Complex numbers practice · Mixed difficulty

Option C (-i) is correct because by converting to exponential form and using geometric series formula, the sum simplifies to -i.

The correct answer is A (0) because the given equation simplifies to |z-2| = |z+2|, which represents the perpendicular bisector of the line joining 2 and -2, i.e., the y-axis. The minimum value of |z| is 0, occurring at the origin.

Option D (None of these) is correct because the multiplicative inverse of $(6+5i)^2$ is $\frac{11}{(61)^2} - \frac{60}{(61)^2}i$, which is not among the given options.

Option C (6,0) is correct because the maximum and minimum values of $|z+1|$ are found by using the triangle inequality on $|(z+4)-3|$.

Option C (1) is correct because the absolute value of the nth term of the geometric progression is 1, as the magnitude of the common ratio is 1.

Option B ($- 160$) is correct because substituting $x = - 5 + 2\sqrt { - 4}$ into the expression ${x^4} + 9{x^3} + 35{x^2} - x + 4$ and simplifying results in -160.

Option A ($\sin^{-1} \frac{1}{\sqrt{3}}$) is correct because it makes the real part of the complex number Z equal to zero, resulting in a purely imaginary number.

Option D (1) is correct because by simplifying the expression inside the square root and then calculating $(x^2+y^2)^2$, it can be shown that the result is 1.

Option B (30) is correct because the minimum value is found by completing the square and recognizing the minimum occurs when z = 1 + 2i.

Option C ($\frac{{\|u|-|v\|}}{{1+|u||v|}}$) is correct because by using properties of complex numbers and the reverse triangle inequality, we can establish the lower bound for |z|.

Option A (The boundary of a circle) is correct because the locus satisfying the given condition represents a circle.

Option D ($\frac{{1}}{{4 \sin 2^{\circ}}}$) is correct because the sum of the imaginary parts simplifies to this expression using trigonometric identities.

The correct answer is a square because all sides are equal and the diagonals are equal, indicating that PQRS is a square.

Option A (8,2) is correct because the maximum and minimum values of $|z+3|$ are found by using the triangle inequality on $|(z+4i)+(3-4i)|$.

Option B ($z_1+z_2+z_3$) is correct because the orthocenter of a triangle with circumcenter at the origin is represented by the sum of the complex numbers representing the vertices.

The correct answer is C (A circle) because the given condition represents a circle in the complex plane.

Option B (A circle) is correct because the locus of the point $z$ that satisfies the given condition is a circle.

Option B (2) is correct because when n=2, ((1+i)/(1-i))^n simplifies to -1, satisfying the given condition.

Option A (-1) is correct because by De Moivre's theorem, the product simplifies to $\cos(\pi) + i \sin(\pi)$, which equals -1.

Option D (-i) is correct because the sum simplifies to -i using the properties of complex numbers and roots of unity.

Option D ($x>3$ and $y$ is any real number) is correct because simplifying the inequality $|z-4|<|z-2|$ leads to the condition $x>3$, with no restrictions on $y$.

Option A (91) is correct because after expanding the complex number and equating real and imaginary parts, y - x equals 91.

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 C (60) is correct because the area of the triangle is related to |z|^2, and given the area is 200, we can find |z| and then |3z|.

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 B (7) is correct because the number of oscillations per minute is proportional to the square root of the horizontal component of the Earth's magnetic field, which depends on the total intensity and the angle of dip.

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 C (${z_3} + i\left( {{z_1} - {z_3}} \right)$) is correct because considering the rotation about 'B' and using the given conditions leads to this result.