Question 1: Palette 1/70

JEE Advanced 2026 MCQ

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
(P) If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2026}}$ and $\frac{1}{(\beta+1)^{2026}}$ is	(1) $x^2 + x + 1 = 0$
(Q) If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2027}}$ and $\frac{1}{(\beta+1)^{2027}}$ is	(2) $x^2 - x + 1 = 0$
(R) If $\gamma$ and $\delta$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $\frac{1}{(\gamma-1)^{2026}} + \frac{1}{(\delta-1)^{2026}}$ is	(3) $x^2 + x - 1 = 0$
(S) If $p$ and $r$ are the distinct roots of the equation $x^2 + x - 1 = 0$, then the value of $\frac{1}{(p+1)^3} + \frac{1}{(r+1)^3}$ is	(4) $-1$
	(5) $-4$

A

(P) $\rightarrow$ (1), (Q) $\rightarrow$ (2), (R) $\rightarrow$ (5), (S) $\rightarrow$ (4)

B

(P) $\rightarrow$ (3), (Q) $\rightarrow$ (1), (R) $\rightarrow$ (4), (S) $\rightarrow$ (5)

C

(P) $\rightarrow$ (1), (Q) $\rightarrow$ (2), (R) $\rightarrow$ (4), (S) $\rightarrow$ (5)

D

(P) $\rightarrow$ (2), (Q) $\rightarrow$ (3), (R) $\rightarrow$ (5), (S) $\rightarrow$ (4)

Question 2: Palette 2/70

JEE Advanced 2026 MSQ

Let $\mathbb{R}$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices

$ S=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right] \quad \text { and } \quad T=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right] . $

Let $a, b, c, d$ be real numbers such that

$ S T=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] $

Let

$ H=\{x+i y: \quad x, y \in \mathbb{R} \text { and } y>0\} . $

Then which of the following statements is (are) TRUE ?

A

$\dfrac{b + i a}{d + i c} = i$

B

If $\omega = \dfrac{-1 + i \sqrt{3}}{2}$, then $\dfrac{a \omega + b}{c \omega + d} = \omega$

C

If $m$ is an integer greater than $2$ such that $(ST)^2 = (ST)^m$, then $m$ is an integer multiple of $8$

D

If $z \in H$, then $\dfrac{az + b}{cz + d} \in H$

Question 3: Palette 3/70

JEE Advanced 2025 MSQ

Let ℝ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$. Let

$S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2| \}.$

Then which of the following statements is (are) TRUE?

A

S is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$

B

S is a circle with centre $\left(\frac{1}{3}, \frac{8}{3} \right)$

C

S is a circle with radius $\frac{\sqrt{2}}{3}$

D

S is a circle with radius $\frac{2\sqrt{2}}{3}$

Question 4: Palette 4/70

JEE Advanced 2024 MSQ

Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

A

$\mathbb{Z} \cup T_1 \cup T_2 \subset S$

B

$T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set.

C

$T_2 \cap(2024, \infty) \neq \phi$

D

For any given $a, b \in \mathbb{Z}, \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in \mathbb{Z}$ if and only if $b=0$, where $i=\sqrt{-1}$.

Question 5: Palette 5/70

JEE Advanced 2023 MCQ

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) $

Question 6: Palette 6/70

JEE Advanced 2022 MCQ

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $ (\bar{z})^{2}+\frac{1}{z^{2}} $ are integers, then which of the following is/are possible value(s) of $|z|$ ?

A

$\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$

B

$\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$

C

$\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$

D

$\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$

Question 7: Palette 7/70

JEE Advanced 2021 MCQ

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\cdots+\theta_{10}=2 \pi$. Define the complex numbers $z_1=e^{i \theta_1}, z_k=z_{k-1} e^{i \theta_k}$ for $k=2,3, \ldots, 10$, 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

Question 8: Palette 8/70

JEE Advanced 2021 MSQ

For any complex number w = c + id, let $\arg (w) \in ( - \pi ,\pi ]$, where $i = \sqrt { - 1} $. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers z = x + iy satisfying $\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}$, the ordered pair (x, y) lies on the circle ${x^2} + {y^2} + 5x - 3y + 4 = 0$, Then which of the following statements is (are) TRUE?

