JEE Main
Mathematics
Complex Numbers
Previous Years Questions

Let $$a,b$$ be two real numbers such that $$ab ... If the center and radius of the circle$$\left| {{{z - 2} \over {z - 3}}} \right| = 2$$are respectively$$(\alpha,\beta)$$and$$\gamma$$, then$$3(\...
The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :
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 :
For two non-zero complex numbers $$z_{1}$$ and $$z_{2}$$, if $$\operatorname{Re}\left(z_{1} z_{2}\right)=0$$ and $$\operatorname{Re}\left(z_{1}+z_{2}\... Let$$z$$be a complex number such that$$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then$$z$$lies on the circle of radius 2 and ce... Let$$\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 - ...
The value of $${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)... Let$$\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}$$and ... If$$z \neq 0$$be a complex number such that$$\left|z-\frac{1}{z}\right|=2$$, then the maximum value of$$|z|$$is : Let$$\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|...
If $$z=2+3 i$$, then $$z^{5}+(\bar{z})^{5}$$ is equal to :
Let $$S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}$$ and $$S_{2}=\left\{z_{2} \in \mathbf{C}:\left|z_{2}-\right| z_{2}+1... Let S be the set of all$$(\alpha, \beta), \pi...
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 $$\left|2 z_{0}^{2}-\b... If$$z=x+i y$$satisfies$$|z|-2=0$$and$$|z-i|-|z+5 i|=0$$, then Let O be the origin and A be the point$${z_1} = 1 + 2i$$. If B is the point$${z_2}$$,$${\mathop{\rm Re}\nolimits} ({z_2}) ...
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 _________...
If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^{4}+x^{3}+x^{2}+x+1=0$$, then $$\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta... For$$\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...
The real part of the complex number $${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$ is equal to :
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...
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:...
The number of points of intersection of $$|z - (4 + 3i)| = 2$$ and $$|z| + |z - 4| = 6$$, z $$\in$$ C, is
The area of the polygon, whose vertices are the non-real roots of the equation $$\overline z = i{z^2}$$ is :
Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| ... Let z1 and z2 be two complex numbers such that$${\overline z _1} = i{\overline z _2}$$and$$\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \righ...
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 ...
Let $$A = \{ z \in C:1 \le |z - (1 + i)| \le 2\}$$ and $$B = \{ z \in A:|z - (1 - i)| = 1\}$$. Then, B :
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 :
If $$S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}$$, then :
If $${\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)$$, then p and q are roots of the equation :
The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with :
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) $$\g... Let C be the set of all complex numbers. Let$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} {S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} ...
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 $$\sum\limits_{k = 0}^... If z and$$\omega$$are two complex numbers such that$$\left| {z\omega } \right| = 1$$and$$\arg (z) - \arg (\omega ) = {{3\pi } \over 2}$$, then$$...
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) = $${\pi \over 2}$$. ...
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...
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$... The area of the triangle with vertices A(z), B(iz) and C(z + iz) is : The least value of |z| where z is complex number which satisfies the inequality $$\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \... Let a complex number z, |z|$$\ne$$1, satisfy$${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$$. Then, th... If$$\alpha$$,$$\beta\in$$R are such that 1$$-$$2i (here i2 =$$-$$1) is a root of z2 +$$\alpha$$z +$$\beta$$= 0, then ($$\alpha-$$... Let the lines (2$$-$$i)z = (2 + i)$$\overline z $$and (2$$+$$i)z + (i$$-$$2)$$\overline z -$$4i = 0, (here i2 =$$-$$1) be normal to a ci... Let z = x + iy be a non-zero complex number such that$${z^2} = i{\left| z \right|^2}$$, where i =$$\sqrt { - 1} $$, then z lies on the : The region represented by {z = x + iy$$ \in $$C : |z| – Re(z)$$ \le $$1} is also given by the inequality: {z = x + iy$$ \in $$C : |z| – Re(z)$$... The value of $${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$ is : 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... 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... 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... 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... The imaginary part of $${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$ can be : The value of $${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)... If z be a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be Let z be complex number such that$$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$and |z| =$${5 \over 2}$$. Then the value of |z + 3i| is :... If the equation, x2 + bx + 45 = 0 (b$$ \in $$R) has conjugate complex roots and they satisfy |z +1| = 2$$\sqrt {10} $$, then If$${{3 + i\sin \theta } \over {4 - i\cos \theta }}$$,$$\theta  \in $$[0, 2$$\theta $$], is a real number, then an argument of sin$$\theta $$... If$${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1$$, where z = x + iy, then the point (x, y) lies on a: Let z$$ \in $$C with Im(z) = 10 and it satisfies$${{2z - n} \over {2z + n}}$$= 2i - 1 for some natural number n. Then : The equation |z – i| = |z – 1|, i =$$\sqrt { - 1} $$, represents : If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) =$${\pi \over 2}$$, then: If a > 0 and z =$${{{{\left( {1 + i} \right)}^2}} \over {a - i}}$$, has magnitude$$\sqrt {{2 \over 5}} $$, then$$\overline z $$is equal to : Let z$$ \in $$C be such that |z| < 1. If$$\omega = {{5 + 3z} \over {5(1 - z)}}$$z, then:- All the points in the set$$S = \left\{ {{{\alpha + i} \over {\alpha - i}}:\alpha \in R} \right\}(i = \sqrt { - 1} )$$lie on a If$$z = {{\sqrt 3 } \over 2} + {i \over 2}\left( {i = \sqrt { - 1} } \right)$$, then (1 + iz + z5 + iz8)9 is equal to... If$$\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... Let z1 and z2 be two complex numbers satisfying | z1 | = 9 and | z2 – 3 – 4i | = 4. Then the minimum value of | z1 – z2 | is :... 