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Complex Numbers - Practice Questions (27)

Question 1: The complex number z satisfies $\frac { Z } { \mathrm { i } } = 3 + \mathrm { i }$ ,then the imagina...

The complex number z satisfies $\frac { Z } { \mathrm { i } } = 3 + \mathrm { i }$ ,then the imaginary part of z is ( )

  • A. A. 3i
  • B. B. -$i$
  • C. C. 3
  • D. D. - 1

Answer: C

Solution: [Knowledge Points] Complex algebraic form of the multiplication operation, the real and imaginary parts of the complex numbers [Analysis] By the multiplication formula of complex numbers to get $z = - 1 + 3 i$, and then by the concept of the imaginary part of the complex number can be solved. [Explanation] $z = ( 3 + \mathrm { i } ) \mathrm { i } = 3 \mathrm { i } + \mathrm { i } ^ { 2 } = - 1 + 3 \mathrm { i }$ , then the imaginary part of $z$ is 3 .

Question 2: If ${ } _ { Z = 2 + \mathrm { i } }$ is known, $\frac { \mathrm { i } } { Z - 1 } = ( )$ is known.

If ${ } _ { Z = 2 + \mathrm { i } }$ is known, $\frac { \mathrm { i } } { Z - 1 } = ( )$ is known.

  • A. A. $\frac { 1 } { 2 } + \frac { 1 } { 2 } \mathrm { i }$
  • B. B. $\frac { 1 } { 2 } - \frac { 1 } { 2 } \mathrm { i }$
  • C. C. $- \frac { 1 } { 2 } + \frac { 1 } { 2 } \mathrm { i }$
  • D. D. $- \frac { 1 } { 2 } - \frac { 1 } { 2 } \mathrm { i }$

Answer: A

Solution: Knowledge Points] Complex algebraic form of multiplication operations, division operations of complex numbers [Analysis] According to the multiplication and division of complex numbers can be solved. [Explanation] By the meaning of the question, $\frac { \mathrm { i } } { z - 1 } = \frac { \mathrm { i } } { 1 + \mathrm { i } } = \frac { \mathrm { i } ( 1 - \mathrm { i } ) } { ( 1 + \mathrm { i } ) ( 1 - \mathrm { i } ) } = \frac { 1 + \mathrm { i } } { 2 } = \frac { 1 } { 2 } + \frac { 1 } { 2 } \mathrm { i }$ is a complex number.

Question 3: If ${ } _ { z = 2 + \mathrm { i } }$ is known, $\frac { \mathrm { i } } { z - 1 } = ( )$ is known.

If ${ } _ { z = 2 + \mathrm { i } }$ is known, $\frac { \mathrm { i } } { z - 1 } = ( )$ is known.

  • A. A. $\frac { 1 } { 2 } + \frac { 1 } { 2 } \mathrm { i }$
  • B. B. $\frac { 1 } { 2 } - \frac { 1 } { 2 } \mathrm { i }$
  • C. C. $- \frac { 1 } { 2 } + \frac { 1 } { 2 } \mathrm { i }$
  • D. D. $- \frac { 1 } { 2 } - \frac { 1 } { 2 } \mathrm { i }$

Answer: A

Solution: Knowledge Points] Complex numbers division operation [Analysis]Using the four operations of complex numbers, you can make a judgment. [Explanation] By the meaning of the question: $\frac { \mathrm { i } } { z - 1 } = \frac { \mathrm { i } } { 1 + \mathrm { i } } = \frac { \mathrm { i } ( 1 - \mathrm { i } ) } { 2 } = \frac { 1 } { 2 } + \frac { 1 } { 2 } \mathrm { i }$ .

Question 4: It is known that the complex number $z = \frac { 1 + \mathrm { i } } { 2 + \mathrm { i } }$ (i is an...

