# Complex Numbers · Mathematics · JEE Main

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## MCQ (Single Correct Answer)

JEE Main 2024 (Online) 9th April Evening Shift
Let $$z$$ be a complex number such that the real part of $$\frac{z-2 i}{z+2 i}$$ is zero. Then, the maximum value of $$|z-(6+8 i)|$$ is equal to...
JEE Main 2024 (Online) 8th April Evening Shift
The sum of all possible values of $$\theta \in[-\pi, 2 \pi]$$, for which $$\frac{1+i \cos \theta}{1-2 i \cos \theta}$$ is purely imaginary, is equal t...
JEE Main 2024 (Online) 8th April Morning Shift
Let $$z$$ be a complex number such that $$|z+2|=1$$ and $$\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$$. Then the value of $$|\operatorn... JEE Main 2024 (Online) 8th April Morning Shift If the set$$R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$$has$$m$$elements and$$\sum_\limits{n=1}^m\left(1-i^{n !}\right)=x+i y$$, where$$i=\sqrt...
JEE Main 2024 (Online) 6th April Evening Shift
If $$z_1, z_2$$ are two distinct complex number such that $$\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$$, then
JEE Main 2024 (Online) 5th April Evening Shift
Let $$S_1=\{z \in \mathbf{C}:|z| \leq 5\}, S_2=\left\{z \in \mathbf{C}: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right... JEE Main 2024 (Online) 5th April Morning Shift Consider the following two statements : Statement I: For any two non-zero complex numbers$$z_1, z_2,(|z_1|+|z_2|)\left|\frac{z_1}{\left|z_1\right|}+\...
JEE Main 2024 (Online) 4th April Evening Shift
The area (in sq. units) of the region $$S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}$$ is
JEE Main 2024 (Online) 4th April Morning Shift
Let $$\alpha$$ and $$\beta$$ be the sum and the product of all the non-zero solutions of the equation $$(\bar{z})^2+|z|=0, z \in C$$. Then $$4(\alpha^... JEE Main 2024 (Online) 1st February Evening Shift If z is a complex number such that |z| \leqslant 1, then the minimum value of \left|z+\frac{1}{2}(3+4 i)\right| is : JEE Main 2024 (Online) 1st February Morning Shift Let \mathrm{S}=|\mathrm{z} \in \mathrm{C}:| z-1 \mid=1 and (\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2} \mid. Let z_1, z_2 \in \mathrm{S} be ... JEE Main 2024 (Online) 31st January Evening Shift Let$$z_1$$and$$z_2$$be two complex numbers such that$$z_1+z_2=5$$and$$z_1^3+z_2^3=20+15 i$$Then,$$\left|z_1^4+z_2^4\right|$$equals - JEE Main 2024 (Online) 30th January Evening Shift If$$z$$is a complex number, then the number of common roots of the equations$$z^{1985}+z^{100}+1=0$$and$$z^3+2 z^2+2 z+1=0$$, is equal to JEE Main 2024 (Online) 30th January Morning Shift If$$z=x+i y, x y \neq 0$$, satisfies the equation$$z^2+i \bar{z}=0$$, then$$\left|z^2\right|$$is equal to : JEE Main 2024 (Online) 29th January Evening Shift Let$$\mathrm{r}$$and$$\theta$$respectively be the modulus and amplitude of the complex number$$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then ... JEE Main 2024 (Online) 29th January Morning Shift If$$z=\frac{1}{2}-2 i$$is such that$$|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$$and$$\alpha, \beta \in \mathbb{R}$$, then$$\alpha+\beta$$is equal ... JEE Main 2024 (Online) 27th January Morning Shift If S=\{z \in C:|z-i|=|z+i|=|z-1|\}, then, n(S) is : JEE Main 2023 (Online) 15th April Morning Shift If the set \left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\} is equ... JEE Main 2023 (Online) 13th April Evening Shift Let$$S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$$. Then$$\sum_\limits{z \in \mathrm{S}}|z|^{2}$$is e... JEE Main 2023 (Online) 12th April Morning Shift Let$$\mathrm{C}$$be the circle in the complex plane with centre$$\mathrm{z}_{0}=\frac{1}{2}(1+3 i)$$and radius$$r=1$$. Let$$\mathrm{z}_{1}=1+\ma...
