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## MCQ (More than One Correct Answer)

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Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both r...
JEE Advanced 2022 Paper 2 Online
For any complex number w = c + id, let $$\arg (w) \in ( - \pi ,\pi ]$$, where $$i = \sqrt { - 1}$$. Let $$\alpha$$ and ...
JEE Advanced 2021 Paper 1 Online
Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRU...
JEE Advanced 2020 Paper 1 Offline
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 t...
JEE Advanced 2018 Paper 2 Offline
For a non-zero complex number z, let arg(z) denote the principal argument with $$-$$ $$\pi$$ &lt; arg(z) $$\le$$ $$\p... JEE Advanced 2018 Paper 1 Offline Let a, b, x and y be real numbers such that a$$-$$b = 1 and y$$ \ne $$0. If the complex number z = x + iy satisfies ... JEE Advanced 2017 Paper 1 Offline Let$$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 \n...
JEE Advanced 2016 Paper 2 Offline
Let $$w = {{\sqrt 3 + i} \over 2}$$ and P = { $${w^n}$$ : n = 1, 2, 3, ...}. Further and , where is the set of all com...
JEE Advanced 2013 Paper 2 Offline
Let $${{z_1}}$$ and $${{z_2}}$$ be two distinct complex number and let z =( 1 - t)$${{z_1}}$$ + t$${{z_2}}$$ for some re...
IIT-JEE 2010 Paper 1 Offline
The value of the sum $$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$ , where i = $$\sqrt { - 1}$$, equals
IIT-JEE 1998
If $$\,\left| {\matrix{ {6i} &amp; { - 3i} &amp; 1 \cr 4 &amp; {3i} &amp; { - 1} \cr {20} &amp; 3 &amp; i ... IIT-JEE 1998 If$${\omega}$$is an imaginary cube root of unity, then$${(1\, + \omega \, - {\omega ^2})^7}$$equals IIT-JEE 1998 The value of$$\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$$is IIT-JEE 1987 If$${{{z_1}}}$$and$${{{z_2}}}$$are two nonzero complex numbers such that$$\left| {{z_1}\, + {z_2}} \right| = \left|...
IIT-JEE 1987
Let $${z_1}$$ and $${z_2}$$ be complex numbers such that $${z_1}$$ $$\ne$$ $${z_2}$$ and $$\left| {{z_1}} \right| =\... IIT-JEE 1986 If$${z_1}$$= a + ib and$${z_2}$$= c + id are complex numbers such that$$\left| {{z_1}} \right| = \left| {{z_2}} \ri...
IIT-JEE 1985

## Numerical

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Let $$\bar{z}$$ denote the complex conjugate of a complex number $$z$$ and let $$i=\sqrt{-1}$$. In the set of complex nu...
JEE Advanced 2022 Paper 1 Online
Let $$z$$ be a complex number with a non-zero imaginary part. If $$\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}$$ is a real n...
JEE Advanced 2022 Paper 1 Online
For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying $${z^... JEE Advanced 2020 Paper 2 Offline Let$$\omega \ne 1$$be a cube root of unity. Then the maximum of the set$$\{ {\left| {a + b\omega + c{\omega ^2}} \r...
JEE Advanced 2019 Paper 1 Offline
For any integer k, let $${a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)... JEE Advanced 2015 Paper 2 Offline Let$$\omega = {e^{{{i\pi } \over 3}}}$$, and a, b, c, x, y, z be non-zero complex numbers such that$$a + b + c = x$$... IIT-JEE 2011 Paper 2 Offline If z is any complex number satisfying$$\,\left| {z - 3 - 2i} \right| \le 2$$, then the minimum value of$$\left| {2z - ...
IIT-JEE 2011 Paper 1 Offline