It is known that the complex number $z = \frac { 1 + \mathrm { i } } { 2 + \mathrm { i } }$ (i is an imaginary unit),then $| z | =$ ()

  • A. A. $\frac { 12 } { 5 }$
  • B. B. $2 + \frac { \sqrt { 10 } } { 5 }$
  • C. C. $\frac { \sqrt { 10 } } { 5 }$
  • D. D. $\frac { 3 \sqrt { 10 } } { 5 }$

Answer: C

Solution: Knowledge Points] Modulus of complex numbers, division of complex numbers [Analysis]Calculate according to the operation law of division of complex numbers and the operation of modulus. [Detailed explanation] Solution 1: By the meaning of the question $z = \frac { 1 + \mathrm { i } } { 2 + \mathrm { i } } = \frac { ( 1 + \mathrm { i } ) ( 2 - \mathrm { i } ) } { ( 2 + \mathrm { i } ) ( 2 - \mathrm { i } ) } = \frac { 3 + \mathrm { i } } { 5 } = \frac { 3 } { 5 } + \frac { \mathrm { i } } { 5 }$ , $z = \frac { 1 + \mathrm { i } } { 2 + \mathrm { i } } = \frac { ( 1 + \mathrm { i } ) ( 2 - \mathrm { i } ) } { ( 2 + \mathrm { i } ) ( 2 - \mathrm { i } ) } = \frac { 3 + \mathrm { i } } { 5 } = \frac { 3 } { 5 } + \frac { \mathrm { i } } { 5 }$ , $z = \frac { 1 + \mathrm { i } } { 2 + \mathrm { i } } = \frac { ( 1 + \mathrm { i } ) ( 2 - \mathrm { i } ) } { ( 2 + \mathrm { i } ) ( 2 - \mathrm { i } ) } = \frac { 3 + \mathrm { i } } { 5 } = \frac { 3 } { 5 } + \frac { \mathrm { i } } { 5 }$ So $| z | = \sqrt { \left( \frac { 3 } { 5 } \right) ^ { 2 } + \left( \frac { 1 } { 5 } \right) ^ { 2 } } = \frac { \sqrt { 10 } } { 5 }$ , .

Question 5: If $z = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } }$ ,then $| z | = ( \quad )$

If $z = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } }$ ,then $| z | = ( \quad )$

  • A. A. 1
  • B. B. $\sqrt { 2 }$
  • C. C. $\sqrt { 3 }$
  • D. D. 4

Answer: A

Solution: [Knowledge Points] Complex division operations, find the mode of complex numbers [Analysis]First use the multiplication and division of complex numbers to find the complex number $Z$, and then find the mode. [Explanation] Because $z = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } = \frac { ( 1 + \mathrm { i } ) ^ { 2 } } { ( 1 - \mathrm { i } ) ( 1 + \mathrm { i } ) } = \frac { 2 \mathrm { i } } { 2 } = \mathrm { i }$ , so $| \mathrm { z } | = | \mathrm { i } | = 1$ .

Question 6: If the complex number ${ } ^ { Z }$ satisfies ${ } ^ { ( 1 - \mathrm { i } ) _ { Z } = 2 - 3 \mathrm...

If the complex number ${ } ^ { Z }$ satisfies ${ } ^ { ( 1 - \mathrm { i } ) _ { Z } = 2 - 3 \mathrm { i } }$ , then the point corresponding to ${ } ^ { Z }$ in the complex plane lies at ( )

  • A. A. first quadrant (of the coordinate plane, where both x and y are positive)
  • B. B. second quadrant (of the coordinate plane, where both x and y are positive)
  • C. C. third quadrant (of the coordinate plane, where both x and y are positive)
  • D. D. fourth quadrant (of the coordinate plane, where both x and y are positive)

Answer: D

Solution: Knowledge Points] Determine the quadrant of the point corresponding to the complex number, the division operation of complex numbers [Analysis]Find the complex number $z$ by dividing the complex number, then find the coordinates of the point in the complex plane, and then find the quadrant of the point. So the point corresponding to $\mathrm { z } _ { \mathrm { Z } }$ in the complex plane is $\left( \frac { 5 } { 2 } , - \frac { 1 } { 2 } \right)$ in the fourth quadrant.

Question 7: The imaginary part of the complex number $z = \frac { - 2 i } { 1 - i }$ is

The imaginary part of the complex number $z = \frac { - 2 i } { 1 - i }$ is

  • A. A. $- \frac { 1 } { 2 } \mathrm { i }$
  • B. B. $- \frac { 1 } { 2 }$
  • C. C. -i
  • D. D. - 1

Answer: D

Solution: [Knowledge Points] Find the real and imaginary parts of complex numbers, division operations of complex numbers [Analysis] The complex number $z$ is simplified to get the imaginary part. $z = \frac { - 2 \mathrm { i } } { 1 - \mathrm { i } } = \frac { - 2 \mathrm { i } ( 1 + \mathrm { i } ) } { 2 } = 1 - \mathrm { i }$, so the imaginary part of the complex number $z$ is - 1 , and the imaginary part of the complex number $z$ is - 1 , and the imaginary part of the complex number $z$ is - 1 .