A

$\alpha$ = $-$1

B

$\alpha$$\beta$ = 4

C

$\alpha$$\beta$ = $-$4

D

$\beta$ = 4

Question 9: Palette 9/70

JEE Advanced 2020 MSQ

Let S be the set of all complex numbers z
satisfying |z² + z + 1| = 1. Then which of the following statements is/are TRUE?

A

$\left| {z + {1 \over 2}} \right|$ $ \le $ ${{1 \over 2}}$ for all z$ \in $S

B

|z| $ \le $ 2 for all z$ \in $S

C

$\left| {z + {1 \over 2}} \right|\, \ge {1 \over 2}$ for all z$ \in $S

D

The set S has exactly four elements

Question 10: Palette 10/70

JEE Advanced 2019 MCQ

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

${\pi \over 2}$

Question 11: Palette 11/70

JEE Advanced 2018 MSQ

Let s, t, r be non-zero complex numbers and L be the set of solutions $z = x + iy(x,y \in R,\,i = \sqrt { - 1} )$ of the equation $sz + t\overline z + r = 0$ where $\overline z $ = x $-$ iy. Then, which of the following statement(s) is(are) TRUE?

A

If L has exactly one element, then |s|$ \ne $|t|

B

If |s| = |t|, then L has infinitely many elements

C

The number of elements in $L \cap \{ z:|z - 1 + i| = 5\} $ is at most 2

D

If L has more than one element, then L has infinitely many elements

Question 12: Palette 12/70

JEE Advanced 2018 MSQ

For a non-zero complex number z, let arg(z) denote the principal argument with $-$ $\pi $ < arg(z) $ \le $ $\pi $. Then, which of the following statement(s) is (are) FALSE?

A

arg($-$1$-$i) = ${\pi \over 4}$, where i = $\sqrt { - 1} $

B

The function f : R $ \to $ ($-$$\pi $, $\pi $), defined by f(t) = arg ($-$1 + it) for all t $ \in $ R, is continuous at all points of R, where i = $\sqrt { - 1} $.

C

For any two non-zero complex numbers z₁ and z₂, arg $\left( {{{{z_1}} \over {{z_2}}}} \right)$$-$ arg (z₁) + arg(z₂) is an integer multiple of 2$\pi $.

D

For any three given distinct complex numbers z₁, z₂ and z₃, the locus of the point z satisfying the condition arg$\left( {{{(z - {z_1})({z_2} - {z_3})} \over {(z - {z_3})({z_2} - {z_1})}}} \right) = \pi $, lies on a straight line.

Question 13: Palette 13/70

JEE Advanced 2017 MSQ

Let a, b, x and y be real numbers such that a $-$ b = 1 and y $ \ne $ 0. If the complex number z = x + iy satisfies ${\mathop{\rm Im}\nolimits} \left( {{{az + b} \over {z + 1}}} \right) = y$, then which of the following is(are) possible value(s) of x?

A

$1 - \sqrt {1 + {y^2}} $

B

$ - 1 - \sqrt {1 - {y^2}} $

C

$1 + \sqrt {1 + {y^2}} $

D

$ - 1 + \sqrt {1 - {y^2}} $

Question 14: Palette 14/70

JEE Advanced 2016 MSQ

Let $a,\,b \in R\,and\,{a^{2\,}} + {b^2} \ne 0$. Suppose
$S = \left\{ {Z \in C:Z = {1 \over {a + ibt}}, + \in R,t \ne 0} \right\}$, where $i = \sqrt { - 1} $. Ifz = x + iy and z $ \in $ S, then (x, y) lies on

