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JEE Main

Complex Numbers

Mathematics

Previous Years Questions

MCQ (Single Correct Answer)

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Let $$\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|...
JEE Main 2022 (Online) 29th July Evening Shift
If $$z \neq 0$$ be a complex number such that $$\left|z-\frac{1}{z}\right|=2$$, then the maximum value of $$|z|$$ is :...
JEE Main 2022 (Online) 29th July Evening Shift
If $$z=2+3 i$$, then $$z^{5}+(\bar{z})^{5}$$ is equal to :
JEE Main 2022 (Online) 29th July Morning Shift
Let $$S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}$$ and $$S_{2}=\left\{z_{2} \in \mathbf{...
JEE Main 2022 (Online) 28th July Morning Shift
Let S be the set of all $$(\alpha, \beta), \pi...
JEE Main 2022 (Online) 27th July Evening Shift
Let the minimum value $$v_{0}$$ of $$v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$$ is attained at $${ }{z}=z_{0}$...
JEE Main 2022 (Online) 27th July Morning Shift
If $$z=x+i y$$ satisfies $$|z|-2=0$$ and $$|z-i|-|z+5 i|=0$$, then
JEE Main 2022 (Online) 26th July Evening Shift
Let O be the origin and A be the point $${z_1} = 1 + 2i$$. If B is the point $${z_2}$$, $${\mathop{\rm Re}\nolimits} ({z...
JEE Main 2022 (Online) 26th July Morning Shift
For $$z \in \mathbb{C}$$ if the minimum value of $$(|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)$$ is $$5 \sqrt{2}$$, then a value Q...
JEE Main 2022 (Online) 25th July Evening Shift
For $$\mathrm{n} \in \mathbf{N}$$, let $$\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\...
JEE Main 2022 (Online) 25th July Morning Shift
If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^{4}+x^{3}+x^{2}+x+1=0$$, then $$\alpha^{2021}+\be...
JEE Main 2022 (Online) 25th July Morning Shift
The real part of the complex number $${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$...
JEE Main 2022 (Online) 30th June Morning Shift
Let $$A = \{ z \in C:1 \le |z - (1 + i)| \le 2\} $$ and $$B = \{ z \in A:|z - (1 - i)| = 1\} $$. Then, B :
JEE Main 2022 (Online) 24th June Morning Shift
Let a circle C in complex plane pass through the points $${z_1} = 3 + 4i$$, $${z_2} = 4 + 3i$$ and $${z_3} = 5i$$. If $$...
JEE Main 2022 (Online) 25th June Morning Shift
Let z1 and z2 be two complex numbers such that $${\overline z _1} = i{\overline z _2}$$ and $$\arg \left( {{{{z_1}} \ove...
JEE Main 2022 (Online) 25th June Evening Shift
Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| ...
JEE Main 2022 (Online) 26th June Morning Shift
The area of the polygon, whose vertices are the non-real roots of the equation $$\overline z = i{z^2}$$ is :
JEE Main 2022 (Online) 27th June Morning Shift
The number of points of intersection of $$|z - (4 + 3i)| = 2$$ and $$|z| + |z - 4| = 6$$, z $$\in$$ C, is...
JEE Main 2022 (Online) 27th June Evening Shift
Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z $$-$$ 1) $$-$$ arg(z + 1) =...
JEE Main 2022 (Online) 29th June Evening Shift
Let $$\alpha$$ and $$\beta$$ be the roots of the equation x2 + (2i $$-$$ 1) = 0. Then, the value of |$$\alpha$$8 + $$\be...
JEE Main 2022 (Online) 29th June Morning Shift
If z is a complex number such that $${{z - i} \over {z - 1}}$$ is purely imaginary, then the minimum value of | z $$-$$ ...
JEE Main 2021 (Online) 31st August Evening Shift
If $$S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}$$, then :
JEE Main 2021 (Online) 27th August Morning Shift
If $${\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)$$, then p and q are roots of the equation :
JEE Main 2021 (Online) 26th August Evening Shift
The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with :
JEE Main 2021 (Online) 26th August Morning Shift
Let C be the set of all complex numbers. Let$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} $$$${S_2} = \{ z \in C|{\mathop{...
JEE Main 2021 (Online) 27th July Morning Shift
Let C be the set of all complex numbers. LetS1 = {z$$\in$$C : |z $$-$$ 2| $$\le$$ 1} and S2 = {z$$\in$$C : z(1 + i) + $$...
JEE Main 2021 (Online) 27th July Evening Shift
Let n denote the number of solutions of the equation z2 + 3$$\overline z $$ = 0, where z is a complex number. Then the v...
JEE Main 2021 (Online) 22th July Evening Shift
If z and $$\omega$$ are two complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$\arg (z) - \arg (\omega ) ...
JEE Main 2021 (Online) 20th July Morning Shift
Let a complex number be w = 1 $$-$$ $${\sqrt 3 }$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$...
JEE Main 2021 (Online) 18th March Evening Shift
If the equation $$a|z{|^2} + \overline {\overline \alpha z + \alpha \overline z } + d = 0$$ represents a circle where ...
JEE Main 2021 (Online) 18th March Morning Shift