Question 8: If $Z _ { 1 } = 2 + 2 \mathrm { i } , Z _ { 2 } = 1 - \mathrm { i }$ ,then $\left| Z _ { 1 } + Z _ {...

If $Z _ { 1 } = 2 + 2 \mathrm { i } , Z _ { 2 } = 1 - \mathrm { i }$ ,then $\left| Z _ { 1 } + Z _ { 2 } \right| =$

  • A. A. $\sqrt { 10 }$
  • B. B. $\sqrt { 13 }$
  • C. C. 3
  • D. D. $\sqrt { 5 }$

Answer: A

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers, find the mode of complex numbers [Analysis]First find $z _ { 1 } + z _ { 2 }$, and then according to the formula of complex modulus can be calculated. so $z _ { 1 } + z _ { 2 } = ( 2 + 2 i ) + ( 1 - i ) = 3 + i$ , and so $\left| z _ { 1 } + z _ { 2 } \right| = \sqrt { 3 ^ { 2 } + 1 ^ { 2 } } = \sqrt { 10 }$ .

Question 9: If ${ } ^ { 2 z + \bar { z } } = ( 1 + 2 \mathrm { i } ) ^ { 2 }$ ,then the real part of ${ } ^ { Z ...

If ${ } ^ { 2 z + \bar { z } } = ( 1 + 2 \mathrm { i } ) ^ { 2 }$ ,then the real part of ${ } ^ { Z }$ is

  • A. A. 1
  • B. B. $\frac { 3 } { 2 }$
  • C. C. - 1
  • D. D. $- \frac { 3 } { 2 }$

Answer: C

Solution: Knowledge]According to the equality conditions for parameters, the concept and calculation of conjugate complex numbers, the algebraic operation of addition and subtraction of complex numbers, the real and imaginary parts of complex numbers [Analysis] The answer can be obtained by using the operation law of complex numbers and the condition of equality of two complex numbers. Then $3 a = - 3$ ,Solve $a = - 1$ ,So the real part of $z$ is - 1 .

Question 10: Calculate the value of $( 2 + 4 i ) + ( 3 - 4 i )$ as ( )

Calculate the value of $( 2 + 4 i ) + ( 3 - 4 i )$ as ( )

  • A. A. 5
  • B. B. - 8 i
  • C. C. 8 i
  • D. D. 5-8i

Answer: A

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis]Using the algebraic form of complex numbers to solve for the addition that is obtained. [Details] $( 2 + 4 \mathrm { i } ) + ( 3 - 4 \mathrm { i } ) = 5$ .

Question 11: It is known that the complex number $Z = 2 + 3 \mathrm { i }$ ,then $| Z - 1 | =$ ( )

It is known that the complex number $Z = 2 + 3 \mathrm { i }$ ,then $| Z - 1 | =$ ( )

  • A. A. $\sqrt { 10 }$
  • B. B. $\sqrt { 13 }$
  • C. C. 2
  • D. D. 4

Answer: A

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers, find the modulus of complex numbers [Analysis] Calculate $z - 1$ and combine with the definition of modulus. $z - 1 = 1 + 3 \mathrm { i }$, then $| z - 1 | = \sqrt { 1 ^ { 2 } + 3 ^ { 2 } } = \sqrt { 10 }$.

Question 12: Given that ${ } ^ { \mathrm { i } }$ is in imaginary units, let the complex number $Z _ { 1 } = 1 - ...

Given that ${ } ^ { \mathrm { i } }$ is in imaginary units, let the complex number $Z _ { 1 } = 1 - \mathrm { i } , Z _ { 2 } = 3 + \mathrm { i }$ ,then ${ } ^ { Z _ { 1 } + Z _ { 2 } } =$ ()

  • A. A. 1
  • B. B. 4
  • C. C. i
  • D. D. 4i

Answer: B

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis] Apply the addition of complex numbers to calculate can be. [Detailed Explanation] Because ${ } ^ { Z _ { 1 } } = 1 - \mathrm { i } , Z _ { 2 } = 3 + \mathrm { i }$ , the So $z _ { 1 } + z _ { 2 } = 1 - \mathrm { i } + 3 + \mathrm { i } = 4$ .