A

the circle with radius ${{1 \over {2a}}}$and centre $\left\{ {{1 \over {2a}},\,0} \right\}\,for\,a > 0\,,b \ne \,0$

B

the circle with radius $-{{1 \over {2a}}}$and centre $\left\{ -{{1 \over {2a}},\,0} \right\}\,for\,a < 0\,,b \ne \,0$

C

the x-axis for $a \ne \,\,0,\,b \ne \,0$

D

the y-axis for $a = \,\,0,\,b \ne \,0$

Question 15: Palette 15/70

JEE Advanced 2014 MCQ

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

Question 16: Palette 16/70

JEE Advanced 2013 MCQ

Let $S = {S_1} \cap {S_2} \cap {S_3}$, where ${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}$ and ${S_3} = \left\{ {z \in C:{\mathop{\rm Re}\nolimits} z > 0} \right\}\,$.

$\,\mathop {\min }\limits_{z \in S} \left| {1 - 3i - z} \right| = $

A

${{2 - \sqrt 3 } \over 2}$

B

${{2 + \sqrt 3 } \over 2}$

C

${{3 - \sqrt 3 } \over 2}$

D

${{3 + \sqrt 3 } \over 2}$

Question 17: Palette 17/70

JEE Advanced 2013 MCQ

Let $S = {S_1} \cap {S_2} \cap {S_3}$, where ${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}$ and ${S_3} = \left\{ {z \in C:{\mathop{\rm Re}\nolimits} z > 0} \right\}\,$.

Area of S =

A

${{10\pi } \over 3}$

B

${{20\pi } \over 3}$

C

${{16\pi } \over 3}$

D

${{32\pi } \over 3}$

Question 18: Palette 18/70

JEE Advanced 2013 MCQ

Let complex numbers $\alpha \,and\,{1 \over {\overline \alpha }}\,$ lie on circles ${\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = {r^2}$ and $\,{\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = 4{r^2}$ respextively. If ${z_0} = {x_0} + i{y_0}$ satisfies the equation $2{\left| {{z_0}} \right|^2}\, = {r^2} + 2,\,then\,\left| a \right| = $

A

${1 \over {\sqrt 2 }}$

B

${1 \over 2}\,$

C

${1 \over {\sqrt 7 }}$

D

${1 \over 3}$

Question 19: Palette 19/70

JEE Advanced 2013 MSQ

Let $\omega=\frac{\sqrt{3}+i}{2}$ and $P=\left\{\omega^n: n=1,2,3, \ldots\right\}$. Further

$\mathrm{H}_1=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{1}{2}\right\}$ and

$\mathrm{H}_2=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{-1}{2}\right\}$, where C is the

set of all complex numbers. If $z_1 \in \mathrm{P} \cap \mathrm{H}_1, z_2 \in$ $\mathrm{P} \cap \mathrm{H}_2$ and O

represents the origin, then $\angle z_1 \mathrm{O} z_2=$

A

${\pi \over 2}$

B

${\pi \over 6}\,$

C

${{2\pi } \over 3}$

D

${{5\pi } \over 6}$

Question 20: Palette 20/70

JEE Advanced 2012 MCQ

Let z be a complex number such that the imaginary part of z is non-zero and $a\, = \,{z^2} + \,z\, + 1$ is real. Then a cannot take the value

A

- 1

B

${1 \over 3}$

C

${1 \over 2}$

D

${3 \over 4}$

Question 21: Palette 21/70

JEE Advanced 2010 MCQ

Match the statements in Column I with those in Column II.

[Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively, the imaginary part and the real part of z.]

Column I

(A) The set of points z satisfying $\left| {z - i} \right|\left. {z\,} \right\|\,\, = \left| {z + i} \right|\left. {\,z} \right\|$ is contained in or equal to
(B) The set of points z satisfying $\left| {z + 4} \right| + \,\left| {z - 4} \right| = 10$ is contained in or equal to
(C) If $\left| w \right|$= 2, then the set of points $z = w - {1 \over w}$ is contained in or equal to
(D) If $\left| w \right|$ = 1, then the set of points $z = w + {1 \over w}$ is contained in or equal to.