Let S1, S2 and S3 be three sets defined asS1 = {z$$\in$$C : |z $$-$$ 1| $$ \le $$ $$\sqrt 2 $$}S2 = {z$$\in$$C : Re((1 $...
JEE Main 2021 (Online) 17th March Evening Shift
The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :
JEE Main 2021 (Online) 17th March Morning Shift
The least value of |z| where z is complex number which satisfies the inequality $$\exp \left( {{{(|z| + 3)(|z| - 1)} \ov...
JEE Main 2021 (Online) 16th March Evening Shift
Let a complex number z, |z| $$\ne$$ 1, satisfy $${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2...
JEE Main 2021 (Online) 16th March Morning Shift
If $$\alpha$$, $$\beta$$ $$\in$$ R are such that 1 $$-$$ 2i (here i2 = $$-$$1) is a root of z2 + $$\alpha$$z + $$\beta$$...
JEE Main 2021 (Online) 25th February Evening Shift
Let the lines (2 $$-$$ i)z = (2 + i)$$\overline z $$ and (2 $$+$$ i)z + (i $$-$$ 2)$$\overline z $$ $$-$$ 4i = 0, (here ...
JEE Main 2021 (Online) 25th February Morning Slot
Let z = x + iy be a non-zero complex number such that $${z^2} = i{\left| z \right|^2}$$, where i = $$\sqrt { - 1} $$ , t...
JEE Main 2020 (Online) 6th September Evening Slot
The region represented by {z = x + iy $$ \in $$ C : |z| – Re(z) $$ \le $$ 1} is also given by the inequality: {z = x + i...
JEE Main 2020 (Online) 6th September Morning Slot
The value of $${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$ is :
JEE Main 2020 (Online) 5th September Evening Slot
If the four complex numbers $$z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)$$ ...
JEE Main 2020 (Online) 5th September Morning Slot
If a and b are real numbers such that $${\left( {2 + \alpha } \right)^4} = a + b\alpha $$ where $$\alpha = {{ - 1 + i\s...
JEE Main 2020 (Online) 4th September Evening Slot
Let $$u = {{2z + i} \over {z - ki}}$$, z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersect...
JEE Main 2020 (Online) 4th September Morning Slot
If z1 , z2 are complex numbers such that Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1| , and arg(z1 - z2) = $${\pi \over 6}$$, ...
JEE Main 2020 (Online) 3rd September Evening Slot
The imaginary part of $${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1...
JEE Main 2020 (Online) 2nd September Evening Slot
The value of $${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\co...
JEE Main 2020 (Online) 2nd September Morning Slot
If z be a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be
JEE Main 2020 (Online) 9th January Evening Slot
Let z be complex number such that $$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$ and |z| = $${5 \over 2}$$. Then the...
JEE Main 2020 (Online) 9th January Morning Slot
If the equation, x2 + bx + 45 = 0 (b $$ \in $$ R) has conjugate complex roots and they satisfy |z +1| = 2$$\sqrt {10} $$...
JEE Main 2020 (Online) 8th January Morning Slot
If $${{3 + i\sin \theta } \over {4 - i\cos \theta }}$$, $$\theta $$ $$ \in $$ [0, 2$$\theta $$], is a real number, then ...
JEE Main 2020 (Online) 7th January Evening Slot
If $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1$$, where z = x + iy, then the point (x, y)...
JEE Main 2020 (Online) 7th January Morning Slot
Let z $$ \in $$ C with Im(z) = 10 and it satisfies $${{2z - n} \over {2z + n}}$$ = 2i - 1 for some natural number n. The...
JEE Main 2019 (Online) 12th April Evening Slot
The equation |z – i| = |z – 1|, i = $$\sqrt { - 1} $$, represents :
JEE Main 2019 (Online) 12th April Morning Slot
If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = $${\pi \over 2}$$ , then:
JEE Main 2019 (Online) 10th April Evening Slot
If a > 0 and z = $${{{{\left( {1 + i} \right)}^2}} \over {a - i}}$$, has magnitude $$\sqrt {{2 \over 5}} $$, then $$...
JEE Main 2019 (Online) 10th April Morning Slot
Let z $$ \in $$ C be such that |z| < 1. If $$\omega = {{5 + 3z} \over {5(1 - z)}}$$z, then:-
JEE Main 2019 (Online) 9th April Evening Slot
All the points in the set $$S = \left\{ {{{\alpha + i} \over {\alpha - i}}:\alpha \in R} \right\}(i = \sqrt { - 1} )$...
JEE Main 2019 (Online) 9th April Morning Slot
If $$z = {{\sqrt 3 } \over 2} + {i \over 2}\left( {i = \sqrt { - 1} } \right)$$, then (1 + iz + z5 + iz8)9 is equal to...
JEE Main 2019 (Online) 8th April Evening Slot
If $$\alpha $$ and $$\beta $$ be the roots of the equation x2 – 2x + 2 = 0, then the least value of n for which $${\left...
JEE Main 2019 (Online) 8th April Morning Slot
Let z1 and z2 be two complex numbers satisfying | z1 | = 9 and | z2 – 3 – 4i | = 4. Then the minimum value of | z1 – z2 ...
JEE Main 2019 (Online) 12th January Evening Slot
If $${{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)$$ is a purely imaginary number and | z | = 2, the...
JEE Main 2019 (Online) 12th January Morning Slot