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Let $$\theta$$1, $$\theta$$2, ........, $$\theta$$10 = 2$$\pi$$. Define the complex numbers z1 = ei$$\theta$$1, zk = zk ...
JEE Advanced 2021 Paper 1 Online
Let S be the set of all complex numbers z satisfying $$\left| {z - 2 + i} \right| \ge \sqrt 5$$. If the complex number ...
JEE Advanced 2019 Paper 1 Offline
Let $${z_k}$$ = $$\cos \left( {{{2k\pi } \over {10}}} \right) + i\,\,\sin \left( {{{2k\pi } \over {10}}} \right);\,k = 1... JEE Advanced 2014 Paper 2 Offline Let$$S = {S_1} \cap {S_2} \cap {S_3}$$, where$${S_1} = \left\{ {z \in C:\left| z \right| &lt; 4} \right\},{S_2} = \lef...
JEE Advanced 2013 Paper 2 Offline
Let $$S = {S_1} \cap {S_2} \cap {S_3}$$, where $${S_1} = \left\{ {z \in C:\left| z \right| &lt; 4} \right\},{S_2} = \lef... JEE Advanced 2013 Paper 2 Offline Let complex numbers$$\alpha \,and\,{1 \over {\overline \alpha }}\,$$lie on circles$${\left( {x - {x_0}} \right)^2} +...
JEE Advanced 2013 Paper 1 Offline
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 ...
IIT-JEE 2012 Paper 1 Offline
Match the statements in Column I with those in Column II. [Note : Here z takes value in the complex plane and Im z ...
IIT-JEE 2010 Paper 2 Offline
Let $$z = \,\cos \,\theta \, + i\,\sin \,\theta$$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\noli... IIT-JEE 2009 Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roo... IIT-JEE 2009 Let A, B, C be three sets of complex numbers as defined below$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, ...
IIT-JEE 2008
Let A, B, C be three sets of complex numbers as defined below $$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, ... IIT-JEE 2008 Let A, B, C be three sets of complex numbers as defined below$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, ...
IIT-JEE 2008
A particle P stats from the point $${z_0}$$ = 1 +2i, where $$i = \sqrt { - 1}$$. It moves horizontally away from origin...
IIT-JEE 2008
If $$\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,$$ then all the values of $${z \over {1 - {z^2}}}$$ lie on
IIT-JEE 2007
A man walks a distance of 3 units from the origin towards the north-east ($$N\,{45^ \circ E }$$) direction. From there, ...
IIT-JEE 2007
If $${{w - \overline w z} \over {1 - z}}$$ is purely real where $$w = \alpha + i\beta ,$$ $$\beta \ne 0$$ and $$z \ne ... IIT-JEE 2006$$a,\,b,\,c$$are integers, not all simultaneously equal and$$\omega $$is cube root of unity$$\left( {\omega \ne 1} ...
IIT-JEE 2005 Screening
If $$\omega$$ $$\left( { \ne 1} \right)$$ be a cube root of unity and $${\left( {1 + {\omega ^2}} \right)^n} = {\left( ... IIT-JEE 2004 Screening If$$\,\left| z \right| = 1$$and$$\omega = {{z - 1} \over {z + 1}}$$(where$$z \ne - 1$$), then$${\mathop{\rm Re}...
IIT-JEE 2003 Screening
Let $$\omega$$ $$= - {1 \over 2} + i{{\sqrt 3 } \over 2},$$ then the value of the det. $$\,\left| {\matrix{ 1 &am... IIT-JEE 2002 For all complex numbers$${z_1},\,{z_2}$$satisfying$$\left| {{z_1}} \right| = 12$$and$$\left| {{z_2} - 3 - 4i} \righ...
IIT-JEE 2002 Screening
The complex numbers $${z_1},\,{z_2}$$ and $${z_3}$$ satisfying $${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt ... IIT-JEE 2001 Screening Let$${z_1}$$and$${z_2}$$be$${n^{th}}$$roots of unity which subtend a right angle at the origin. Then$$n$$must be... IIT-JEE 2001 Screening If$${z_1},\,{z_2}$$and$${z_3}$$are complex numbers such that$$\left| {{z_1}} \right| = \left| {{z_2}} \right| = \le...
IIT-JEE 2000 Screening
If $$\arg \left( z \right) &lt; 0,$$ then $$\arg \left( { - z} \right) - \arg \left( z \right) =$$
IIT-JEE 2000 Screening
$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - ... IIT-JEE 1999 For positive integers$${n_1},\,{n_2}$$the value of the expression$${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 +...
IIT-JEE 1996
Let $$z$$ and $$\omega$$ be two non zero complex numbers such that $$\left| z \right| = \left| \omega \right|$$ and...
IIT-JEE 1995 Screening
Let $$z$$ and $$\omega$$ be two complex numbers such that $$\left| z \right| \le 1,$$ $$\left| \omega \right| \le 1$$...
IIT-JEE 1995 Screening
If $$\omega \,\left( { \ne 1} \right)$$ is a cube root of unity and $${\left( {1 + \omega } \right)^7} = A + B\,\omega ... IIT-JEE 1995 Screening$${\rm{z }} \ne {\rm{0}}$$is a complex number Column I (A) Re z = 0 (B) Arg$$z = {\pi \over 4}$$Column II ... IIT-JEE 1992 If$$a,\,b,\,c$$and$$u,\,v,\,w$$are complex numbers representing the vertics of two triangles such that$$c = \left( ...
