Mathematics
Complex Numbers
Previous Years Questions

## Numerical

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...
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 $$|z|... Let$$\bar{z}$$denote the complex conjugate of a complex number$$z$$and let$$i=\sqrt{-1}$$. In the set of complex numbers, the number of distinct ... For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying$${z^4} - |z{|^4} = 4i{z^2}$$, wher... Let$$\omega \ne 1$$be a cube root of unity. Then the maximum of the set$$\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$$distinct non-... For any integer k, let$${a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)$$, where$$i = \sqrt { - 1} \,...
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
Let $$\omega = {e^{{{i\pi } \over 3}}}$$, and a, b, c, x, y, z be non-zero complex numbers such that $$a + b + c = x$$ $$a + b\omega + c{\omega ... ## MCQ (Single Correct Answer) Let z be a complex number satisfying |z|^3+2 z^2+4 \bar{z}-8=0, where \bar{z} denotes the complex conjugate of z. Let the imaginary part of z... Let$$\theta$$1,$$\theta$$2, ........,$$\theta$$10 = 2$$\pi$$. Define the complex numbers z1 = ei$$\theta$$1, zk = zk$$-$$1ei$$\theta$$k for k = ... Let S be the set of all complex numbers z satisfying$$\left| {z - 2 + i} \right| \ge \sqrt 5 $$. If the complex number z0 is such that$${1 \over {\l...
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. ...
Let $$S = {S_1} \cap {S_2} \cap {S_3}$$, where $${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\... Let$$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}\...
Let complex numbers $$\alpha \,and\,{1 \over {\overline \alpha }}\,$$ lie on circles $${\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \righ... Let z be a complex number such that the imaginary part of z is non-zero and$$a\, = \,{z^2} + \,z\, + 1$$is real. Then a cannot take the value Match the statements in Column I with those in Column II. [Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively... Let$$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...
Let $$z = \,\cos \,\theta \, + i\,\sin \,\theta$$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\t... A particle P stats from the point$${z_0}$$= 1 +2i, where$$i = \sqrt { - 1} $$. It moves horizontally away from origin by 5 unit and then vertically... The number of elements in the set$$A \cap B \cap C$$is Let z be any point in$$A \cap B \cap C$$Then,$${\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}$$lies between : Let z be any point$$A \cap B \cap C$$and let w be any point satisfying$$\left| {w - 2 - i} \right| < 3\,$$. Then,$$\left| z \right| - \left| w...
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...
If $$\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,$$ then all the values of $${z \over {1 - {z^2}}}$$ lie on
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...
$$a,\,b,\,c$$ are integers, not all simultaneously equal and $$\omega$$ is cube root of unity $$\left( {\omega \ne 1} \right),$$ then minimum value ...
If $$\omega$$ $$\left( { \ne 1} \right)$$ be a cube root of unity and $${\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n},... If$$\,\left| z \right| = 1$$and$$\omega = {{z - 1} \over {z + 1}}$$(where$$z \ne - 1$$), then$${\mathop{\rm Re}\nolimits} \left( \omega \rig...
Let $$\omega$$ $$= - {1 \over 2} + i{{\sqrt 3 } \over 2},$$ then the value of the det. $$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & ... For all complex numbers$${z_1},\,{z_2}$$satisfying$$\left| {{z_1}} \right| = 12$$and$$\left| {{z_2} - 3 - 4i} \right| = 5,$$the minimum value ... Let$${z_1}$$and$${z_2}$$be$${n^{th}}$$roots of unity which subtend a right angle at the origin. Then$$n$$must be of the form The complex numbers$${z_1},\,{z_2}$$and$${z_3}$$satisfying$${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,$$are the vert... If$$\arg \left( z \right) < 0,$$then$$\arg \left( { - z} \right) - \arg \left( z \right) = $$If$${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| {...
$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \ov... For positive integers$${n_1},\,{n_2}$$the value of the expression$${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\l...
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...
Let $$z$$ and $$\omega$$ be two complex numbers such that $$\left| z \right| \le 1,$$ $$\left| \omega \right| \le 1$$ and $$\left| {z + i\omega } \... Let$$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\,\...
$${\rm{z }} \ne {\rm{0}}$$ 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$... 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 $$... If$$z = x + iy$$and$$\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$$then$$\,\left| \omega \right| = 1$$implies that, in the comple... The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order if and only if... If$$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},$$then The inequality |z-4| < |z-2| represents the region given by The complex numbers$$z = x + iy$$which satisfy the equation$$\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$$lie on The smallest positive integer n for which$${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$$is If the cube roots of unity are$$1,\,\omega ,\,{\omega ^2},$$then the roots of the equation$${\left( {x - 1} \right)^3} + 8 = 0$$are ## MCQ (More than One Correct Answer) Let \bar{z} denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary parts of$$... 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... Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE? 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 $$sz + t\overline ... For a non-zero complex number z, let arg(z) denote the principal argument with$$-\pi $$< arg(z)$$ \le \pi $$. Then, which of the follo... Let a, b, x and y be real numbers such that a$$-$$b = 1 and y$$ \ne $$0. If the complex number z = x + iy satisfies$${\mathop{\rm Im}\nolimits} ... 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 $$i = \... Let$$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... 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 <... If $${\omega}$$ is an imaginary cube root of unity, then $${(1\, + \omega \, - {\omega ^2})^7}$$ equals If $$\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$$ ... The value of the sum $$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$ , where i = $$\sqrt { - 1}$$, equals If $${{{z_1}}}$$ and $${{{z_2}}}$$ are two nonzero complex numbers such that $$\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{... The value of$$\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$$is Let$${z_1}$$and$${z_2}$$be complex numbers such that$${z_1} \ne {z_2}$$and$$\left| {{z_1}} \right| =\,\left| {{z_2}} \right|$$. If ... If$${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 ... ## Subjective 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... Find the centre and radius of circle given by $$\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$$ where, $${\rm{z = x + iy, ... If$${z_1}$$and$${z_2}$$are two complex numbers such that$$\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,$$then prove that$$\,\le... 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 $$\lef... Let a complex number$$\alpha ,\,\alpha \ne 1$$, be a root of the equation$${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, where p, q are distinct primes. S... For complex numbers z and w, prove that$${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$if and only if$$ z = w\,or\,z\overline {\,w} = 1$...
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...
Find all non-zero complex numbers Z satisfying $$\overline Z = i{Z^2}$$.
If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.
If $$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that $${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(A... Let$${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...
Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is $${1 \over 2}\,{\left| z \right|^2}$$ .

## True or False

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