Column II

(p) an ellipse with eccentricity ${4 \over 5}$
(q) the set of points z satisfying Im z = 0
(r) the set of points z satisfying $\left| {{\rm{Im }}\,{\rm{z }}} \right| \le 1$
(s) the set of points z satisfying $\,\left| {{\mathop{\rm Re}\nolimits} \,\,z} \right| < 2$
(t) the set of points z satisfying $\left| {\,z} \right| \le 3$

A

(A) - q, s ; (B) - p ; (C) - p, t ; (D) - q, r, s, t

B

(A) - q, r ; (B) - p ; (C) - p, s, t ; (D) - q, r, s, t

C

(A) - p, r ; (B) - p ; (C) - p, t ; (D) -q, r, s, t

D

(A) - p ; (B) - q ; (C) - r, s ; (D) -q, r, s, t

Question 22: Palette 22/70

JEE Advanced 2010 MSQ

Let ${{z_1}}$ and ${{z_2}}$ be two distinct complex number and let z =( 1 - t)${{z_1}}$ + t${{z_2}}$ for some real number t with 0 < t < 1. IfArg (w) denote the principal argument of a non-zero complex number w, then

A

$\left| {z - {z_1}} \right| + \left| {z - {z_2}} \right| = \left| {{z_1} - {z_2}} \right|$

B

Arg $(z - {z_1})$ = Arg$(z - {z_2})$

C

$\left| {\matrix{ {z - {z_1}} & {\overline z - {{\overline z }_1}} \cr {{z_2} - {z_1}} & {{{\overline z }_2} - {{\overline z }_1}} \cr } } \right|$ = 0

D

Arg $(z - {z_1})$ = Arg$({z_2} - {z_1})$

Question 23: Palette 23/70

JEE Advanced 2010 MSQ

Let $z_1$ and $z_2$ be two distinct complex numbers let $z=(1-t) z_1+t z_2$ for some real number t with $0 < t < 1$.

If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$, then :

A

$\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$

B

$\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z-z_2\right)$

C

$\left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right|=0$

D

$\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z_2-z_1\right)$

Question 24: Palette 24/70

JEE Advanced 2009 MCQ

Let $z = x + iy$ be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation $\overline z {z^3} + z{\overline z ^3} = 350$ is

A

48

B

32

C

40

D

80

Question 25: Palette 25/70

JEE Advanced 2009 MCQ

Let $z = \,\cos \,\theta \, + i\,\sin \,\theta $ . Then the value of $\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\theta \, = {2^ \circ }$ is

A

${1 \over {\sin \,{2^ \circ }}}$

B

${1 \over {3\sin \,{2^ \circ }}}$

C

${1 \over {2\sin \,{2^ \circ }}}$

D

${1 \over {4\sin \,{2^ \circ }}}$

Question 26: Palette 26/70

JEE Advanced 2008 MCQ

A particle P stats from the point ${z_0}$ = 1 +2i, where $i = \sqrt { - 1} $. It moves horizontally away from origin by 5 unit and then vertically away from origin by 3 units to reach a point ${z_1}$. From ${z_1}$ the particle moves $\sqrt 2 $ units in the direction of the vector $\hat i + \hat j$ and then it moves through an angle ${\pi \over 2}$ in anticlockwise direction on a circle with centre at origin, to reach a point ${z_2}$. The point ${z_2}$ is given by

A

6 + 7i

B

-7 + 6i

C

7 + 6i

D

- 6 + 7i

Question 27: Palette 27/70

JEE Advanced 2008 MCQ

Let z be any point in $A \cap B \cap C$

Then, ${\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}$ lies between :