IIT-JEE 1985
The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order if and only if...
IIT-JEE 1983
If $$z = x + iy$$ and $$\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$$ then $$\,\left| \omega \right| = 1... IIT-JEE 1983 The inequality |z-4| &lt; |z-2| represents the region given by IIT-JEE 1982 If$$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^...
IIT-JEE 1982
The complex numbers $$z = x + iy$$ which satisfy the equation $$\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$$ lie o...
IIT-JEE 1981
The smallest positive integer n for which $${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$$ is
IIT-JEE 1980
If the cube roots of unity are $$1,\,\omega ,\,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3... IIT-JEE 1979 ## Subjective More If one the vertices of the square circumscribing the circle$$\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i...
IIT-JEE 2005
Find the centre and radius of circle given by $$\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$$ ...
IIT-JEE 2004
Prove that there exists no complex number z such that $$\left| z \right| &lt; {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{... IIT-JEE 2003 If$${z_1}$$and$${z_2}$$are two complex numbers such that$$\,\left| {{z_1}} \right| &lt; 1 &lt; \left| {{z_2}} \righ...
IIT-JEE 2003
Let a complex number $$\alpha ,\,\alpha \ne 1$$, be a root of the equation $${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, whe...
IIT-JEE 2002
For complex numbers z and w, prove that $${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$ if and only if $$z =... IIT-JEE 1999 Let$${z_1}$$and$${z_2}$$be roots of the equation$${z^2} + pz + q = 0\,$$, where the coefficients p and q may be co... IIT-JEE 1997 Find all non-zero complex numbers Z satisfying$$\overline Z = i{Z^2}$$. IIT-JEE 1996 If$$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that$${\left| {Z - W} \right|^2} \le {(\left| Z \right...
IIT-JEE 1995
If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.
IIT-JEE 1995
Let $${z_1}$$ = 10 + 6i and $${z_2}$$ = 4 + 6i. If Z is any complex number such that the argument of $${{(z - {z_1})} \o... IIT-JEE 1990 Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is$${1 \over 2}...
IIT-JEE 1986
If 1, $${{a_1}}$$, $${{a_2}}$$......,$${a_{n - 1}}$$ are the n roots of unity, then show that (1- $${{a_1}}$$) (1- $${{a... IIT-JEE 1984 Prove that the complex numbers$${{z_1}}$$,$${{z_2}}$$and the origin form an equilateral triangle only if$$z_1^2 + z_...
IIT-JEE 1983
Let the complex number $${{z_1}}$$, $${{z_2}}$$ and $${{z_3}}$$ be the vertices of an equilateral triangle. Let $${{z_0}... IIT-JEE 1981 Find the real values of x and y for which the following equation is satisfied$$\,{{(1 + i)x - 2i} \over {3 + i}} + {{(2...
IIT-JEE 1980
If x + iy = $$\sqrt {{{a + ib} \over {c + id}}}$$, prove that $${({x^2} + {y^2})^2} = {{{a^2} + {b^2}} \over {{c^2} + {... IIT-JEE 1979 Express$${1 \over {1 - \cos \,\theta + 2i\sin \theta }}$$in the form x + iy. IIT-JEE 1978 If x = a + b, y = a$$\gamma $$+ b$$\beta $$and z = a$$\beta $$+b$$\gamma $$where$$\gamma $$and$$\beta $$are th... IIT-JEE 1978 ## Fill in the Blanks More The value of the expression$$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( ...
IIT-JEE 1996
Suppose Z1, Z2, Z3 are the vertices of an equilateral triangle inscribed in the circle $$\left| Z \right| = 2.$$ If Z1 =...
IIT-JEE 1994
$$ABCD$$ is a rhombus. Its diagonals $$AC$$ and $$BD$$ intersect at the point $$M$$ and satisfy $$BD$$ = 2$$AC$$. If the...
IIT-JEE 1993
If $$a,\,b,\,c,$$ are the numbers between 0 and 1 such that the ponts $${z_1} = a + i,{z_2} = 1 + bi$$ and $${z_3} = 0$$...
IIT-JEE 1989
For any two complex numbers $${z_1},{z_2}$$ and any real number a and b. $$\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\le... IIT-JEE 1988 If the expression$$\${{\left[ {\sin \left( {{x \over 2}} \right) + \cos {x \over 2} + i\,\tan \left( x \right)} \right]}...
IIT-JEE 1987

## True or False

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The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.
IIT-JEE 1988
If three complex numbers are in A.P. then they lie on a circle in the complex plane.
IIT-JEE 1985
If the complex numbers, $${Z_1},{Z_2}$$ and $${Z_3}$$ represent the vertics of an equilateral triangle such that $$\lef... IIT-JEE 1984 For complex number$${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...
IIT-JEE 1981

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