the geometric series and using trigonometric identities.

Option B (3) is correct because simplifying the given equation using the properties of cube roots of unity leads to ω^n = 1, which is satisfied when n is a multiple of 3, and the least positive integral value is 3.

Option A ($X$-axis) is correct because the condition |z-ai| = |z+ai| implies that z is equidistant from ai and -ai, which means z lies on the x-axis (y=0).

Option B (3) is correct because it is the radius of the circle represented by the given equation, calculated after converting the equation to the standard form of a circle.

Option D ($-1$) is correct because the expression simplifies to $-1$ using the properties of complex cube roots of unity.

Option A ($\frac{{1 - x}}{2}$) is correct because performing partial fraction decomposition and solving for f(x) yields this result.

The correct answer is D (${x^2} + x + 1 = 0$) because ${\alpha ^{19}}$ and ${\beta ^7}$ simplify to $\omega$ and ${\omega ^2}$ respectively, which are the roots of the given equation.

Option B ($\{0,3\}$) is correct because the function f(n) evaluates to either 0 or 3 depending on whether n is a multiple of 3 or not.

Option D (Less than one) is correct because the absolute value of the sum is less than the sum of the absolute values, and since each |a_k| < 1 and the sum of λ_k is 1, the result is less than 1.

None of these is the correct answer. The correct answer is $\sqrt{\frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}}$ because by taking the modulus squared of both sides of the equation, we can find ${x^2} + {y^2}$.

Option D ($2\,\cos \,n\alpha$) is correct because it is derived by using De Moivre's theorem and simplifying the expression.

Option A (a straight line) is correct because the equation simplifies to x=0, which represents a straight line on the complex plane.

Option D (3) is correct because by substituting y=0 into the given equation and solving for x, we find that the modulus of z is 3.

Option B ($\frac{4}{3}$) is correct because solving the equation ${z^2} + 3\overline z = 0$ gives 4 solutions, and the sum of the infinite geometric series with the common ratio $\frac{1}{4}$ is $\frac{4}{3}$.

Option C (A pair of straight lines) is correct because the equation can be factored into two linear equations representing straight lines.

Option D (None of the above) is correct because the given complex numbers are collinear, and therefore cannot form a triangle.

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

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