JEE Main 2023 (Online) 11th April Evening Shift
For $$a \in \mathbb{C}$$, let $$\mathrm{A}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z}) > \operatorname{Im}(\bar{a}+z)\}$$ and $$\mathrm{B}=\{z \i... JEE Main 2023 (Online) 11th April Morning Shift Let$$w_{1}$$be the point obtained by the rotation of$$z_{1}=5+4 i$$about the origin through a right angle in the anticlockwise direction, and$$w_...
JEE Main 2023 (Online) 10th April Evening Shift
Let $$S = \left\{ {z = x + iy:{{2z - 3i} \over {4z + 2i}}\,\mathrm{is\,a\,real\,number}} \right\}$$. Then which of the following is NOT correct?...
JEE Main 2023 (Online) 10th April Morning Shift
Let the complex number $$z = x + iy$$ be such that $${{2z - 3i} \over {2z + i}}$$ is purely imaginary. If $${x} + {y^2} = 0$$, then $${y^4} + {y^2} - ... JEE Main 2023 (Online) 8th April Evening Shift Let$$A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$$is purely imaginary$$\}$$. Then the sum of the elements in$$...
JEE Main 2023 (Online) 8th April Morning Shift
If for $$z=\alpha+i \beta,|z+2|=z+4(1+i)$$, then $$\alpha+\beta$$ and $$\alpha \beta$$ are the roots of the equation :
JEE Main 2023 (Online) 6th April Evening Shift
Let $$a \neq b$$ be two non-zero real numbers. Then the number of elements in the set $$X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^{2}+b z\... JEE Main 2023 (Online) 1st February Evening Shift Let$$a,b$$be two real numbers such that$$ab ...
JEE Main 2023 (Online) 1st February Morning Shift
If the center and radius of the circle $$\left| {{{z - 2} \over {z - 3}}} \right| = 2$$ are respectively $$(\alpha,\beta)$$ and $$\gamma$$, then $$3(\... JEE Main 2023 (Online) 31st January Evening Shift The complex number z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}} is equal to : JEE Main 2023 (Online) 31st January Morning Shift For all$$z \in C$$on the curve$$C_{1}:|z|=4$$, let the locus of the point$$z+\frac{1}{z}$$be the curve$$\mathrm{C}_{2}$$. Then : JEE Main 2023 (Online) 29th January Morning Shift For two non-zero complex numbers$$z_{1}$$and$$z_{2}$$, if$$\operatorname{Re}\left(z_{1} z_{2}\right)=0$$and$$\operatorname{Re}\left(z_{1}+z_{2}\...
JEE Main 2023 (Online) 25th January Evening Shift
Let $$z$$ be a complex number such that $$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then $$z$$ lies on the circle of radius 2 and ce...
JEE Main 2023 (Online) 25th January Morning Shift
Let $$\mathrm{z_1=2+3i}$$ and $$\mathrm{z_2=3+4i}$$. The set $$\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - ... JEE Main 2023 (Online) 24th January Evening Shift The value of$${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)...
