1
JEE Advanced 2023 Paper 1 Online
+3
-1
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) $|z-\bar{z}|^2$ is equal to (2) 4
(R) $|z|^2+|z+\bar{z}|^2$ is equal to (3) 8
(S) $|z+1|^2$ is equal to (4) 10
(5) 7

The correct option is:
A
$$(P) \rightarrow(1) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4)$$
B
$$(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(3) \quad(S) \rightarrow(5)$$
C
$$(P) \rightarrow(2) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(5) \quad(S) \rightarrow(1)$$
D
$$(P) \rightarrow(2) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4)$$
2
JEE Advanced 2021 Paper 1 Online
+3
-1 Let $$\theta$$1, $$\theta$$2, ........, $$\theta$$10 = 2$$\pi$$. Define the complex numbers z1 = ei$$\theta$$1, zk = zk $$-$$ 1ei$$\theta$$k for k = 2, 3, ......., 10, where i = $$\sqrt { - 1}$$. Consider the statements P and Q given below :

$$P:\left| {{z_2} - {z_1}} \right| + \left| {{z_3} - {z_2}} \right| + ..... + \left| {{z_{10}} - {z_9}} \right| + \left| {{z_1} - {z_{10}}} \right| \le 2\pi$$

$$Q:\left| {z_2^2 - z_1^2} \right| + \left| {z_3^2 - z_2^2} \right| + .... + \left| {z_{10}^2 - z_9^2} \right| + \left| {z_1^2 - z_{10}^2} \right| \le 4\pi$$

Then,
A
P is TRUE and Q is FALSE
B
Q is TRUE and P is FALSE
C
both P and Q are TRUE
D
both P and Q are FALSE
3
JEE Advanced 2019 Paper 1 Offline
+3
-1 Let S be the set of all complex numbers z satisfying $$\left| {z - 2 + i} \right| \ge \sqrt 5$$. If the complex number z0 is such that $${1 \over {\left| {{z_0} - 1} \right|}}$$ is the maximum of the set $$\left\{ {{1 \over {\left| {{z_0} - 1} \right|}}:z \in S} \right\}$$, then the principal argument of $${{4 - {z_0} - {{\overline z }_0}} \over {{z_0} - {{\overline z }_0} + 2i}}$$ is
A
$${\pi \over 4}$$
B
$${3\pi \over 4}$$
C
$$-$$$${\pi \over 2}$$
D
$${\pi \over 2}$$
4
JEE Advanced 2014 Paper 2 Offline
+4
-1
Let $${z_k}$$ = $$\cos \left( {{{2k\pi } \over {10}}} \right) + i\,\,\sin \left( {{{2k\pi } \over {10}}} \right);\,k = 1,2....,9$$

List-I

P. For each $${z_k}$$ = there exits as $${z_j}$$ such that $${z_k}$$.$${z_j}$$ = 1
Q. There exists a $$k \in \left\{ {1,2,....,9} \right\}$$ such that $${z_1}.z = {z_k}$$ has no solution z in the set of complex numbers
R. $${{\left| {1 - {z_1}} \right|\,\left| {1 - {z_2}} \right|\,....\left| {1 - {z_9}} \right|} \over {10}}$$ equals
S. $$1 - \sum\limits_{k = 1}^9 {\cos \left( {{{2k\pi } \over {10}}} \right)}$$ equals

List-II

1. True
2. False
3. 1
4. 2
A
P = 1, Q = 2, R = 4, S = 3
B
P = 2, Q = 1, R = 3, S = 4
C
P = 1, Q = 2, R = 3, S = 4
D
P =2, Q = 1, R = 4, S = 3
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