Complex numbers practice · Medium

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.

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

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 (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 C is correct because it accurately determines the least value of n for which the given condition is satisfied, using complex number properties.

Option A (1) is correct because substituting the root into the polynomial and equating the real part to zero leads to the given expression being equal to -1, and taking the absolute value gives 1.

Option C ($\theta$) is correct because using the property $\bar{z} = \frac{1}{z}$ when $|z|=1$, we can simplify the expression to $\arg(z) = \theta$.

Option C ($1+c \cos \theta$) is the correct answer because after simplification and using the given equation, the expression reduces to $1+c \cos \theta$.

Option A ($\frac{{3\pi}}{{4}}$) is correct because the given conditions lead to $\arg(z) = \frac{{3\pi}}{{4}}$ after substituting and simplifying the arguments.

Option B ($(a^2 + b^2)(|z_1|^2 + |z_2|^2)$) is correct because expanding and simplifying the given expression using properties of complex numbers leads to this result.

Option B (I is true, II is false) is correct because statement I is based on the property of complex number magnitudes, while statement II is only true when z1 and z2 have the same argument.

Option B (y² + 4x - 4 = 0) is correct because substituting z = 1 + iα into z² = x + iy and eliminating α results in the equation y² + 4x - 4 = 0.

Option A (5) is correct because simplifying the given equation leads to |z|^2 = 25, thus |z| = 5.

Option C (16) is correct because by simplifying the expression using the properties of complex cube roots of unity, the result is 16.

Option B (32) is correct because substituting x=i into the binomial expansion and taking the magnitude squared results in 32.

Option D is correct because $\left|\frac{{z-1}}{{z+1}}\right|=1$ represents a straight line, while the other options represent circles.

Option A ($-1$) is correct because it is the simplified form of the given expression using properties of complex cube roots of unity.

Option D (None of these) is correct because the given conditions lead to contradictory requirements for x and y, indicating no solution exists for z.

Option A (2) is correct because the minimum value is obtained when any two are zero and the third is a minimum magnitude integer i.e. 1.

Option C (A circle) is correct because the given condition leads to the equation of a circle when $z$ is expressed as $x+iy$.

Option C ($-\frac{5 \pi}{12}$) is correct because calculating the argument of each complex number and using the properties of arguments leads to the principal argument being $-\frac{5 \pi}{12}$.

Option B ($|z-7-5 i|=\sqrt{2}$) is correct because it represents the locus of the complex number $z$ satisfying the given argument condition.

Option D ($i$) is correct because the sum is simplified using De Moivre's theorem and the properties of geometric progressions.

Option A (0) is correct because the given equations lead to an impossible condition, indicating no solution for z.

Option A (0) is correct because the given condition implies that z_1/z_2 is a complex cube root of unity, which leads to the equation z_1² + z_2² - z_1z_2 = 0.

Option D ($4z_1+z_2+z_3=0$) is correct because the midpoint of AG being the origin leads to the given relationship between the complex numbers.

Option A (1) is correct because the product simplifies to 1 using the properties of complex cube roots of unity.

Option D ($- 2$) is correct because after substituting $z = x - iy$ and ${z^{1/3}} = p + iq$ and simplifying, the expression $\left( {\frac{x}{p} + \frac{y}{q}} \right)/\left( {{p^2} + {q^2}} \right)$ evaluates to -2.

Option B ($\frac{2 \pi}{3}$ or $2 \pi$ or $\frac{10 \pi}{3}$) is correct because it represents the possible values of the argument of $z^{2/3}$ when $z=-1$.

Option A ($1 - i, - 1 + i$) is correct because these are the vertices obtained by rotating the given vertex by 90 and 270 degrees around the origin.