Modulus and argument of a complex number — practice questions

Question 1 · difficulty L2 · understanding

Let $z_1$ and $z_2$ be two distinct complex numbers let $z=(1-t) z_1+t z_2$ for some real number t with $0 < t < 1$. If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$, then :

• A. $\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$
• B. $\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z-z_2\right)$
• C. $\left|$$\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}$$\right|=0$
• D. $\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z_2-z_1\right)$

Question 2 · difficulty L3 · understanding

Let $\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}$. Then $\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal to :

• A. 42
• B. 23
• C. 27
• D. 38

Question 3 · difficulty L3 · understanding

Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to:

• A. 20
• B. 26
• C. 12
• D. 16

Question 4 · difficulty L3 · understanding

If $$z_1, z_2$$ are two distinct complex number such that $$\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$$, then

• A. either $$z_1$$ lies on a circle of radius $$\frac{1}{2}$$ or $$z_2$$ lies on a circle of radius 1.
• B. $$z_1$$ lies on a circle of radius $$\frac{1}{2}$$ and $$z_2$$ lies on a circle of radius 1.
• C. either $$z_1$$ lies on a circle of radius 1 or $$z_2$$ lies on a circle of radius $$\frac{1}{2}$$.
• D. both $$z_1$$ and $$z_2$$ lie on the same circle.

Question 5 · difficulty L3 · understanding

Consider the following two statements : Statement I: For any two non-zero complex numbers $$z_1, z_2,(|z_1|+|z_2|)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right) \text {, and }$$ Statement II : If $$x, y, z$$ are three distinct complex numbers and $$\mathrm{a}, \mathrm{b}, \mathrm{c}$$ are three positive real numbers such that $$\frac{\mathrm{a}}{|y-z|}=\frac{\mathrm{b}}{|z-x|}=\frac{\mathrm{c}}{|x-y|}$$, then $$\frac{\mathrm{a}^2}{y-z}+\frac{\mathrm{b}^2}{z-x}+\frac{\mathrm{c}^2}{x-y}=1$$. Between the above two statements,

• A. both Statement I and Statement II are incorrect.
• B. Statement I is correct but Statement II is incorrect.
• C. Statement I is incorrect but Statement II is correct.
• D. both Statement I and Statement II are correct.

Question 6 · difficulty L3 · understanding

If $z$ is a complex number such that $|z| \leqslant 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is :

• A. 2
• B. $\frac{5}{2}$
• C. $\frac{3}{2}$
• D. 3

Question 7 · difficulty L3 · understanding

Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then $$(\mathrm{r}, \theta)$$ is equal to

• A. $$\left(2 \sec \frac{11 \pi}{8}, \frac{11 \pi}{8}\right)$$
• B. $$\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)$$
• C. $$\left(2 \sec \frac{5 \pi}{8}, \frac{3 \pi}{8}\right)$$
• D. $$\left(2 \sec \frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$$

Question 8 · difficulty L3 · understanding

Let $$w_{1}$$ be the point obtained by the rotation of $$z_{1}=5+4 i$$ about the origin through a right angle in the anticlockwise direction, and $$w_{2}$$ be the point obtained by the rotation of $$z_{2}=3+5 i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_{1}-w_{2}$$ is equal to :

• A. $$-\pi+\tan ^{-1} \frac{8}{9}$$
• B. $$-\pi+\tan ^{-1} \frac{33}{5}$$
• C. $$\pi-\tan ^{-1} \frac{8}{9}$$
• D. $$\pi-\tan ^{-1} \frac{33}{5}$$

Question 9 · difficulty L3 · understanding

If for $$z=\alpha+i \beta,|z+2|=z+4(1+i)$$, then $$\alpha+\beta$$ and $$\alpha \beta$$ are the roots of the equation :

• A. $$x^{2}+2 x-3=0$$
• B. $$x^{2}+3 x-4=0$$
• C. $$x^{2}+x-12=0$$
• D. $$x^{2}+7 x+12=0$$

Question 10 · difficulty L3 · understanding

Let $$z$$ be a complex number such that $$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then $$z$$ lies on the circle of radius 2 and centre :

• A. (0, $$-$$2)
• B. (0, 0)
• C. (0, 2)
• D. (2, 0)