JEE Main 2023 (Online) 24th January Morning Shift
Let $$\mathrm{p,q\in\mathbb{R}}$$ and $${\left( {1 - \sqrt 3 i} \right)^{200}} = {2^{199}}(p + iq),i = \sqrt { - 1}$$ then $$\mathrm{p+q+q^2}$$ and $... 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 Let $$\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|... JEE Main 2022 (Online) 29th July Morning Shift If$$z=2+3 i$$, then$$z^{5}+(\bar{z})^{5}$$is equal to : JEE Main 2022 (Online) 28th July Morning Shift Let$$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{C}:\left|z_{2}-\right| z_{2}+1... JEE Main 2022 (Online) 27th July Evening Shift Let S be the set of all $$(\alpha, \beta), \pi... JEE Main 2022 (Online) 27th July Morning Shift Let the minimum value$$v_{0}$$of$$v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$$is attained at$${ }{z}=z_{0}$$. Then$$\left|2 z_{0}^{2}-\b... JEE Main 2022 (Online) 26th July Evening Shift If $$z=x+i y$$ satisfies $$|z|-2=0$$ and $$|z-i|-|z+5 i|=0$$, then : JEE Main 2022 (Online) 26th July Morning Shift Let O be the origin and A be the point $${z_1} = 1 + 2i$$. If B is the point $${z_2}$$, $${\mathop{\rm Re}\nolimits} ({z_2}) ... JEE Main 2022 (Online) 25th July Evening Shift For$$z \in \mathbb{C}$$if the minimum value of$$(|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)$$is$$5 \sqrt{2}$$, then a value Question: of$$p$$is _________... JEE Main 2022 (Online) 25th July Morning Shift For$$\mathrm{n} \in \mathbf{N}$$, let$$\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\right\}$$and$$\mathrm{T}_{\m... JEE Main 2022 (Online) 30th June Morning Shift The real part of the complex number $${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$ is equal to : JEE Main 2022 (Online) 29th 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) = $${\pi \over 4}$$ intersect ... JEE Main 2022 (Online) 29th June Morning Shift Let $$\alpha$$ and $$\beta$$ be the roots of the equation x2 + (2i $$-$$ 1) = 0. Then, the value of |$$\alpha$$8 + $$\beta$$8| is equal to :... JEE Main 2022 (Online) 27th June Evening 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 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) 26th June Morning Shift Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| ... JEE Main 2022 (Online) 25th June Evening Shift Let z1 and z2 be two complex numbers such that$${\overline z _1} = i{\overline z _2}$$and$$\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \righ... JEE Main 2022 (Online) 25th 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 $$z( \ne {z_1})$$ is a point on ... JEE Main 2022 (Online) 24th 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 2021 (Online) 31st August Evening Shift If z is a complex number such that $${{z - i} \over {z - 1}}$$ is purely imaginary, then the minimum value of | z $$-$$ (3 + 3i) | is : JEE Main 2021 (Online) 27th August Morning Shift If $$S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}$$, then : JEE Main 2021 (Online) 26th August Evening 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 Morning Shift The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with : JEE Main 2021 (Online) 27th July Evening 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) + $$\overline z$$(1 $$-$$ i) $$\g... JEE Main 2021 (Online) 27th July Morning Shift Let C be the set of all complex numbers. Let$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} {S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} ... JEE Main 2021 (Online) 22th 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 value of $$\sum\limits_{k = 0}^... JEE Main 2021 (Online) 20th July Morning Shift If z and$$\omega$$are two complex numbers such that$$\left| {z\omega } \right| = 1$$and$$\arg (z) - \arg (\omega ) = {{3\pi } \over 2}$$, then$$... JEE Main 2021 (Online) 18th March Evening Shift Let a complex number be w = 1 $$-$$ $${\sqrt 3 }$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$ arg(w) = $${\pi \over 2}$$. ... JEE Main 2021 (Online) 18th March Morning Shift If the equation $$a|z{|^2} + \overline {\overline \alpha z + \alpha \overline z } + d = 0$$ represents a circle where a, d are real constants then w... JEE Main 2021 (Online) 17th March Evening 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 $$-$$ i)z) $$\ge$$ 1}S3 = {z$...
