Key Concepts and Formulas

• Geometric Interpretation of Complex Numbers: A complex number $z = x + iy$ can be represented as a point $(x, y)$ in the complex plane. The modulus $|z| = \sqrt{x^2 + y^2}$ represents the distance from the origin to the point, and the argument $\arg(z)$ represents the angle between the positive real axis and the line connecting the origin to the point.
• Rotation in the Complex Plane: Multiplying a complex number $z$ by $e^{i\theta} = \cos\theta + i\sin\theta$ rotates $z$ counterclockwise by an angle $\theta$ about the origin. If $z_1$ and $z_2$ represent two points, then $\frac{z_2}{z_1} = re^{i\theta}$ implies $|z_2| = r|z_1|$ and $\arg(z_2) = \arg(z_1) + \theta$.
• Argument of a Complex Number: For $z = x + iy$, $\arg(z)$ depends on the quadrant in which the point $(x, y)$ lies. Specifically:
  • Quadrant I ($x>0, y>0$): $\arg(z) = \tan^{-1}(\frac{y}{x})$
  • Quadrant II ($x<0, y>0$): $\arg(z) = \pi - \tan^{-1}(|\frac{y}{x}|)$
  • Quadrant III ($x<0, y<0$): $\arg(z) = -\pi + \tan^{-1}(\frac{y}{x})$
  • Quadrant IV ($x>0, y<0$): $\arg(z) = -\tan^{-1}(|\frac{y}{x}|)$

Step 1: Analyzing the Geometric Conditions and Setting up the Equation

We are given that $O$ is the origin, $A$ is $z_1 = 1 + 2i$, $B$ is $z_2$ with $\Re(z_2) < 0$, and $\triangle OAB$ is a right-angled isosceles triangle with $OB$ as the hypotenuse. Since $OB$ is the hypotenuse, the right angle is at $A$, so $\angle OAB = 90^\circ$. Also, since the triangle is isosceles, $OA = AB$, which implies $|z_1| = |z_2 - z_1|$. Because $\angle OAB = 90^\circ$, the vector $\vec{AB}$ is obtained by rotating $\vec{AO}$ by $90^\circ$ either clockwise or counterclockwise. Since $OA = AB$, the scaling factor is 1. Therefore, $\frac{z_2 - z_1}{0 - z_1} = e^{\pm i\frac{\pi}{2}} = \pm i$. This gives us two possibilities:

• $\frac{z_2 - z_1}{-z_1} = i \implies z_2 - z_1 = -iz_1 \implies z_2 = z_1 - iz_1 = z_1(1 - i)$
• $\frac{z_2 - z_1}{-z_1} = -i \implies z_2 - z_1 = iz_1 \implies z_2 = z_1 + iz_1 = z_1(1 + i)$

Step 2: Determining the Correct Value of $z_2$

We have $z_1 = 1 + 2i$. Let's compute $z_2$ for both possibilities and use the condition $\Re(z_2) < 0$ to determine the correct value.

• Case 1: $z_2 = z_1(1 - i) = (1 + 2i)(1 - i) = 1 - i + 2i - 2i^2 = 1 + i + 2 = 3 + i$. Since $\Re(z_2) = 3 > 0$, this case is invalid.
• Case 2: $z_2 = z_1(1 + i) = (1 + 2i)(1 + i) = 1 + i + 2i + 2i^2 = 1 + 3i - 2 = -1 + 3i$. Since $\Re(z_2) = -1 < 0$, this case is valid.

Thus, $z_2 = -1 + 3i$.

Step 3: Evaluating Option A: $\arg {z_2} = \pi - {\tan ^{ - 1}}3$

Since $z_2 = -1 + 3i$, it lies in the second quadrant. Therefore, $\arg(z_2) = \pi - \tan^{-1}(|\frac{3}{-1}|) = \pi - \tan^{-1}(3)$. This matches option (A), so option (A) is TRUE.

Step 4: Evaluating Option B: $\arg ({z_1} - 2{z_2}) = - {\tan ^{ - 1}}{4 \over 3}$

$z_1 - 2z_2 = (1 + 2i) - 2(-1 + 3i) = 1 + 2i + 2 - 6i = 3 - 4i$. This lies in the fourth quadrant. $\arg(z_1 - 2z_2) = -\tan^{-1}(|\frac{-4}{3}|) = -\tan^{-1}(\frac{4}{3})$. This matches option (B), so option (B) is TRUE.

Step 5: Evaluating Option C: $|{z_2}| = \sqrt {10}$

$|z_2| = |-1 + 3i| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$. This matches option (C), so option (C) is TRUE.

Step 6: Evaluating Option D: $|2{z_1} - {z_2}| = 5$

$2z_1 - z_2 = 2(1 + 2i) - (-1 + 3i) = 2 + 4i + 1 - 3i = 3 + i$. $|2z_1 - z_2| = |3 + i| = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}$. Since $\sqrt{10} \neq 5$, option (D) is FALSE.

Common Mistakes & Tips

• Remember to consider both clockwise and counterclockwise rotations when dealing with geometric problems in the complex plane.
• Always verify the condition $\Re(z_2) < 0$ to choose the correct solution for $z_2$.
• Be careful when calculating the argument of a complex number, paying attention to the quadrant in which it lies.

Summary

We analyzed the geometric conditions to derive the value of $z_2 = -1 + 3i$. Then, we checked each option. Options (A), (B), and (C) are true. Option (D) is false because $|2z_1 - z_2| = \sqrt{10} \neq 5$. Thus, the answer is option (D).

The final answer is $\boxed{D}$.