Complex numbers practice · Easy

Option C (Right angled and isosceles) is correct because the sides PQ and RP are equal in length, and the Pythagorean theorem holds, indicating a right-angled isosceles triangle.

Option B ($m,n,k$ are integers) is correct because it ensures that the polynomial is divisible by ${x^2} + x + 1$ by making $\omega$ and $\omega^2$ roots of the polynomial.

Option D (None of these) is correct because the expression simplifies to $2i^n$, which is not represented by any of the other options.

Option A (A circle) is correct because the given condition on the imaginary part of the complex expression leads to an equation that represents a circle in the complex plane.

Option C (1) is correct because the real part of z is equal to the real part of its conjugate, and the conjugate of z is given as 1+i, so Re(z) = 1.

Option D ($-{2^7}i$) is correct because, after simplifying z and raising it to the power of 14 using De Moivre's theorem, the result is $-{2^7}i$.

Option C (-1) is correct because after substituting $z = \cos \theta \pm i \sin \theta$ into the expression and simplifying, the result is -1.

Option D (None of these) is correct because the only integral solution to the equation is x=0, which is not a non-zero solution.

Option B (4) is correct because, by equating the imaginary parts of the given equation, we directly find that Im(z) = 4.

Option C ($\beta \in(1, \infty)$) is correct because the product of the roots, which is $\beta$, must be greater than 1 due to the roots being distinct and lying on Re(z) = 1.

Option B (6) is correct because it is the smallest integer value of n for which both z^n = 1 and (1+z)^n = 1 are satisfied, given the constraints on z.

The correct answer is B ($x=4n$, where $n$ is any positive integer) because it satisfies the condition $\left(\frac{1+i}{1-i}\right)^{x}=1$ by simplifying to $i^x = 1$, which holds true for multiples of 4.

Option B (7) is correct because the maximum value is found using the triangle inequality and the given condition |z-i| <= 2.

Option D (Straight line) is correct because the argument being constant implies a straight line emanating from the point 'a'.

Option B ($-1, 1-2 \omega, 1-2 \omega^2$) is correct because these are the roots obtained by solving the given equation using the cube roots of unity.

Option A (55) is correct because the expression simplifies to the square root of 3025, which equals 55.

Option A (1000) is correct because by finding the minimum value of v and the corresponding z_0, we can substitute these values into the given expression and evaluate it to 1000.

Option B ($\mathrm{Im\\;}\left( {{{{z}}^2}} \right) = 0$) is correct because if the real part of z is 0, then z is purely imaginary, and the square of a purely imaginary number is a real number, thus having an imaginary part of 0.

The correct answer is B (on a straight line parallel to imaginary-axis) because the equation |w| = |w-1| represents the perpendicular bisector of the line joining 0 and 1 in the complex plane, which is a vertical line.

Option C (3) is correct because the expression simplifies to (a² + b²)(|z_1|^2 + |z_2|^2), and since (a, b) lies on the unit circle, a² + b² = 1, resulting in 1 * 3 = 3.

Option C ((a²⁺b²)(|z1|^2+|z2|^2)) is correct because expanding the given expression and simplifying using the properties of complex numbers leads to the result (a²⁺b²)(|z1|^2+|z2|^2).

Option C ($x$-axis) is correct because the locus of the complex number z satisfying the given condition is the x-axis.

Option B ($\frac{5}{3}$) is correct because the center of the circle is found by converting the complex equation to Cartesian form and comparing it with the standard circle equation.

Option B ($z$ lies on real axis) is correct because the condition $|w| = 1$ implies that the imaginary part of $z$ must be zero, meaning $z$ is real.

Option D (-2) is correct because the expression simplifies to -2.

Option D (0) is correct because after solving the logarithmic equation, the solution x=2 is extraneous.

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

Option C (4) is correct because it is the smallest positive integer $n$ for which ${\left( {\frac{{1 + i}}{{1 - i}}} \right)^n} = 1$.

Option B ($\sqrt{5}$) is correct because it is derived from the parallelogram law relating the sum and difference of two complex numbers to their magnitudes.

Option C (a hyperbola) is correct because the locus of a point where the difference of its distances from two fixed points is constant defines a hyperbola.