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 : Let z be a complex number such that |z| + z = 3 + i (where i = $$\sqrt { - 1}$$). Then |z| is equal to 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 ... 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... 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... 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 : ... Let $$\alpha$$ and $$\beta$$ be two roots of the equation x2 + 2x + 2 = 0 , then $$\alpha ^{15}$$ + $$\beta ^{15}$$ is equal to : Let A = $$\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\... The least positive integer n for which$${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$$is : If$$\alpha ,\beta \in C$$are the distinct roots of the equation x2 - x + 1 = 0, then$${\alpha ^{101}} + {\beta ^{107}}$$is equal to... If |z$$-$$3 + 2i|$$ \le $$4 then the difference between the greatest value and the least value of |z| is : The set of all$$\alpha  \in $$R, for which w =$${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$is purely imaginary number, for all z ... Let$$\omega $$be a complex number such that 2$$\omega $$+ 1 = z where z =$$\sqrt {-3} $$. If$$\left| {\matrix{ 1 & 1 & 1 \cr 1 &a... 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... A value of $$\theta \,$$ for which $${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$$ is purely imaginary, is : 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 $${{{z_1} - 2... If z is a complex number such that$$\,\left| z \right| \ge 2\,$$, then the minimum value of$$\,\,\left| {z + {1 \over 2}} \right|$$: If z is a complex number of unit modulus and argument$$\theta $$, then arg$$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$equals : If$$z \ne 1$$and$$\,{{{z^2}} \over {z - 1}}\,$$is real, then the point represented by the complex number z lies : Let$$\alpha \,,\beta $$be real and z be a complex number. If$${z^2} + \alpha z + \beta = 0$$has two distinct roots on the line Re z = 1, then it ... If$$\omega ( \ne 1)$$is a cube root of unity, and$${(1 + \omega )^7} = A + B\omega \,$$. Then$$(A,B)$$equals The number of complex numbers z such that$$\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|$$equals The conjugate of a complex number is$${1 \over {i - 1}}$$then that complex number is Let R be the real line. Consider the following subsets of the plane$$R \times R$$:$$S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right... If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is If $${z^2} + z + 1 = 0$$, where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right... The value of$$\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)} $$is If$${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... If the cube roots of unity are 1, $$\omega \,,\,{\omega ^2}$$ then the roots of the equation $${(x - 1)^3}$$ + 8 = 0, are If $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$, then $$z$$ lies on Let z and w be complex numbers such that $$\overline z + i\overline w = 0$$ and arg zw = $$\pi$$. Then arg z equals If $$z = x - iy$$ and $${z^{{1 \over 3}}} = p + iq$$, then $${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$ ... If $$\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$$, then z lies on If $$z$$ and $$\omega$$ are two non-zero complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$Arg(z) - Arg(\omega ) = {\pi \over 2},$$ ... 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 $${Z... If$${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$then z and w are two nonzero complex numbers such that$$\,\left| z \right| = \left| w \right|$$and Arg z + Arg w =$$\pi $$then z equals If$$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by The locus of the centre of a circle which touches the circle$$\left| {z - {z_1}} \right| = a$$and$$\left| {z - {z_2}} \right| = b\,$$externally (... ## Numerical Let$$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)$$is equal to __________.... Let$$\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=... 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 $$z^... Let$$S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$$. Then$$\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$$is equal to ____... Let$$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\} $$. Let$$|z - 4i|$$attains minimum and maximum values, respectively, at ... Sum of squares of modulus of all the complex numbers z satisfying$$\overline z = i{z^2} + {z^2} - z$$is equal to ___________. The number of elements in the set {z = a + ib$$\in$$C : a, b$$\in$$Z and 1 If$${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... 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... 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 ... 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 $${\left|... The least positive integer n such that$${{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}},i = \sqrt { - 1} $$is a positive integer, is ___________. Let$$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... 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... 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... Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{ 0 & i \cr 1 & 0 \cr } } \right)}^n}\left( {\matrix{ a & b \cr c &... Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is... Let z and$$\omega$$be two complex numbers such that$$\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1$$and Re($$\omeg... Let z be those complex numbers which satisfy| z + 5 | $$\le$$ 4 and z(1 + i) + $$\overline z$$(1 $$-$$ i) $$\ge$$ $$-$$10, i = $$\sqrt { - 1}$$.... Let $$i = \sqrt { - 1}$$. If $${{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 + i\sqrt 3 } \right)}^{21}}} \... If the least and the largest real values of a, for which the equation z +$$\alpha $$|z – 1| + 2i = 0 (z$$ \in $$C and i =$$\sqrt { - 1} $$) has a ... If$${\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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