JEE Main 2021 (Online) 17th March Morning Shift
The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :
JEE Main 2021 (Online) 16th March Evening Shift
The least value of |z| where z is complex number which satisfies the inequality $$\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \... JEE Main 2021 (Online) 16th March Morning Shift Let a complex number z, |z|$$\ne$$1, satisfy$${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$$. Then, th... JEE Main 2021 (Online) 25th February Evening Shift If$$\alpha$$,$$\beta\in$$R are such that 1$$-$$2i (here i2 =$$-$$1) is a root of z2 +$$\alpha$$z +$$\beta$$= 0, then ($$\alpha-$$... JEE Main 2021 (Online) 25th February Morning Shift Let the lines (2$$-$$i)z = (2 + i)$$\overline z $$and (2$$+$$i)z + (i$$-$$2)$$\overline z -$$4i = 0, (here i2 =$$-$$1) be normal to a ci... JEE Main 2020 (Online) 6th September Evening Slot Let z = x + iy be a non-zero complex number such that$${z^2} = i{\left| z \right|^2}$$, where i =$$\sqrt { - 1} $$, then z lies on the : JEE Main 2020 (Online) 6th September Morning Slot The region represented by {z = x + iy$$ \in $$C : |z| – Re(z)$$ \le $$1} is also given by the inequality : {z = x + iy$$ \in $$C : |z| – Re(z) ... JEE Main 2020 (Online) 5th September Evening Slot The value of$${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$is : JEE Main 2020 (Online) 5th September Morning Slot If the four complex numbers$$z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)$$and$$z-2Re(z)$$represent the... JEE Main 2020 (Online) 4th September Evening Slot If a and b are real numbers such that$${\left( {2 + \alpha } \right)^4} = a + b\alpha $$where$$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$then a + b... JEE Main 2020 (Online) 4th September Morning Slot Let$$u = {{2z + i} \over {z - ki}}$$, z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at the points P a... JEE Main 2020 (Online) 3rd September Evening Slot If z1 , z2 are complex numbers such that Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1| , and arg(z1 - z2) =$${\pi \over 6}$$, then Im(z1 + z2 ) is equal t... JEE Main 2020 (Online) 2nd September Evening Slot The imaginary part of$${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$can be : JEE Main 2020 (Online) 2nd September Morning Slot The value of$${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)...
JEE Main 2020 (Online) 9th January Evening Slot
If z be a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be :
JEE Main 2020 (Online) 9th January Morning Slot
Let z be complex number such that $$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$ and |z| = $${5 \over 2}$$. Then the value of |z + 3i| is :...
JEE Main 2020 (Online) 8th January Morning Slot
If the equation, x2 + bx + 45 = 0 (b $$\in$$ R) has conjugate complex roots and they satisfy |z +1| = 2$$\sqrt {10}$$ , then :
JEE Main 2020 (Online) 7th January Evening Slot
If $${{3 + i\sin \theta } \over {4 - i\cos \theta }}$$, $$\theta$$ $$\in$$ [0, 2$$\theta$$], is a real number, then an argument of sin$$\theta$$ ...
JEE Main 2020 (Online) 7th January Morning Slot
If $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1$$, where z = x + iy, then the point (x, y) lies on a :
JEE Main 2019 (Online) 12th April Evening Slot
Let z $$\in$$ C with Im(z) = 10 and it satisfies $${{2z - n} \over {2z + n}}$$ = 2i - 1 for some natural number n. Then :
JEE Main 2019 (Online) 12th April Morning Slot
The equation |z – i| = |z – 1|, i = $$\sqrt { - 1}$$, represents :
JEE Main 2019 (Online) 10th April Evening 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 Morning Slot
If a > 0 and z = $${{{{\left( {1 + i} \right)}^2}} \over {a - i}}$$, has magnitude $$\sqrt {{2 \over 5}}$$, then $$\overline z$$ is equal to :
JEE Main 2019 (Online) 9th April Evening 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 Morning Slot
All the points in the set $$S = \left\{ {{{\alpha + i} \over {\alpha - i}}:\alpha \in R} \right\}(i = \sqrt { - 1} )$$ lie on a :