Question 13: $i$ is an imaginary unit, then $1 + i + i ^ { 2 } + i ^ { 3 } = i$ ( )

$i$ is an imaginary unit, then $1 + i + i ^ { 2 } + i ^ { 3 } = i$ ( )

  • A. A. 1
  • B. B. i
  • C. C. 1-i
  • D. D. 0

Answer: D

Solution: [Knowledge Points] Imaginary number unit i and its properties, the algebraic operations of addition and subtraction of complex numbers [Detailed Explanation]Test Question Analysis: According to the question, $1 + \mathrm { i } + \mathrm { i } ^ { 2 } + \mathrm { i } ^ { 3 } = 1 + \mathrm { i } - 1 - \mathrm { i } = 0$ , $1 + \mathrm { i } + \mathrm { i } ^ { 2 } + \mathrm { i } ^ { 3 } = 1 + \mathrm { i } - 1 - \mathrm { i } = 0$ , $1 + \mathrm { i } + \mathrm { i } ^ { 2 } + \mathrm { i } ^ { 3 } = 1 + \mathrm { i } - 1 - \mathrm { i } = 0$ Therefore, we can know the answer is 0, choose D. Points: the operations of complex numbers Point: the main test is the operation of imaginary units, belongs to the basic questions

Question 14: $\frac { \mathrm { i } } { 1 - \mathrm { i } } + \frac { 1 - \mathrm { i } } { \mathrm { i } } =$

$\frac { \mathrm { i } } { 1 - \mathrm { i } } + \frac { 1 - \mathrm { i } } { \mathrm { i } } =$

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

Answer: C

Solution: Knowledge Points] Algebraic operations of addition and subtraction of complex numbers, division operations of complex numbers [Analysis]Using the complex division algorithm to simplify can be. [Details] $\frac { i } { 1 - i } + \frac { 1 - i } { i } = \frac { i ( 1 + i ) } { ( 1 - i ) ( 1 + i ) } + \frac { ( 1 - i ) i } { i ^ { 2 } } = - \frac { 1 } { 2 } + \frac { 1 } { 2 } i - 1 - i = - \frac { 3 } { 2 } - \frac { 1 } { 2 } i$ The correct choice for this question: C

Question 15: $( 2 + 3 i ) + ( - 1 - 2 i ) =$

$( 2 + 3 i ) + ( - 1 - 2 i ) =$

  • A. A. $1 + i$
  • B. B. 1-i
  • C. C. $- 1 + i$
  • D. D. $- 1 - i$

Answer: A

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis]Calculate the result according to the addition of complex numbers. [Explanation] $( 2 + 3 i ) + ( - 1 - 2 i ) = ( 2 - 1 ) + ( 3 - 2 ) i = 1 + i$ ,choose A.

Question 16: It is known that the complex number $z _ { 1 } = 3 + 4 i , z _ { 2 } = 3 - 4 i$ ,then $z _ { 1 } + z...

It is known that the complex number $z _ { 1 } = 3 + 4 i , z _ { 2 } = 3 - 4 i$ ,then $z _ { 1 } + z _ { 2 } =$ ( )

  • A. A. $8 i$
  • B. B. 6
  • C. C. $6 + 8 i$
  • D. D. .6-8i

Answer: B

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis]According to the law of addition of complex numbers can be found. [Details] $z _ { 1 } + z _ { 2 } = ( 3 + 4 i ) + ( 3 - 4 i ) = ( 3 + 3 ) + ( 4 - 4 ) i = 6$ .

Question 17: It is known that the complex number $z = 1 - i$ ,then $\left| z + 2 i ^ { 3 } \right| =$ ( )

It is known that the complex number $z = 1 - i$ ,then $\left| z + 2 i ^ { 3 } \right| =$ ( )

  • A. A. $\sqrt { 10 }$
  • B. B. 6
  • C. C. $2 \sqrt { 3 }$
  • D. D. 7

Answer: A

Solution: Knowledge]Find the mode of complex numbers, complex addition and subtraction of algebraic operations [Analysis] According to the algorithm of complex numbers and the modulus of the algorithm to solve. [Details] Solution: $\because z = 1 - i$ $\therefore \left| z + 2 \mathrm { i } ^ { 3 } \right| = | 1 - \mathrm { i } + 2 \times ( - 1 ) \times \mathrm { i } | = | 1 - 3 \mathrm { i } | = \sqrt { 1 ^ { 2 } + ( - 3 ) ^ { 2 } } = \sqrt { 10 }$

