Complex Numbers · Mathematics · JEE Advanced

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MCQ (More than One Correct Answer)

JEE Advanced 2024 Paper 1 Online
Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\... JEE Advanced 2022 Paper 2 Online Let$\bar{z}$denote the complex conjugate of a complex number$z$. If$z$is a non-zero complex number for which both real and imaginary parts of $$... JEE Advanced 2021 Paper 1 Online For any complex number w = c + id, let$$\arg (w) \in ( - \pi ,\pi ]$$, where$$i = \sqrt { - 1} $$. Let$$\alpha$$and$$\beta$$be real numbers such... JEE Advanced 2020 Paper 1 Offline Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE? JEE Advanced 2018 Paper 2 Offline Let s, t, r be non-zero complex numbers and L be the set of solutions$$z = x + iy(x,y \in R,\,i = \sqrt { - 1} )$$of the equation$$sz + t\overline ... JEE Advanced 2018 Paper 1 Offline For a non-zero complex number z, let arg(z) denote the principal argument with $$-$$ $$\pi$$ < arg(z) $$\le$$ $$\pi$$. Then, which of the follo... JEE Advanced 2017 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 $${\mathop{\rm Im}\nolimits} ... JEE Advanced 2016 Paper 2 Offline Let$$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 \ne 0} \right\}$$, where$$i = \... JEE Advanced 2013 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 complex numbers. If $${z_1} \in P... IIT-JEE 2010 Paper 1 Offline Let$${{z_1}}$$and$${{z_2}}$$be two distinct complex number and let z =( 1 - t)$${{z_1}}$$+ t$${{z_2}}$$for some real number t with 0 < t <... IIT-JEE 1998 If$${\omega}$$is an imaginary cube root of unity, then$${(1\, + \omega \, - {\omega ^2})^7}$$equals IIT-JEE 1998 The value of the sum$$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$, where i =$$\sqrt { - 1} $$, equals IIT-JEE 1998 If$$\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$$... IIT-JEE 1987 If$${{{z_1}}}$$and$${{{z_2}}}$$are two nonzero complex numbers such that$$\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{... IIT-JEE 1987 The value of $$\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$$ is IIT-JEE 1986 Let $${z_1}$$ and $${z_2}$$ be complex numbers such that $${z_1}$$ $$\ne$$ $${z_2}$$ and $$\left| {{z_1}} \right| =\,\left| {{z_2}} \right|$$. If ... IIT-JEE 1985 If $${z_1}$$ = a + ib and $${z_2}$$ = c + id are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1$$ and $${\mathop{\rm ... Numerical JEE Advanced 2024 Paper 1 Online Let f(x)=x^4+a x^3+b x^2+c be a polynomial with real coefficients such that f(1)=-9. Suppose that i \sqrt{3} is a root of the equation 4 x^3+3 ... JEE Advanced 2023 Paper 1 Online Let A=\left\{\frac{1967+1686 i \sin \theta}{7-3 i \cos \theta}: \theta \in \mathbb{R}\right\}. If A contains exactly one positive integer n, the... JEE Advanced 2022 Paper 1 Online Let$$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}} $$is a real number, then the value of$$|z|... JEE Advanced 2022 Paper 1 Online Let $$\bar{z}$$ denote the complex conjugate of a complex number $$z$$ and let $$i=\sqrt{-1}$$. In the set of complex numbers, the number of distinct ... JEE Advanced 2020 Paper 2 Offline 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 $${z^4} - |z{|^4} = 4i{z^2}$$, wher... JEE Advanced 2019 Paper 1 Offline Let $$\omega \ne 1$$ be a cube root of unity. Then the minimum of the set $$\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$$ distinct non-... JEE Advanced 2015 Paper 2 Offline For any integer k, let $${a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)$$, where $$i = \sqrt { - 1} \,... IIT-JEE 2011 Paper 1 Offline If z is any complex number satisfying$$\,\left| {z - 3 - 2i} \right| \le 2$$, then the minimum value of$$\left| {2z - 6 + 5i} \right|$$is IIT-JEE 2011 Paper 2 Offline Let$$\omega = {e^{{{i\pi } \over 3}}}$$, and a, b, c, x, y, z be non-zero complex numbers such that$$a + b + c = xa + b\omega + c{\omega ... MCQ (Single Correct Answer) JEE Advanced 2023 Paper 1 Online Let$z$be a complex number satisfying$|z|^3+2 z^2+4 \bar{z}-8=0$, where$\bar{z}$denotes the complex conjugate of$z$. Let the imaginary part of$z...
