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JEE Advanced 2022 Paper 1
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© examsnet.com
Question : 4
Total: 54
Let
z
be a complex number with non-zero imaginary part. If
2
+
3
z
+
4
z
2
2
−
3
z
+
4
z
2
is a real number, then the value of
|
z
|
2
is_____
Your Answer:
Validate
Solution:
Given that
z
≠
z
Let
α
=
2
+
3
z
+
4
z
2
2
−
3
z
+
4
z
2
=
(
2
−
3
z
+
4
z
2
)
+
6
z
2
−
3
z
+
4
z
2
∴
α
=
1
+
6
z
2
−
3
z
+
4
z
2
If
α
is a real number, then
α
=
α
⇒
z
2
−
3
z
+
4
z
2
=
z
2
−
3
z
+
4
z
2
∴
2
(
z
−
z
)
=
4
z
z
(
z
−
z
)
⇒
(
z
−
z
)
(
2
−
4
z
z
)
=
0
As
z
≠
z
(
Given
)
⇒
z
z
=
2
4
=
1
2
⇒
|
z
|
2
=
0.50
© examsnet.com
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