Question 18: Plural $( 1 - \mathrm { i } ) - ( 2 + \mathrm { i } ) + 3 \mathrm { i } _ { \text {η­‰δΊŽ( )} }$

Plural $( 1 - \mathrm { i } ) - ( 2 + \mathrm { i } ) + 3 \mathrm { i } _ { \text {η­‰δΊŽ( )} }$

  • A. A. $- 1 + \mathrm { i }$
  • B. B. 1-i
  • C. C. i
  • D. D. -$i$

Answer: A

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis]Follow the laws of addition and subtraction of complex numbers to solve. [Details] $( 1 - \mathrm { i } ) - ( 2 + \mathrm { i } ) + 3 \mathrm { i } = ( 1 - 2 ) + ( - \mathrm { i } - \mathrm { i } + 3 \mathrm { i } ) = - 1 + \mathrm { i }$

Question 19: The complex number $Z = 1 - 2 \mathrm { i }$ (where i is an imaginary unit), then $| z + 3 \mathrm {...

The complex number $Z = 1 - 2 \mathrm { i }$ (where i is an imaginary unit), then $| z + 3 \mathrm { i } | =$ ( )

  • A. A. $\sqrt { 2 }$
  • B. B. 2
  • C. C. $\sqrt { 10 }$
  • D. D. 5

Answer: A

Solution: [Knowledge Points] Finding the modulus of a complex number,Algebraic operations of addition and subtraction of complex numbers [Analysis] $z + 3 \mathrm { i } = 1 + \mathrm { i }$, according to the complex number of modulus $| z | = \sqrt { a ^ { 2 } + b ^ { 2 } }$ substitution calculation. [Explain]$\because z + 3 \mathrm { i } = 1 + \mathrm { i }$ , then $| z + 3 \mathrm { i } | = | 1 + \mathrm { i } | = \sqrt { 1 ^ { 2 } + 1 ^ { 2 } } = \sqrt { 2 }$

Question 20: Calculation: $( 5 - 6 \mathrm { i } ) + ( - 2 - \mathrm { i } ) - ( 3 + 4 \mathrm { i } ) = ( \quad ...

Calculation: $( 5 - 6 \mathrm { i } ) + ( - 2 - \mathrm { i } ) - ( 3 + 4 \mathrm { i } ) = ( \quad )$

  • A. A. 3
  • B. B. 4
  • C. C. - 11 i
  • D. D. -i

Answer: C

Solution: Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis] First remove the brackets, apply the addition and subtraction of complex numbers to simplify the complex numbers. [Details] $( 5 - 6 \mathrm { i } ) + ( - 2 - \mathrm { i } ) - ( 3 + 4 \mathrm { i } ) = 5 - 6 \mathrm { i } - 2 - \mathrm { i } - 3 - 4 \mathrm { i } = - 11 \mathrm { i }$ .

Question 21: It is known that the complex number $z$ satisfies $i \cdot z - 1 = i$ ,then $| z | =$

It is known that the complex number $z$ satisfies $i \cdot z - 1 = i$ ,then $| z | =$

  • A. A. 4
  • B. B. $2 \sqrt { 2 }$
  • C. C. 2
  • D. D. $\sqrt { 2 }$

Answer: D

Solution: Knowledge Points] Modulus of complex numbers, division of complex numbers [Analysis]According to the given conditions, use the division of complex numbers to find $Z$, and then use the definition of complex modulus to solve. By ${ } _ { \mathrm { i } \cdot \mathrm { Z } - 1 = \mathrm { i } }$, we get $\mathrm { Z } = \frac { 1 + \mathrm { i } } { \mathrm { i } } = 1 - \mathrm { i }$, so $| \mathrm { Z } | = \sqrt { 1 ^ { 2 } + ( - 1 ) ^ { 2 } } = \sqrt { 2 }$.

Question 22: If the complex number ${ } _ { Z }$ satisfies $\frac { Z + \mathrm { i } } { Z } = 1 - \mathrm { i }...

If the complex number ${ } _ { Z }$ satisfies $\frac { Z + \mathrm { i } } { Z } = 1 - \mathrm { i } ^ { 2 }$ ,then ${ } _ { Z } =$

  • A. A. i
  • B. B. 1
  • C. C. -i
  • D. D. - 1

Answer: A

Solution: Knowledge Points] Complex division operations, the equality of complex numbers [Analysis] According to the multiplication and division of complex numbers can be calculated. [Detailed Explanation]Because $\frac { z + \mathrm { i } } { z } = 1 - \mathrm { i } ^ { 2 } = 2$ ,so ${ } _ { z + \mathrm { i } = 2 z }$ ,so $z = i$ ,so $z = i$ . Solve $z = i$ .