JEE Advanced 2021 Paper 1 Online
Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\cdots+\theta_{10}=2 \pi$. Define the...
JEE Advanced 2019 Paper 1 Offline
Let S be the set of all complex numbers z satisfying $$\left| {z - 2 + i} \right| \ge \sqrt 5$$. If the complex number z0 is such that $${1 \over {\l... JEE Advanced 2014 Paper 2 Offline Let$${z_k}$$=$$\cos \left( {{{2k\pi } \over {10}}} \right) + i\,\,\sin \left( {{{2k\pi } \over {10}}} \right);\,k = 1,2....,9$$List-I P. ... JEE Advanced 2013 Paper 2 Offline Let$$S = {S_1} \cap {S_2} \cap {S_3}$$, where$${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\...
JEE Advanced 2013 Paper 2 Offline
Let $$S = {S_1} \cap {S_2} \cap {S_3}$$, where $${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\... JEE Advanced 2013 Paper 1 Offline Let complex numbers$$\alpha \,and\,{1 \over {\overline \alpha }}\,$$lie on circles$${\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \righ...
IIT-JEE 2012 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 a cannot take the value
IIT-JEE 2010 Paper 2 Offline
Match the statements in Column I with those in Column II. [Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively...
IIT-JEE 2009 Paper 1 Offline
Let $$z = \,\cos \,\theta \, + i\,\sin \,\theta$$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\t... IIT-JEE 2009 Paper 1 Offline Let$$z = x + iy$$be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation$$\over...
IIT-JEE 2008 Paper 2 Offline
A particle P stats from the point $${z_0}$$ = 1 +2i, where $$i = \sqrt { - 1}$$. It moves horizontally away from origin by 5 unit and then vertically...
IIT-JEE 2008 Paper 1 Offline
The number of elements in the set $$A \cap B \cap C$$ is
IIT-JEE 2008 Paper 1 Offline
Let z be any point $$A \cap B \cap C$$ and let w be any point satisfying $$\left| {w - 2 - i} \right| < 3\,$$. Then, $$\left| z \right| - \left| w... IIT-JEE 2008 Paper 1 Offline Let z be any point in$$A \cap B \cap C$$Then,$${\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}$$lies between : IIT-JEE 2007 A man walks a distance of 3 units from the origin towards the north-east ($$N\,{45^ \circ E }$$) direction. From there, he walks a distance of 4 units... IIT-JEE 2007 If$$\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,$$then all the values of$${z \over {1 - {z^2}}}$$lie on IIT-JEE 2006 If$${{w - \overline w z} \over {1 - z}}$$is purely real where$$w = \alpha + i\beta ,\beta \ne 0$$and$$z \ne 1,$$then the set of the value... IIT-JEE 2005 Screening$$a,\,b,\,c$$are integers, not all simultaneously equal and$$\omega $$is cube root of unity$$\left( {\omega \ne 1} \right),$$then minimum value ... IIT-JEE 2004 Screening If$$\omega \left( { \ne 1} \right)$$be a cube root of unity and$${\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n},$... IIT-JEE 2003 Screening If $$\,\left| z \right| = 1$$ and $$\omega = {{z - 1} \over {z + 1}}$$ (where $$z \ne - 1$$), then $${\mathop{\rm Re}\nolimits} \left( \omega \rig... IIT-JEE 2002 Let$$\omega  = - {1 \over 2} + i{{\sqrt 3 } \over 2},$$then the value of the det.$$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & ... IIT-JEE 2002 Screening For all complex numbers $${z_1},\,{z_2}$$ satisfying $$\left| {{z_1}} \right| = 12$$ and $$\left| {{z_2} - 3 - 4i} \right| = 5,$$ the minimum value ... IIT-JEE 2001 Screening The complex numbers $${z_1},\,{z_2}$$ and $${z_3}$$ satisfying $${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,$$ are the vert... 