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JEE Advanced

Complex Numbers

Mathematics

Previous Years Questions

MCQ (More than One Correct Answer)

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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 $$-$$ $$\pi $$ < 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
If $$\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i ...
IIT-JEE 1998
The value of the sum $$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$ , where i = $$\sqrt { - 1} $$, equals
IIT-JEE 1998
If $${\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

MCQ (Single Correct Answer)

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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| < 4} \right\},{S_2} = \lef...
JEE Advanced 2013 Paper 2 Offline
Let $$S = {S_1} \cap {S_2} \cap {S_3}$$, where $${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \lef...
JEE Advanced 2013 Paper 2 Offline
Let complex numbers $$\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 = 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 $$z = \,\cos \,\theta \, + i\,\sin \,\theta $$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\noli...
IIT-JEE 2009
Let A, B, C be three sets of complex numbers as defined below $$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
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
The complex numbers $${z_1},\,{z_2}$$ and $${z_3}$$ satisfying $${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt ...
IIT-JEE 2001 Screening
If $$\arg \left( z \right) < 0,$$ then $$\arg \left( { - z} \right) - \arg \left( z \right) = $$
IIT-JEE 2000 Screening
If $${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
$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - ...
IIT-JEE 1999
For positive integers $${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
If $$\omega \,\left( { \ne 1} \right)$$ is a cube root of unity and $${\left( {1 + \omega } \right)^7} = A + B\,\omega $...
IIT-JEE 1995 Screening
Let $$z$$ and $$\omega $$ be two complex numbers such that $$\left| z \right| \le 1,$$ $$\left| \omega \right| \le 1$$...
IIT-JEE 1995 Screening
$${\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 $$\,\left| \omega \right| = 1...
IIT-JEE 1983
The inequality |z-4| < |z-2| represents the region given by
IIT-JEE 1982
If $$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

Numerical

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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

Subjective

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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| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{...
IIT-JEE 2003
If $${z_1}$$ and $${z_2}$$ are two complex numbers such that $$\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \righ...
IIT-JEE 2003
Let a complex number $$\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

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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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