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Joint Entrance Examination
Mathematics
Complex Numbers
Question 313
Question 313 of 455
Medium
MCQ
If
z
z
z
is a complex number such that
∣
z
∣
≥
2
|z|\ge2
∣
z
∣
≥
2
, then the minimum value of
∣
z
+
1
2
∣
\left|z+\dfrac12\right|
z
+
2
1
:
A
is strictly greater than
5
2
\dfrac52
2
5
B
is strictly greater than
3
2
\dfrac32
2
3
but less than
5
2
\dfrac52
2
5
C
is equal to
5
2
\dfrac52
2
5
D
lies in the interval (1,2)
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Official solution
∣
∣
z
∣
−
1
2
∣
≤
∣
z
+
1
2
∣
≤
∣
z
∣
+
1
2
\left||z|-\dfrac12\right|\le\left|z+\dfrac12\right|\le|z|+\dfrac12
∣
z
∣
−
2
1
≤
z
+
2
1
≤
∣
z
∣
+
2
1
. With
∣
z
∣
≥
2
|z|\ge2
∣
z
∣
≥
2
:
3
2
<
∣
z
+
1
2
∣
<
5
2
\dfrac32<\left|z+\dfrac12\right|<\dfrac52
2
3
<
z
+
2
1
<
2
5
, and this range includes values in
(
1
,
2
)
(1,2)
(
1
,
2
)
; the correct interval statement is (1,2).
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