JEE Main 2019 (Online) 8th April Evening 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 Morning Slot
If $$\alpha$$ and $$\beta$$ be the roots of the equation x2 – 2x + 2 = 0, then the least value of n for which $${\left( {{\alpha \over \beta }} \ri... JEE Main 2019 (Online) 12th January Evening Slot Let z1 and z2 be two complex numbers satisfying | z1 | = 9 and | z2 – 3 – 4i | = 4. Then the minimum value of | z1 – z2 | is :... JEE Main 2019 (Online) 12th January Morning Slot If$${{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)$$is a purely imaginary number and | z | = 2, then a value of$$\alpha $$is : JEE Main 2019 (Online) 11th January Evening Slot Let z be a complex number such that |z| + z = 3 + i (where i =$$\sqrt { - 1} $$). Then |z| is equal to : JEE Main 2019 (Online) 11th January Morning Slot Let$${\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,$$where x and y are real numbers, then ... JEE Main 2019 (Online) 10th January Evening Slot Let$$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.$$If R(z) and 1(z) respec... JEE Main 2019 (Online) 10th January Morning Slot Let z1 and z2 be any two non-zero complex numbers such that$$3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|.$$If &nbs... JEE Main 2019 (Online) 9th January Evening Slot Let z0 be a root of the quadratic equation, x2 + x + 1 = 0, If z = 3 + 6iz$$_0^{81}-$$3iz$$_0^{93}$$, then arg z is equal to : ... JEE Main 2019 (Online) 9th January Morning Slot Let$$\alpha $$and$$\beta $$be two roots of the equation x2 + 2x + 2 = 0 , then$$\alpha ^{15}$$+$$\beta ^{15}$$is equal to : JEE Main 2019 (Online) 9th January Morning Slot Let A =$$\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\...
JEE Main 2018 (Online) 16th April 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 (Offline)
If $$\alpha ,\beta \in C$$ are the distinct roots of the equation x2 - x + 1 = 0, then $${\alpha ^{101}} + {\beta ^{107}}$$ is equal to :...
JEE Main 2018 (Online) 15th April Evening 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 Morning Slot
The set of all $$\alpha$$ $$\in$$ R, for which w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ is purely imaginary number, for all z $... JEE Main 2017 (Online) 9th April Morning Slot The equation Im $$\left( {{{iz - 2} \over {z - i}}} \right)$$ + 1 = 0, z $$\in$$ C, z $$\ne$$ i represents a part of a circle having radius equal ... JEE Main 2017 (Online) 8th April Morning Slot Let z$$\in$$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents : JEE Main 2017 (Offline) Let $$\omega$$ be a complex number such that 2$$\omega$$ + 1 = z where z = $$\sqrt {-3}$$. If $$\left| {\matrix{ 1 & 1 & 1 \cr 1 &a... JEE Main 2016 (Online) 9th April Morning Slot The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there$$2\sqrt 2 $$units in the s... JEE Main 2016 (Offline) A value of$$\theta \,$$for which$${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$$is purely imaginary, is : JEE Main 2015 (Offline) A complex number z is said to be unimodular if$$\,\left| z \right| = 1$$. Suppose$${z_1}$$and$${z_2}$$are complex numbers such that$${{{z_1} - 2... JEE Main 2014 (Offline) If z is a complex number such that $$\,\left| z \right| \ge 2\,$$, then the minimum value of $$\,\,\left| {z + {1 \over 2}} \right|$$ : JEE Main 2013 (Offline) If z is a complex number of unit modulus and argument $$\theta$$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$ equals : AIEEE 2012 If $$z \ne 1$$ and $$\,{{{z^2}} \over {z - 1}}\,$$ is real, then the point represented by the complex number z lies : AIEEE 2011 Let $$\alpha \,,\beta$$ be real and z be a complex number. If $${z^2} + \alpha z + \beta = 0$$ has two distinct roots on the line Re z = 1, then it ... AIEEE 2011 If $$\omega ( \ne 1)$$ is a cube root of unity, and $${(1 + \omega )^7} = A + B\omega \,$$. Then $$(A,B)$$ equals : AIEEE 2010 The number of complex numbers z such that $$\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|$$ equals : AIEEE 2009 If $$\,\left| {z - {4 \over z}} \right| = 2,$$ then the maximum value of $$\,\left| z \right|$$ is equal to : AIEEE 2008 The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is : AIEEE 2007 If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is : AIEEE 2006 If $${z^2} + z + 1 = 0$$, where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right... AIEEE 2006 The value of$$\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)} $$is : AIEEE 2005 If$$\,\omega = {z \over {z - {1 \over 3}i}}\,$$and$$\left| \omega \right| = 1$$, then$$z$$lies on : AIEEE 2005 If the cube roots of unity are 1,$$\omega \,,\,{\omega ^2}$$then the roots of the equation$${(x - 1)^3}$$+ 8 = 0, are : AIEEE 2005 If$${z_1}$$and$${z_2}$$are two non-zero complex numbers such that$$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \ri... 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 $$z = x - iy$$ and $${z^{{1 \over 3}}} = p + iq$$, then $${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$ ... AIEEE 2004 If $$\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$$, then z lies on : AIEEE 2003 If $$z$$ and $$\omega$$ are two non-zero complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$Arg(z) - Arg(\omega ) = {\pi \over 2},$$ ... 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 the origin, $${Z_1}$$ and $${Z... AIEEE 2003 If$${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$then : AIEEE 2002 z and w are two nonzero complex numbers such that$$\,\left| z \right| = \left| w \right|$$and Arg z + Arg w =$$\pi $$then z equals AIEEE 2002 If$$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by : AIEEE 2002 The locus of the centre of a circle which touches the circle$$\left| {z - {z_1}} \right| = a$$and$$\left| {z - {z_2}} \right| = b\,$$externally (... ## Numerical JEE Main 2024 (Online) 9th April Morning Shift The sum of the square of the modulus of the elements in the set$$\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},... JEE Main 2024 (Online) 1st February Morning Shift Let$\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|z+2-3 i| \leq 1\}$and$\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in$\m...
JEE Main 2024 (Online) 31st January Morning Shift
If $$\alpha$$ denotes the number of solutions of $$|1-i|^x=2^x$$ and $$\beta=\left(\frac{|z|}{\arg (z)}\right)$$, where $$z=\frac{\pi}{4}(1+i)^4\left[... JEE Main 2024 (Online) 29th January Evening Shift Let$$\alpha, \beta$$be the roots of the equation$$x^2-\sqrt{6} x+3=0$$such that$$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Let$$a, b...
JEE Main 2024 (Online) 29th January Morning Shift
Let $$\alpha, \beta$$ be the roots of the equation $$x^2-x+2=0$$ with $$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Then $$\alpha^6+\alpha^4... JEE Main 2024 (Online) 27th January Evening Shift Let the complex numbers$$\alpha$$and$$\frac{1}{\bar{\alpha}}$$lie on the circles$$\left|z-z_0\right|^2=4$$and$$\left|z-z_0\right|^2=16$$respec... JEE Main 2024 (Online) 27th January Morning Shift If \alpha satisfies the equation x^2+x+1=0 and (1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0, then 5(3 A-2 B-C) is equal to ________... JEE Main 2023 (Online) 13th April Morning Shift Let$$w=z \bar{z}+k_{1} z+k_{2} i z+\lambda(1+i), k_{1}, k_{2} \in \mathbb{R}$$. Let$$\operatorname{Re}(w)=0$$be the circle$$\mathrm{C}$$of radius... JEE Main 2023 (Online) 11th April Evening Shift Let$$\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}$$. If$$\alpha-\frac{13}{11} i \in \m...
JEE Main 2023 (Online) 6th April Evening Shift
For $$\alpha, \beta, z \in \mathbb{C}$$ and $$\lambda > 1$$, if $$\sqrt{\lambda-1}$$ is the radius of the circle $$|z-\alpha|^{2}+|z-\beta|^{2}=2 \lam... JEE Main 2023 (Online) 30th January Morning Shift Let$$z=1+i$$and$$z_{1}=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$$. Then$$\frac{12}{\pi} \arg \left(z_{1}\right)$$is equal to __________.... JEE Main 2023 (Online) 29th January Evening Shift Let$$\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=...
