Mathematics & Numerical Analysis

Inequality analysis for real numbers:
Given $0 \lt a \lt b$:
Multiplying the inequality $a \lt b$ by the strictly positive real number $a$ preserves the inequality:

This proves Option 4 is correct.

Testing other options:

• Since $a \lt b$, taking reciprocals reverses the inequality $\frac{1}{a} \gt \frac{1}{b}$ (false).
• Multiplying $a \lt b$ by $b \gt 0$ gives $ab \lt b^2$ (false).
• $a^2 \lt ab$ means $ab \gt a^2$ (false).

Properties of complex conjugates:

• The conjugate of a product is the product of the conjugates: $\overline{Z_1 Z_2} = \overline{Z_1}\, \overline{Z_2}$. (Option 3 is strictly true).
• The field of complex numbers $\mathbb{C}$ is not an ordered field, meaning inequality relations ($\lt , \le, \gt , \ge$) are undefined for general complex numbers. Therefore, Options 1, 2, and 4 are mathematically meaningless.

Calculations for the principal argument $\text{Arg}(Z) \in (-\pi, \pi]$:

• Multiply the two complex numbers:
  $$Z_1 Z_2 = (-2 + 2i)(3i) = -6i + 6i^2 = -6 - 6i$$
• Notice that for $Z = -6 - 6i$, both real and imaginary parts are negative, placing it in the third quadrant.
• The principal argument in the third quadrant is:
  $$\text{Arg}(Z_1 Z_2) = -\pi + \arctan\left(\frac{-6}{-6}\right) = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4}$$

Official NTA Answer Key & Algebraic Analysis:
Official Answer Key: Option 4 (-3, -7) is designated as the correct response in the official NTA key.
Step-by-step simplification: Expanding the terms:

Evaluating the rational term with intended integer coordinates $\frac{1 - 8i}{2 - i} = 2 - 3i$ yields:

Equating real and imaginary parts gives $x = -3$ and $y = -7$. Option 4 is therefore the officially designated answer.