A

25 and 29

B

30 and 34

C

35 and 39

D

40 and 44

Question 28: Palette 28/70

JEE Advanced 2008 MCQ

Let z be any point $A \cap B \cap C$ and let w be any point satisfying $\left| {w - 2 - i} \right| < 3\,$. Then, $\left| z \right| - \left| w \right| + 3$ lies between :

A

- 6 and 3

B

- 3 and 6

C

- 6 and 6

D

- 3 and 9

Question 29: Palette 29/70

JEE Advanced 2008 MCQ

The number of elements in the set $A \cap B \cap C$ is

A

0

B

1

C

2

D

$\infty $

Question 30: Palette 30/70

JEE Advanced 2007 MCQ

If $\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,$ then all the values of ${z \over {1 - {z^2}}}$ lie on

A

a line not passing through the origin

B

$\left| z \right|\, = \,\sqrt 2 $

C

the x-axis

D

the y-axis

Question 31: Palette 31/70

JEE Advanced 2007 MCQ

A man walks a distance of 3 units from the origin towards the north-east ($N\,{45^ \circ E }$) direction. From there, he walks a distance of 4 units towards the north-west $\left( {N\,{{45}^ \circ }\,W} \right)$ direction to reach a point P. Then the position of P in the Argand plane is

A

$3{e^{i\pi /4}} + 4i$

B

$\left( {3 - 4i} \right){e^{i\pi /4}}$

C

$\left( {4 + 3i} \right){e^{i\pi /4}}$

D

$\left( {3 + 4i} \right){e^{i\pi /4}}$

Question 32: Palette 32/70

JEE Advanced 2007 MCQ

If $|z|=1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^{2}}$ lie on

A

a line not passing through the origin

B

$|z|=\sqrt{2}$

C

the X-axis

D

the Y-axis

Question 33: Palette 33/70

JEE Advanced 2007 MCQ

A man walks a distance of 3 units from the origin towards the north-east (N 45$^\circ$E) direction. From there, he walks a distance of 4 units towards the north-west (N 45$^\circ$W) direction to reach a point P. Then the position of P in the Argand plane is

A

$3{e^{i\pi /4}} + 4i$

B

$(3 - 4i){e^{i\pi /4}}$

C

$(4 + 3i){e^{i\pi /4}}$

D

$(3 + 4i){e^{i\pi /4}}$

Question 34: Palette 34/70

JEE Advanced 2006 MCQ

If $w=\alpha+\mathrm{i} \beta$, where $\beta \neq 0$ and $z \neq 1$, satisfies the condition that $\left(\frac{w-\bar{w} z}{1-z}\right)$ is purely real, then the set of values of $z$ is:

A

$\{z:|z|=1\}$

B

$\{z: z=\vec{z}\}$

C

$\{z: z \neq z\}$

D

$\{z:|z|=1, z \neq 1 \mid\}$

Question 35: Palette 35/70

JEE Advanced 2006 MCQ

If $P$ is a point on $C_1$ and $Q$ in another point on $\mathrm{C}_2$, then $\frac{\mathrm{PA}^2+\mathrm{PB}^2+\mathrm{PC}^2+\mathrm{PD}^2}{\mathrm{QA}^2+\mathrm{QB}^2+\mathrm{QC}^2+\mathrm{QD}^2}$ is equal to :

A

0.75

B

1.25

C

1

D

0.5

Question 36: Palette 36/70

JEE Advanced 2005 MCQ

$a,\,b,\,c$ are integers, not all simultaneously equal and $\omega $ is cube root of unity $\left( {\omega \ne 1} \right),$ then minimum value of $\left| {a + b\omega + c{\omega ^2}} \right|$ is

A

0

B

1

C

${{\sqrt 3 } \over 2}$

D

${1 \over 2}$

Question 37: Palette 37/70

JEE Advanced 2005 MCQ

If one of the vertices of the square circumscribing the circle $|z-1|=\sqrt{2}$ is $(2+\sqrt{3 i})$. Find the other vertices of square.