Question 23: Let $i$ be an imaginary unit, then the complex number $\frac { 1 - i } { 2 + i } =$

Let $i$ be an imaginary unit, then the complex number $\frac { 1 - i } { 2 + i } =$

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

Answer: C

Solution: Knowledge Points] Complex numbers division operation [Analysis] Use the division operation of complex numbers to calculate the solution. [Details] $\frac { 1 - \mathrm { i } } { 2 + \mathrm { i } } = \frac { ( 1 - \mathrm { i } ) ( 2 - \mathrm { i } ) } { ( 2 + \mathrm { i } ) ( 2 - \mathrm { i } ) } = \frac { 1 - 3 \mathrm { i } } { 5 } = \frac { 1 } { 5 } - \frac { 3 } { 5 } \mathrm { i }$ .

Question 24: It is known that the complex number $Z _ { 1 } = 6 - 5 \mathrm { i } , Z _ { 2 } = 3 + 2 \mathrm { i...

It is known that the complex number $Z _ { 1 } = 6 - 5 \mathrm { i } , Z _ { 2 } = 3 + 2 \mathrm { i }$ , where ${ } ^ { \mathrm { i } }$ is in imaginary units, then $Z _ { 1 } + Z _ { 2 } =$

  • A. A. 9-3i
  • B. B. $9 + 3 \mathrm { i }$
  • C. C. 9-7i
  • D. D. $9 + 7 \mathrm { i }$

Answer: A

Solution: [Knowledge Points] Algebraic operations of addition and subtraction of complex numbers [Analysis]According to the addition of complex numbers can be solved. [Explanation] Because ${ } ^ { z _ { 1 } = 6 - 5 \mathrm { i } } , z _ { 2 } = 3 + 2 \mathrm { i }$ ,then $z _ { 1 } + z _ { 2 } = ( 6 - 5 \mathrm { i } ) + ( 3 + 2 \mathrm { i } ) = 9 - 3 \mathrm { i }$ .

Question 25: It is known that the complex number $z$ satisfies $z = \frac { ( 1 + \mathrm { i } ) ^ { 2 } } { 2 }...

It is known that the complex number $z$ satisfies $z = \frac { ( 1 + \mathrm { i } ) ^ { 2 } } { 2 }$ ,then $z ^ { 2023 } =$

  • A. A. - 1
  • B. B. 1
  • C. C. -i
  • D. D. i

Answer: C

Solution: Knowledge point] Multiplication of complex numbers [Analysis]First simplify the known z, then use the periodicity of i to solve the problem. [Explanation] According to the question, $z = \mathrm { i } , ~ \therefore z ^ { 2023 } = \mathrm { i } ^ { 2022 } \cdot \mathrm { i } = - \mathrm { i }$ .

Question 26: If the complex number $z$ satisfies $i z = - 1 + i$ ,then $| z | =$

If the complex number $z$ satisfies $i z = - 1 + i$ ,then $| z | =$

  • A. A. $\frac { 1 } { 2 }$
  • B. B. $\frac { \sqrt { 2 } } { 2 }$
  • C. C. $\sqrt { 2 }$
  • D. D. 2

Answer: C

Solution: Knowledge points] to find the mode of the complex number, the division operation of complex numbers [Analysis]Use the complex division method to find the complex number $z$, and then find the modulus can be. [Explanation] From $z = \frac { i - 1 } { i } = \frac { i ( i - 1 ) } { i ^ { 2 } } = i + 1$ , then $| z | = \sqrt { 2 }$ .

Question 27: If ${ } ^ { Z } = ( 1 - i ) ( 2 - i )$ , then the imaginary part of $z$ is

If ${ } ^ { Z } = ( 1 - i ) ( 2 - i )$ , then the imaginary part of $z$ is

  • A. A. - 3 i
  • B. B. 3
  • C. C. - 3
  • D. D. - 1 references

Answer: C

Solution: [Knowledge Points] Finding the real and imaginary parts of complex numbers, multiplication of complex numbers in algebraic form. [Analysis]First simplify the complex number $z$, then find the imaginary part of $z$. Since $z = ( 1 - i ) ( 2 - i ) = 1 - 3 i$, the imaginary part of $z$ is - 3.
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Complex Numbers

倍数

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