Let z be a complex number such that |z| + z = 3 + i (where i = $$\sqrt { - 1} $$). Then |z| is equal to
JEE Main 2019 (Online) 11th January Evening Slot
Let $${\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,$$ where x...
JEE Main 2019 (Online) 11th January Morning Slot
Let $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)...
JEE Main 2019 (Online) 10th January Evening Slot
Let z1 and z2 be any two non-zero complex numbers such that   $$3\left| {{z_1}} \right| = 4\left| {{z_2}} \rig...
JEE Main 2019 (Online) 10th January Morning Slot
Let z0 be a root of the quadratic equation, x2 + x + 1 = 0, If z = 3 + 6iz$$_0^{81}$$ $$-$$ 3iz$$_0^{93}$$, then arg z ...
JEE Main 2019 (Online) 9th January Evening Slot
Let A = $$\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}...
JEE Main 2019 (Online) 9th January Morning Slot
Let $$\alpha $$ and $$\beta $$ be two roots of the equation x2 + 2x + 2 = 0 , then $$\alpha ^{15}$$ + $$\beta ^{15}$$ is...
JEE Main 2019 (Online) 9th January Morning Slot
The least positive integer n for which $${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$$ is :
JEE Main 2018 (Online) 16th April Morning Slot
If |z $$-$$ 3 + 2i| $$ \le $$ 4 then the difference between the greatest value and the least value of |z| is :
JEE Main 2018 (Online) 15th April Evening Slot
The set of all $$\alpha $$ $$ \in $$ R, for which w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ is purely...
JEE Main 2018 (Online) 15th April Morning Slot
If $$\alpha ,\beta \in C$$ are the distinct roots of the equation x2 - x + 1 = 0, then $${\alpha ^{101}} + {\beta ^{107...
JEE Main 2018 (Offline)
Let $$\omega $$ be a complex number such that 2$$\omega $$ + 1 = z where z = $$\sqrt {-3} $$. If $$\left| {\matrix{ 1...
JEE Main 2017 (Offline)
The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from ther...
JEE Main 2016 (Online) 9th April Morning Slot
A value of $$\theta \,$$ for which $${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$$ is purely imaginary, ...
JEE Main 2016 (Offline)
A complex number z is said to be unimodular if $$\,\left| z \right| = 1$$. Suppose $${z_1}$$ and $${z_2}$$ are complex n...
JEE Main 2015 (Offline)
If z is a complex number such that $$\,\left| z \right| \ge 2\,$$, then the minimum value of $$\,\,\left| {z + {1 \over ...
JEE Main 2014 (Offline)
If z is a complex number of unit modulus and argument $$\theta $$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}...
JEE Main 2013 (Offline)
If $$z \ne 1$$ and $$\,{{{z^2}} \over {z - 1}}\,$$ is real, then the point represented by the complex number z lies :
AIEEE 2012
If $$\omega ( \ne 1)$$ is a cube root of unity, and $${(1 + \omega )^7} = A + B\omega \,$$. Then $$(A,B)$$ equals
AIEEE 2011
Let $$\alpha \,,\beta $$ be real and z be a complex number. If $${z^2} + \alpha z + \beta = 0$$ has two distinct roots ...
AIEEE 2011
The number of complex numbers z such that $$\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|$$ e...
AIEEE 2010
Let R be the real line. Consider the following subsets of the plane $$R \times R$$ : $$S = \left\{ {(x,y):y = x + 1\,\,...
AIEEE 2008
The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is
AIEEE 2008
If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is
AIEEE 2007
The value of $$\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)} ...
AIEEE 2006
If $${z^2} + z + 1 = 0$$, where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^...
AIEEE 2006
If $${z_1}$$ and $${z_2}$$ are two non-zero complex numbers such that $$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}...
AIEEE 2005
If the cube roots of unity are 1, $$\omega \,,\,{\omega ^2}$$ then the roots of the equation $${(x - 1)^3}$$ + 8 = 0, ar...
AIEEE 2005
If $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$, then $$z$$ lies on
AIEEE 2005
If $$\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$$, then z lies on
AIEEE 2004
If $$z = x - iy$$ and $${z^{{1 \over 3}}} = p + iq$$, then $${{\left( {{x \over p} + {y \over q}} \right)} \over {\left...
AIEEE 2004
Let z and w be complex numbers such that $$\overline z + i\overline w = 0$$ and arg zw = $$\pi $$. Then arg z equals ...
AIEEE 2004
If $${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$ then
AIEEE 2003
Let $${Z_1}$$ and $${Z_2}$$ be two roots of the equation $${Z^2} + aZ + b = 0$$, Z being complex. Further , assume that ...
AIEEE 2003
If $$z$$ and $$\omega $$ are two non-zero complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$Arg(z) - Arg...
AIEEE 2003
The locus of the centre of a circle which touches the circle $$\left| {z - {z_1}} \right| = a$$ and$$\left| {z - {z_2}} ...
AIEEE 2002
If $$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by
AIEEE 2002
z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ ...
AIEEE 2002