A

$\left( {1 - 2\sqrt 3 } \right) + i,\left( {1 + \sqrt 3 } \right) - i, - \sqrt 3 i$

B

$\left( {1 - \sqrt 3 } \right) + i,\left( {2 + \sqrt 3 } \right) - i, - i$

C

$\left( {1 - \sqrt 3 } \right) + i,\left( {1 + 2\sqrt 3 } \right) - i, - \sqrt 5 i$

D

$\left( {1 - \sqrt 3 } \right) + i,\left( {1 + \sqrt 3 } \right) - i, - \sqrt 3 i$

Question 38: Palette 38/70

JEE Advanced 2004 MCQ

If $\omega $ $\left( { \ne 1} \right)$ be a cube root of unity and ${\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n},$ then the least positive value of n is

A

2

B

3

C

5

D

6

Question 39: Palette 39/70

JEE Advanced 2003 MCQ

If $\,\left| z \right| = 1$ and $\omega = {{z - 1} \over {z + 1}}$ (where $z \ne - 1$), then ${\mathop{\rm Re}\nolimits} \left( \omega \right)$ is

A

0

B

$ - {1 \over {{{\left| {z + 1} \right|}^2}}}$

C

$\left| {{z \over {z + 1}}} \right|.{1 \over {{{\left| {z + 1} \right|}^2}}}$

D

$\,{{\sqrt 2 } \over {{{\left| {z + 1} \right|}^2}}}$

Question 40: Palette 40/70

JEE Advanced 2002 MCQ

Let $\omega $ $ = - {1 \over 2} + i{{\sqrt 3 } \over 2},$ then the value of the det.
$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|$ is

A

$3\omega $

B

$3\omega \left( {\omega - 1} \right)$

C

$3{\omega ^2}$

D

$3\omega \left( {1 - \omega } \right)$

Question 41: Palette 41/70

JEE Advanced 2002 MCQ

For all complex numbers ${z_1},\,{z_2}$ satisfying $\left| {{z_1}} \right| = 12$ and $\left| {{z_2} - 3 - 4i} \right| = 5,$
the minimum value of $\left| {{z_1} - {z_2}} \right|$ is

A

0

B

2

C

7

D

17

Question 42: Palette 42/70

JEE Advanced 2001 MCQ

The complex numbers ${z_1},\,{z_2}$ and ${z_3}$ satisfying ${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,$ are the vertices of a triangle which is

A

of area zero

B

right-angled isosceles

C

equilateral

D

obtuse-angled isosceles

Question 43: Palette 43/70

JEE Advanced 2001 MCQ

Let ${z_1}$ and ${z_2}$ be ${n^{th}}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form

A

$4k + 1$

B

$4k + 2$

C

$4k + 3$

D

$4k$

Question 44: Palette 44/70

JEE Advanced 2000 MCQ

If ${z_1},\,{z_2}$ and ${z_3}$ are complex numbers such that $\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right| = 1,$ then $\left| {{z_1} + {z_2} + {z_3}} \right|$ is

A

equal to 1

B

less than 1

C

greater than 3

D

equal to 3

Question 45: Palette 45/70

JEE Advanced 2000 MCQ

If $\arg \left( z \right) < 0,$ then $\arg \left( { - z} \right) - \arg \left( z \right) = $

A

$\pi $

B

$ - \pi $

C

$ - {\pi \over 2}$

D

${\pi \over 2}$

Question 46: Palette 46/70

JEE Advanced 1999 MCQ

$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$ is equal to

A

$1 - i\sqrt 3 $

B

$ - 1 + i\sqrt 3 $

C

$i\sqrt 3 $

D

$ - i\sqrt 3 $

Question 47: Palette 47/70

JEE Advanced 1998 MCQ

If $\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$ , then

A

x = 3, y = 2

B

x = 1, y = 3

C

x = 0, y = 3

D

x = 0, y = 0

Question 48: Palette 48/70

JEE Advanced 1998 MCQ

If ${\omega}$ is an imaginary cube root of unity, then ${(1\, + \omega \, - {\omega ^2})^7}$ equals

