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JEE Advanced 2023 Paper 1 Solutions
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Section:
Mathematics
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© examsnet.com
Question : 51
Total: 51
Let
z
be a complex number satisfying
|
z
|
3
+
2
z
2
+
4
‌
bar
‌
z
−
8
=
0
, where
bar
‌
z
denotes the complex conjugate of
z
. Let the imaginary part of
z
be nonzero.Match each entry in List-I to the correct entries in List-II.
List-I
List-II
(P)
|
z
|
2
is equal to
(1) 12
(Q)
|
z
−
z
|
2
is equal to
(2) 4
(R)
|
z
|
2
+
|
z
+
z
|
2
is equal to
(3) 8
(S)
|
z
+
1
|
2
is equal to
(4) 10
(5) 7
The correct option is
[JEE Adv 2023 P1]
(P)
→
(1) (Q)
→
(3) (R)
→
(5) (S)
→
(4)
(
P
)
→
(
2
)
(
Q
)
→
(
1
)
(
R
)
→
(3) (S)
→
(5)
(
P
)
→
(
2
)
(
Q
)
→
(
4
)
(
R
)
→
(
5
)
(
S
)
→
(1)
(
P
)
→
(
2
)
(
Q
)
→
(
3
)
(
R
)
→
(
5
)
(
S
)
→
(4)
Validate
Solution:
|
z
|
3
+
2
z
2
+
4
z
−
8
=
0
|
z
|
3
+
2
z
2
+
4
z
−
8
=
0
────────────────
2
(
z
2
−
z
2
)
+
4
(
z
−
z
)
=
0
(
z
−
z
)
[
2
(
z
+
z
)
−
4
]
=
0
∵
z
=
z
not possible or
4
x
=
4
⇒
x
=
1
z
=
1
+
λ
i
⇒
|
z
|
=
√
1
+
λ
2
⇒
z
=
1
−
λ
i
(
1
+
λ
2
)
3
∕
2
+
2
(
1
−
λ
2
+
2
λ
i
)
+
4
(
1
−
λ
i
)
−
8
=
0
⇒
(
1
+
λ
2
)
3
∕
2
+
2
(
1
−
λ
2
)
=
4
(
1
+
λ
2
)
3
∕
2
=
2
(
1
+
λ
2
)
(
1
+
λ
2
)
[
√
1
+
λ
2
−
2
]
=
0
⇒
λ
2
=
3
Now
(P)
|
z
|
2
=
1
+
λ
2
=
1
+
3
=
4
(Q)
|
z
−
z
|
2
=
|
1
+
λ
i
−
(
1
−
λ
i
)
|
2
=
|
2
λ
i
|
2
=
4
λ
2
=
12
(R)
|
z
|
2
+
|
z
+
z
|
2
=
4
+
|
(
1
+
λ
i
)
+
(
1
−
λ
i
)
|
2
=
4
+
4
=
8
(S)
|
z
+
1
|
2
=
|
1
+
λ
i
+
1
|
2
=
4
+
λ
2
=
4
+
3
=
7
∴
P
→
(
2
)
,
Q
→
(
1
)
,
R
→
(
3
)
,
S
→
(
5
)
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