Complex Numbers
Complex Inequalities — Set Membership
nta_pyq_2023_apr
Grade 11
Question:
For $a\in\mathbb{C}$, let $A=\{z\in\mathbb{C}:\ \text{Re}(a+\bar{z})>\text{Im}(\bar{a}+z)\}$ and $B=\{z\in\mathbb{C}:\ \text{Re}(a+\bar{z})<\text{Im}(\bar{a}+z)\}$. Then among the two statements: (S1): If $\text{Re}(a),\text{Im}(a)>0$, then $A$ contains all real numbers. (S2): If $\text{Re}(a),\text{Im}(a)<0$, then $B$ contains all real numbers.
Only (S2) is true
Only (S1) is true
Both are true
Both are false
Step-by-Step Solution
Key Concept: For real $z$ (i.e., $y_2=0$): $A=\{z: x_1+y_1+x_2>0\}$ and $B=\{z: x_1+y_1+x_2<0\}$. Neither covers all of $\mathbb{R}$ regardless of the sign of $x_1,y_1$.
Both (S1) and (S2) are false — neither half-plane contains all real numbers.
Correct Answer: 4