1. Let $i$ be the imaginary unit. Then the complex number $( 1 - i ) ( 1 + 2 i ) =$
(A) $3 + 3 i$
(B) $- 1 + 3 i$
(C) $3 + \mathrm { i }$
(D) $- 1 + i$

1. $\mathrm { i } ( 2 - \mathrm { i } ) =$
A. $1 + 2 \mathrm { i }$
B. $1 - 2 \mathrm { i }$
C. $- 1 + 2 \mathrm { i }$
D. $- 1 - 2 \mathrm { i }$

11. Let $i$ be the imaginary unit. Then $i - \frac { 1 } { i } =$ \_\_\_\_.

2. Let $i$ be the imaginary unit, then the complex number $i ^ { 2 } - \frac { 2 } { i } =$
A. $-i$ B. $-3i$
C. $i$ D. $3 i$

9. $i$ is the imaginary unit. Calculate $\frac { 1 - 2 i } { 2 + i }$ and the result is $\_\_\_\_$.

Among the following expressions, which has a result that is a pure imaginary number?
A. $i(1 + i)^2$
B. $i^2(1 - i)$
C. $(1 + i)^2$
D. $i(1 + i)$

$(1+i)(2+i) = $
A. $1-i$
B. $1+3i$
C. $3+i$
D. $3+3i$

1. $\frac { 3 + i } { 1 + i } =$
A. $1 + 2 i$
B. $1 - 2 \mathrm { i }$
C. $2 + \mathrm { i }$
2. $A = \{ 1,2,4 \}$
$$B = \left\{ x \mid x ^ { 2 } - 4 x + m = 0 \right\}$$
If $A \cap B = \{ 1 \}$, then $B =$
A. $\{ 1 , - 3 \}$
B. $\{ 1,0 \}$
C. $\{ 1,3 \}$
D. $2 - i$
3. In the ancient Chinese mathematical classic ``Suanfa Tongzong'', there is the following problem: ``Looking from afar at a seven-story pagoda, with lights doubling at each level, totaling 381 lights, how many lights are at the top?'' This means: a seven-story pagoda has a total of 381 lights, and the number of lights at each lower level is twice that of the level above it. The number of lights at the top of the pagoda is
A. 1
B. 3
C. 5 [Figure]

$i ( 2 + 3 i ) =$
A. $3 - 2 \mathrm { i }$
B. $3 + 2 i$
C. $- 3 - 2 \mathrm { i }$
D. $- 3 + 2 \mathrm { i }$

$\frac { 1 + 2 i } { 1 - 2 i } =$
A. $- \frac { 4 } { 5 } - \frac { 3 } { 5 }$ i
B. $- \frac { 4 } { 5 } + \frac { 3 } { 5 } \mathrm { i }$
C. $- \frac { 3 } { 5 } - \frac { 4 } { 5 } \mathrm { i }$
D. $- \frac { 3 } { 5 } + \frac { 4 } { 5 }$ i

$( 1 + i ) ( 2 - i ) =$
A. $- 3 - \mathrm { i }$
B. $- 3 + \mathrm { i }$
C. $3 - i$
D. $3 + i$

2. C
Solution: $z ( \bar { z } - i ) = ( 2 - i ) ( 2 + 2 i ) = 6 + 2 i$, so the answer is $C$.

If $z = - 1 + \sqrt { 3 } \mathrm { i }$ , then $\frac { z } { z \bar { z } - 1 } =$
A. $- 1 + \sqrt { 3 } \mathrm { i }$
B. $- 1 - \sqrt { 3 } \mathrm { i }$
C. $- \frac { 1 } { 3 } + \frac { \sqrt { 3 } } { 3 } \mathrm { i }$
D. $- \frac { 1 } { 3 } - \frac { \sqrt { 3 } } { 3 } \mathrm { i }$

Let $z = \frac { 2 + i } { 1 + i ^ { 2 } + i ^ { 5 } }$, then $\bar { z } =$
A. $1 - 2 i$
B. $1 + 2 i$
C. $2 - i$
D. $2 + i$

Given $z = 1 + \mathrm{i}$, then $\frac{1}{z-1} = $ ( )
A. $-i$
B. $i$
C. $-1$
D. $1$

Let $i = \sqrt { - 1 }$. Given that the complex number $\frac { 1 - 7 i } { - 1 + i } = a + b i$, where $a, b$ are real numbers. Then $a =$ (10–1)(10–2), $b =$ (10–3).

Let $\bar{z}$ denote the conjugate of $z$. For the complex number $z = 2 + i$, $$\frac{z}{\bar{z}-1}$$ Which of the following is this expression equal to?
A) $\frac{1}{2} + \frac{3}{2}i$
B) $\frac{2}{3} - \frac{3}{2}i$
C) $1 + 3i$
D) $2 - 3i$
E) $3 + i$

In the set of complex numbers, the result of the operation
$$( 3 - i ) ( 2 - i ) ( 1 + i ) ( 2 + i ) ( 3 + i )$$
is $\mathbf { a } + \mathbf { b i }$. What is the sum $a + b$?
A) 80
B) 84
C) 90
D) 96
E) 100

Let a be a real number. In complex numbers,
$$\frac { 1 - a i } { a - i } = i$$
the equality is given.
Accordingly, what is a?
A) 4 B) 3 C) 2 D) 1 E) 0

In the set of complex numbers
$$\frac { ( 4 - 2 i ) \cdot ( 6 + 3 i ) } { ( 1 - i ) \cdot ( 1 + i ) }$$
What is the result of this operation?
A) 15
B) 12
C) 10
D) 9
E) 6

In the set of complex numbers
$$\frac { i \cdot ( 2 - i ) \cdot ( 2 - 4 i ) } { ( 1 - i ) \cdot ( 1 + i ) }$$
what is the result of the operation?
A) 2
B) 5
C) 10
D) $2 i$
E) $5 i$