A

$128\omega $

B

$ - 128\omega $

C

$128{\omega ^2}$

D

$ - 128{\omega ^2}$

Question 49: Palette 49/70

JEE Advanced 1998 MCQ

The value of the sum $\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$ , where i = $\sqrt { - 1} $, equals

A

i

B

i - 1

C

- i

D

0

Question 50: Palette 50/70

JEE Advanced 1996 MCQ

For positive integers ${n_1},\,{n_2}$ the value of the expression ${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$
where $i = \sqrt { - 1} $ is real number if and only if

A

${n_1} = {n_2} + 1$

B

${n_1} = {n_2} - 1$

C

${n_1} = {n_2}$

D

${n_1} > 0,\,{n_2} > 0$

Question 51: Palette 51/70

JEE Advanced 1995 MCQ

Let $z$ and $\omega $ be two complex numbers such that
$\left| z \right| \le 1,$ $\left| \omega \right| \le 1$ and $\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2$ then $z$ equals

A

$1$ or $i$

B

$i$ or $-i$

C

$1$ or $ - 1$

D

$i$ or $ - 1$

Question 52: Palette 52/70

JEE Advanced 1995 MCQ

Let $z$ and $\omega $ be two non zero complex numbers such that
$\left| z \right| = \left| \omega \right|$ and ${\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,$ then $z$ equals

A

$\omega $

B

$ - \omega $

C

$\overline \omega $

D

$ - \overline \omega $

Question 53: Palette 53/70

JEE Advanced 1995 MCQ

If $\omega \,\left( { \ne 1} \right)$ is a cube root of unity and ${\left( {1 + \omega } \right)^7} = A + B\,\omega $ then $A$ and $B$ are respectively

A

0, 1

B

1, 1

C

1, 0

D

-1, 1

Question 54: Palette 54/70

JEE Advanced 1992 MCQ

${\rm{z }} \ne {\rm{0}}$ is a complex number

Column I

(A) Re z = 0
(B) Arg $z = {\pi \over 4}$

Column II

(p) Re${z^2}$ = 0
(q) Im${z^2}$ = 0
(r) Re${z^2}$ = Im${z^2}$

A

(A) - q, (B) - p

B

(A) - p, (B) - q

C

(A) - r, (B) - p

D

(A) - p, (B) - r

Question 55: Palette 55/70

JEE Advanced 1988 MCQ

The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.

A

TRUE

B

FALSE

Question 56: Palette 56/70

JEE Advanced 1987 MCQ

If ${{{z_1}}}$ and ${{{z_2}}}$ are two nonzero complex numbers such that $\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{z_2}} \right|\,$, then Arg ${z_1}$ - Arg ${z_2}$ is equal to

A

$ - \pi $

B

$ - {\pi \over 2}$

C

0

D

${\pi \over 2}$

Question 57: Palette 57/70

JEE Advanced 1987 MCQ

The value of $\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$ is

A

- 1

B

0

C

- i

D

i

Question 58: Palette 58/70

JEE Advanced 1986 MSQ

Let ${z_1}$ and ${z_2}$ be complex numbers such that ${z_1}$ $ \ne $ ${z_2}$ and $\left| {{z_1}} \right| =\,\left| {{z_2}} \right|$. If ${z_1}$ has positive real and ${z_2}$ has negative imaginary part, then ${{{z_1}\, + \,{z_2}} \over {{z_1}\, - \,{z_2}}}$ may be

A

zero

B

real and positive

C

real and negative

D

purely imaginary

Question 59: Palette 59/70

JEE Advanced 1985 MCQ

If $a,\,b,\,c$ and $u,\,v,\,w$ are complex numbers representing the vertics of two triangles such that $c = \left( {1 - r} \right)a + rb$ and $w = \left( {1 - r} \right)u + rv,$ where $w = \left( {1 - r} \right)u + rv,$ is a complex number, then the two triangles