Numerical

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Let $$\mathrm{z}=a+i b, b \neq 0$$ be complex numbers satisfying $$z^{2}=\bar{z} \cdot 2^{1-z}$$. Then the least value ...
JEE Main 2022 (Online) 28th July Evening Shift
Let $$S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$$. Then $$\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorn...
JEE Main 2022 (Online) 27th July Morning Shift
Let S = {z $$\in$$ C : |z $$-$$ 3| $$\le$$ 1 and z(4 + 3i) + $$\overline z $$(4 $$-$$ 3i) $$\le$$ 24}. If $$\alpha$$ + i...
JEE Main 2022 (Online) 24th June Evening Shift
If $${z^2} + z + 1 = 0$$, $$z \in C$$, then $$\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {...
JEE Main 2022 (Online) 26th June Evening Shift
The number of elements in the set {z = a + ib $$\in$$ C : a, b $$\in$$ Z and 1 ...
JEE Main 2022 (Online) 28th June Morning Shift
Sum of squares of modulus of all the complex numbers z satisfying $$\overline z = i{z^2} + {z^2} - z$$ is equal to ____...
JEE Main 2022 (Online) 28th June Evening Shift
Let $$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\} $$. Let $$|z - 4i|$$ attains minimum and max...
JEE Main 2022 (Online) 29th June Morning Shift
If for the complex numbers z satisfying | z $$-$$ 2 $$-$$ 2i | $$\le$$ 1, the maximum value of | 3iz + 6 | is attained a...
JEE Main 2021 (Online) 1st September Evening Shift
A point z moves in the complex plane such that $$\arg \left( {{{z - 2} \over {z + 2}}} \right) = {\pi \over 4}$$, then ...
JEE Main 2021 (Online) 31st August Morning Shift
The least positive integer n such that $${{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}},i = \sqrt { - 1} $$ is a positive in...
JEE Main 2021 (Online) 26th August Evening Shift
Let $$z = {{1 - i\sqrt 3 } \over 2}$$, $$i = \sqrt { - 1} $$. Then the value of $$21 + {\left( {z + {1 \over z}} \right)...
JEE Main 2021 (Online) 26th August Morning Shift
The equation of a circle is Re(z2) + 2(Im(z))2 + 2Re(z) = 0, where z = x + iy. A line which passes through the center of...
JEE Main 2021 (Online) 25th July Evening Shift
If the real part of the complex number $$z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\p...
JEE Main 2021 (Online) 27th July Evening Shift
Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{ 0 & i \cr 1 & 0 \cr } } \right)}^n}\left( {\mat...
JEE Main 2021 (Online) 25th July Morning Shift
Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the ...
JEE Main 2021 (Online) 18th March Morning Shift
Let z and $$\omega$$ be two complex numbers such that $$\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}...
JEE Main 2021 (Online) 16th March Morning Shift
Let z be those complex numbers which satisfy| z + 5 | $$ \le $$ 4 and z(1 + i) + $$\overline z $$(1 $$-$$ i) $$ \ge $$ $...
JEE Main 2021 (Online) 26th February Evening Shift
Let $$i = \sqrt { - 1} $$. If $${{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 ...
JEE Main 2021 (Online) 24th February Evening Slot
If the least and the largest real values of a, for which the equation z + $$\alpha $$|z – 1| + 2i = 0 (z $$ \in $$ C and...
JEE Main 2021 (Online) 24th February Morning Slot
If $${\left( {{{1 + i} \over {1 - i}}} \right)^{{m \over 2}}} = {\left( {{{1 + i} \over {1 - i}}} \right)^{{n \over 3}}}...
JEE Main 2020 (Online) 3rd September Morning Slot

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