First, multiple choice questions
1. The false proposition in the following proposition is ()
A. the modulus of a complex number is a non-negative real number.
The necessary and sufficient condition for a complex number to be equal to zero is that its modulus is equal to zero.
C. the modulus equality of two complex numbers is a necessary condition for the equality of these two complex numbers.
D. complex number z1>; The necessary and sufficient condition of z2 is | z 1 | > | z2 |
[answer] d
【 Analysis 】 ① The module | z | = A2+B2 ≥ 0 of any complex number Z = A+Bi (A, b∈R) will always hold. ∴ A is correct;
(2) What is the condition that complex number equals z = 0? a=0b=0。 ? | z |z|=0, so b is correct;
③ if z 1 = a 1+b 1i, z2 = a2+b2i (a 1, b 1, a2, b2∈R).
If z 1 = z2, then a 1 = a2, b 1 = B2, ∴| z 1 | = | z2 |
On the other hand, from | z 1 | = | z2|, we cannot deduce z 1 = z2.
For example, when z 1 = 1+3i, z2 = 1-3i | z 1 | = | z2 |, then c is correct;
④ Two complex numbers that are not all zero cannot compare sizes, but the modules of any two complex numbers can always compare sizes. D is wrong, so I chose D.
2. Given a and b∈R, the positional relationship between two points corresponding to complex numbers A-Bi and -A-Bi on the complex plane is ().
A. About X-axis symmetry
B. About Y-axis symmetry
C. symmetry about the origin
D. symmetry of straight line y = x
[answer] b
[Analysis] The two corresponding points of complex numbers A-bi and -a-bi on the complex plane are (a, -b) and (-a, -b) which are symmetric about Y. 。
3. The following conclusion is correct ()
On the complex plane, the x axis is called the real axis and the y axis is called the imaginary axis.
B. No two complex numbers can compare sizes.
C. if the real number a corresponds to the pure imaginary number ai, then the real number set and the pure imaginary number set are in one-to-one correspondence.
The square root of D.- 1 is i.
[Answer] A.
[Analysis] Two imaginary numbers cannot be compared to exclude B. When a = 0, ai is a real number, excluding C, and the square root of-1 is I, excluding D, so choose A.
4. The point corresponding to the complex number z = (a2-2a)+(a2-a-2) i is on the imaginary axis, then ().
A.a ≠ 2 or a≠ 1
B.a ≠ 2 or a ≦- 1
C.a = 2 or a = 0.
Digital and analog =0
[answer] d
[Resolution] According to the meaning of the question, A2-2A = 0, A2-A-2 ≠ 0.
The solution is a = 0.
5. There is _ _ _ _ _ in the following formula. ()
①3i>2i ②|2+3i| >|-2-3i |③I2 & gt; (-i)2
④ | z | = | z| (where z is the complex number of the * * * yoke of the complex number z)
a . 0 b . 1
C.2 D.3
[answer] b
[Resolution] The imaginary numbers 3i and 2i are not comparable in size; |2+3i|= 13,|-2-3i|= 13,∴|2+3i|=|-2-3i|; I2 =- 1,(-I) 2 =- 1,∴ I2 = (-I) 2。 Let z = A+Bi (A, b∈R), then | z | = A2+B2, | z | = A2.
6. if the complex number z 1 = a+2i (a ∈ r), z2 = 2+i and | z 1 | < |z2|, then the range of a is ().
A.( 1,+∞)b .(-∞,- 1)
C.(- 1, 1)d .(-∞,- 1)∞( 1,+∞)
[answer] c
[resolution] ∵| z1| <| z2 |, ∴ a2+4 < 5,∴a2+4<; 5,
∴- 1 < a < 1。 So, C.
7. In the complex plane, the complex number represented by the vector OA→ is 1+I, and the vector O ′ A→ is obtained by shifting OA→ to the right by one unit, so the complex numbers corresponding to the point A→ are () respectively.
A. 1+i, 1+I b . 2+i,2+I
C. 1+i,2+I ` d . 2+I, 1+i
[answer] c
[Resolution] After the vector OA→ the starting point O'( 1, 0) is shifted to the right by one unit,
∫OA′→= OO ′→+ O′A′→= OO ′→+ OA→=( 1,0)+( 1, 1)=(2, 1),
∴ Point A' corresponds to the complex number 2+I, o' a' → = OA →
∴O'A'→ the corresponding complex number is1+i. Therefore, C.
At the age of 8.23
A. The first quadrant B. The second quadrant
C. The third quadrant D. The fourth quadrant
[answer] d
[resolution ]∫23 < m < 1, ∴ 3m-2 > 0, m- 1 0 and m2-3m-3 >; 0,∴m= 15.
14. If the complex number z satisfies | z+3-3i | = 3, then the maximum and minimum values of |z| are _ _ _ _ _ _ _.
[Answer] 33, 3
[Analysis] | z+3-3i | = 3 represents a circle with c (-3,3) as the center and 3 as the radius, so |z| represents the distance from the point on the circle to the origin. Obviously, the maximum value of |z| is | oc |+3 = 23+3 = 33, and the minimum value is | oc |-3 =.
Third, answer questions.
15. If the inequality m2-(m2-3m) i.
[Resolution] According to the meaning of the question, m2-3m = 0m2-4m+3 = 0m2.
∴ m = 0 or m = 3m = 3 or m = 1 | m | < 10,
When m = 3, the original inequality holds.
16. If the point corresponding to the complex number z = (m2+m-1)+(4m2-8m+3) i (m ∈ r) is in the first quadrant, then the value range of the realistic number m is determined.
[Resolution] ∫z =(m2+m- 1)+(4 m2-8m+3)I,
∴z=(m2+m- 1)-(4m2-8m+3)i.
M2+M- 1 > 04m 2-8m+3 & lt; 0, the solution is-1+52.
That is, the range of the real number m is-1+52.
17. Given a∈R, in which quadrant does the complex number Z = (A2-2A+4)-(A2-2A+2) I correspond? What is the locus of the corresponding point of complex number Z?
[Resolution] A2-2A+4 = (A- 1) 2+3 ≥ 3,-(A2-2A+2) =-(a-1) 2-1.
The real part of z is positive and the imaginary part is negative.
The corresponding point of the complex number z is in the fourth quadrant.
Let z = x+yi (x, y∈R), then x = a2-2a+4y =-(a2-2a+2).
Y =-x+2 (x ≥ 3) By eliminating A2-2a,
∴ The locus of the corresponding point of complex number Z is a ray, and its equation is y =-x+2 (x ≥ 3).
18. The known complex numbers Z 1 = 3-i and Z 2 =- 12+32i.
(1) Find the values of |z 1| and |z2| and compare their sizes;
(2) Let z∈C, what is the locus of the point Z that satisfies the condition |z2|≤|z|≤|z 1|?
[Resolution] (1) | z 1 | = | 3+I |
=(3)2+ 12=2
|z2|=- 12-32i= 1。
∴|z 1|>|z2|.
(2) From |z2|≤|z|≤|z 1|, 1≤|z|≤2.
Because |z|≥ 1 represents the set of all points outside the circle | z | = 1.
|z|≤2 represents the set of all points in a circle | z | = 2.
∴ 1≤|z|≤2 represents a ring as shown in the figure.