A

have the same area

B

are similar

C

are congruent

D

none of these

Question 60: Palette 60/70

JEE Advanced 1985 MSQ

If ${z_1}$ = a + ib and ${z_2}$ = c + id are complex numbers such that $\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1$ and ${\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0$, then the pair of complex numbers ${w_1}$ = a + ic and ${w_2}$ = b+ id satisfies -

A

$\left| {{w_1}} \right| = 1\,$

B

$\left| {{w_2}} \right| = 1\,$

C

${\mathop{\rm Re}\nolimits} ({w_1}\,{\overline w _2}) = 0$

D

none of these

Question 61: Palette 61/70

JEE Advanced 1985 MCQ

If three complex numbers are in A.P. then they lie on a circle in the complex plane.

A

TRUE

B

FALSE

Question 62: Palette 62/70

JEE Advanced 1984 MCQ

If the complex numbers, ${Z_1},{Z_2}$ and ${Z_3}$ represent the vertics of an equilateral triangle such that
$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$ then ${Z_1} + {Z_2} + {Z_3} = 0.$

A

TRUE

B

FALSE

Question 63: Palette 63/70

JEE Advanced 1983 MCQ

If $z = x + iy$ and $\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$ then $\,\left| \omega \right| = 1$ implies that, in the complex plane,

A

$z$ lies on the imaginary axis

B

$z$ lies on the real axis

C

$z$ lies on the unit circle

D

none of these

Question 64: Palette 64/70

JEE Advanced 1983 MCQ

The points z₁, z₂, z₃, z₄ in the complex plane are the vertices of a parallelogram taken in order if and only if

A

z₁ + z₄ = z₂ + z₃

B

z₁ + z₃ = z₂ + z₄

C

z₁ + z₂ = z₃ + z₄

D

None of these

Question 65: Palette 65/70

JEE Advanced 1982 MCQ

The inequality |z-4| < |z-2| represents the region given by

A

${\mathop{\rm Re}\nolimits} \left( z \right) \ge 0\,\,$

B

${\mathop{\rm Re}\nolimits} \left( z \right) < 0$

C

${\mathop{\rm Re}\nolimits} \left( z \right) > 0$

D

none of these

Question 66: Palette 66/70

JEE Advanced 1982 MCQ

If $z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},$ then

A

${\mathop{\rm Re}\nolimits} \left( z \right) = 0$

B

${\rm I}m\left( z \right) = 0$

C

${\mathop{\rm Re}\nolimits} \left( z \right) > 0,\,{\rm I}m\left( z \right) > 0\,$

D

${\mathop{\rm Re}\nolimits} \left( z \right) > 0,\,{\rm I}m\left( z \right) < 0$

Question 67: Palette 67/70

JEE Advanced 1981 MCQ

The complex numbers $z = x + iy$ which satisfy the equation $\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$ lie on

A

the x-axis

B

the straight line y=5

C

a circle passing through the origin

D

none of these

Question 68: Palette 68/70

JEE Advanced 1981 MCQ

For complex number ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2},$ we write ${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$
Then for all complex numbers $z\,\,with\,\,1 \cap z,$ we have ${{1 - z} \over {1 + z}} \cap 0.$

A

TRUE

B

FALSE

Question 69: Palette 69/70

JEE Advanced 1980 MCQ

The smallest positive integer n for which ${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$ is

A

$n = 8$

B

$n = 16$

C

$n = 12$

D

none of these

Question 70: Palette 70/70

JEE Advanced 1979 MCQ

If the cube roots of unity are $1,\,\omega ,\,{\omega ^2},$ then the roots of the equation ${\left( {x - 1} \right)^3} + 8 = 0$ are

A

$ - 1,1 + 2\omega ,\,1 + 2{\omega ^2}$

B

$ - 1,1 - 2\omega ,\,1 - 2{\omega ^2}$

C

$ - 1, - 1, - 1$

D

None of these