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 of the form IIT-JEE 2000 Screening If $${z_1},\,{z_2}$$ and $${z_3}$$ are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {... IIT-JEE 2000 Screening If$$\arg \left( z \right) < 0,$$then$$\arg \left( { - z} \right) - \arg \left( z \right) = $$IIT-JEE 1999$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \ov... IIT-JEE 1996 For positive integers $${n_1},\,{n_2}$$ the value of the expression $${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\l... IIT-JEE 1995 Screening Let$$z$$and$$\omega $$be two complex numbers such that$$\left| z \right| \le 1,\left| \omega \right| \le 1$$and$$\left| {z + i\omega } \... IIT-JEE 1995 Screening If $$\omega \,\left( { \ne 1} \right)$$ is a cube root of unity and $${\left( {1 + \omega } \right)^7} = A + B\,\omega$$ then $$A$$ and $$B$$ are res... IIT-JEE 1995 Screening Let $$z$$ and $$\omega$$ be two non zero complex numbers such that $$\left| z \right| = \left| \omega \right|$$ and $${\rm A}rg\,z + {\rm A}rg\,\... IIT-JEE 1992$${\rm{z }} \ne {\rm{0}}$$is a complex number Column I (A) Re z = 0 (B) Arg$$z = {\pi \over 4}$$Column II (p) Re$${z^2}$$= 0 (q) Im... IIT-JEE 1985 If$$a,\,b,\,c$$and$$u,\,v,\,w$$are complex numbers representing the vertics of two triangles such that$$c = \left( {1 - r} \right)a + rb$$and$$... IIT-JEE 1983 If $$z = x + iy$$ and $$\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$$ then $$\,\left| \omega \right| = 1$$ implies that, in the comple... IIT-JEE 1983 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 1982 If $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},$$ then IIT-JEE 1982 The inequality |z-4| < |z-2| represents the region given by IIT-JEE 1981 The complex numbers $$z = x + iy$$ which satisfy the equation $$\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$$ lie on IIT-JEE 1980 The smallest positive integer n for which $${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$$ is IIT-JEE 1979 If the cube roots of unity are $$1,\,\omega ,\,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0$$ are Subjective IIT-JEE 2005 If one the vertices of the square circumscribing the circle $$\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i$$. Find the other vertices of... IIT-JEE 2004 Find the centre and radius of circle given by $$\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$$ where, $${\rm{z = x + iy, ... IIT-JEE 2003 If$${z_1}$$and$${z_2}$$are two complex numbers such that$$\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,$$then prove that$$\,\le... IIT-JEE 2003 Prove that there exists no complex number z such that $$\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1$$ where $$\lef... IIT-JEE 2002 Let a complex number$$\alpha ,\,\alpha \ne 1$$, be a root of the equation$${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, where p, q are distinct primes. S... IIT-JEE 1999 For complex numbers z and w, prove that$${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$if and only if$$ z = w\,or\,z\overline {\,w} = 1$...
IIT-JEE 1997
Let $${z_1}$$ and $${z_2}$$ be roots of the equation $${z^2} + pz + q = 0\,$$ , where the coefficients p and q may be complex numbers. Let A and B rep...
IIT-JEE 1996
Find all non-zero complex numbers Z satisfying $$\overline Z = i{Z^2}$$.
IIT-JEE 1995
If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.
IIT-JEE 1995
If $$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that $${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(A... IIT-JEE 1990 Let$${z_1}$$= 10 + 6i and$${z_2}$$= 4 + 6i. If Z is any complex number such that the argument of$${{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \o...
IIT-JEE 1986
Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is $${1 \over 2}\,{\left| z \right|^2}$$ .
IIT-JEE 1984

True or False

IIT-JEE 1988
The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.
IIT-JEE 1985
If three complex numbers are in A.P. then they lie on a circle in the complex plane.
IIT-JEE 1984
If the complex numbers, $${Z_1},{Z_2}$$ and $${Z_3}$$ represent the vertics of an equilateral triangle such that $$\left| {{Z_1}} \right| = \left| {{... IIT-JEE 1981 For complex number$${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} \le {x_2}\,\,and\,\,{y_1} \l...
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