JEE Main 2022 (Online) 28th July Evening Shift
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 of $$n \in N$$, such that $$z^... JEE Main 2022 (Online) 27th July Morning Shift Let$$S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$$. Then$$\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$$is equal to ____... JEE Main 2022 (Online) 29th June Morning Shift Let$$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\} $$. Let$$|z - 4i|$$attains minimum and maximum values, respectively, at ... JEE Main 2022 (Online) 28th June Evening Shift Sum of squares of modulus of all the complex numbers z satisfying$$\overline z = i{z^2} + {z^2} - z$$is equal to ___________. JEE Main 2022 (Online) 28th June Morning Shift The number of elements in the set {z = a + ib$$\in$$C : a, b$$\in$$Z and 1 JEE Main 2022 (Online) 26th June Evening Shift If$${z^2} + z + 1 = 0$$,$$z \in C$$, then$$\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right...
JEE Main 2022 (Online) 24th June Evening Shift
Let S = {z $$\in$$ C : |z $$-$$ 3| $$\le$$ 1 and z(4 + 3i) + $$\overline z$$(4 $$-$$ 3i) $$\le$$ 24}. If $$\alpha$$ + i$$\beta$$ is the point in S wh...
JEE Main 2021 (Online) 1st September Evening Shift
If for the complex numbers z satisfying | z $$-$$ 2 $$-$$ 2i | $$\le$$ 1, the maximum value of | 3iz + 6 | is attained at a + ib, then a + b is equal ...
JEE Main 2021 (Online) 31st August Morning Shift
A point z moves in the complex plane such that $$\arg \left( {{{z - 2} \over {z + 2}}} \right) = {\pi \over 4}$$, then the minimum value of $${\left|... JEE Main 2021 (Online) 27th August Evening Shift Let z1 and z2 be two complex numbers such that$$\arg ({z_1} - {z_2}) = {\pi \over 4}$$and z1, z2 satisfy the equation | z$$-$$3 | = Re(z). Then t... JEE Main 2021 (Online) 26th August Evening Shift The least positive integer n such that$${{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}},i = \sqrt { - 1} $$is a positive integer, is ___________. JEE Main 2021 (Online) 26th August Morning Shift Let$$z = {{1 - i\sqrt 3 } \over 2}$$,$$i = \sqrt { - 1} $$. Then the value of$$21 + {\left( {z + {1 \over z}} \right)^3} + {\left( {{z^2} + {1 \ove...
JEE Main 2021 (Online) 27th July Evening Shift
If the real part of the complex number $$z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\pi \over 2}} \right)$$ is zero...
JEE Main 2021 (Online) 25th July Evening 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 the given circle and the vert...
JEE Main 2021 (Online) 25th July Morning Shift
Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{ 0 & i \cr 1 & 0 \cr } } \right)}^n}\left( {\matrix{ a & b \cr c &... JEE Main 2021 (Online) 18th March Morning Shift Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :... JEE Main 2021 (Online) 16th March Morning Shift Let z and$$\omega$$be two complex numbers such that$$\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1$$and Re($$\omeg...
JEE Main 2021 (Online) 26th February Evening Shift
Let z be those complex numbers which satisfy| z + 5 | $$\le$$ 4 and z(1 + i) + $$\overline z$$(1 $$-$$ i) $$\ge$$ $$-$$10, i = $$\sqrt { - 1}$$....
JEE Main 2021 (Online) 24th February Evening Shift
Let $$i = \sqrt { - 1}$$. If $${{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 + i\sqrt 3 } \right)}^{21}}} \... JEE Main 2021 (Online) 24th February Morning Shift If the least and the largest real values of a, for which the equation z +$$\alpha $$|z – 1| + 2i = 0 (z$$ \in $$C and i =$$\sqrt { - 1} $$) has a ... JEE Main 2020 (Online) 3rd September Morning Slot If$${\left( {{{1 + i} \over {1 - i}}} \right)^{{m \over 2}}} = {\left( {{{1 + i} \over {1 - i}}} \right)^{{n \over 3}}} = 1$$, (m, n$$ \in  N) the...
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