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Complex Numbers

106 Questions Start JEE Advanced Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
2004 Q51 JEE Advanced Numerical
14 Mar 2026
Find the centre and radius of circle given by $\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$

where, ${\rm{z = x + iy, }}\alpha {\rm{ = }}\,{\alpha _1}{\rm{ + i}}{\alpha _2}{\rm{,}}\,\beta = {\beta _1}{\rm{ + i}}{\beta _2}{\rm{ }}$

2003 Q52 JEE Advanced MCQ
14 Mar 2026
If $\,\left| z \right| = 1$ and $\omega = {{z - 1} \over {z + 1}}$ (where $z \ne - 1$), then ${\mathop{\rm Re}\nolimits} \left( \omega \right)$ is
A.
0
B.
$ - {1 \over {{{\left| {z + 1} \right|}^2}}}$
C.
$\left| {{z \over {z + 1}}} \right|.{1 \over {{{\left| {z + 1} \right|}^2}}}$
D.
$\,{{\sqrt 2 } \over {{{\left| {z + 1} \right|}^2}}}$
2003 Q53 JEE Advanced Numerical
14 Mar 2026
If ${z_1}$ and ${z_2}$ are two complex numbers such that $\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,$ then prove that $\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1$.
2003 Q54 JEE Advanced Numerical
14 Mar 2026
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 $\left| {{a_r}} \right| < 2$.
2002 Q55 JEE Advanced MCQ
14 Mar 2026
Let $\omega $ $ = - {1 \over 2} + i{{\sqrt 3 } \over 2},$ then the value of the det.
$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|$ is
A.
$3\omega $
B.
$3\omega \left( {\omega - 1} \right)$
C.
$3{\omega ^2}$
D.
$3\omega \left( {1 - \omega } \right)$
2002 Q56 JEE Advanced MCQ
14 Mar 2026
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 of $\left| {{z_1} - {z_2}} \right|$ is
A.
0
B.
2
C.
7
D.
17
2002 Q57 JEE Advanced Numerical
14 Mar 2026
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. Show that either $1 + \alpha + {\alpha ^2} + .... + {\alpha ^{p - 1}} = 0\,or\,1 + \alpha + {\alpha ^2} + .... + {\alpha ^{q - 1}} = 0$, but not both together.
2001 Q58 JEE Advanced MCQ
14 Mar 2026
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 vertices of a triangle which is
A.
of area zero
B.
right-angled isosceles
C.
equilateral
D.
obtuse-angled isosceles
2001 Q59 JEE Advanced MCQ
14 Mar 2026
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
A.
$4k + 1$
B.
$4k + 2$
C.
$4k + 3$
D.
$4k$
2000 Q60 JEE Advanced MCQ
14 Mar 2026
If ${z_1},\,{z_2}$ and ${z_3}$ are complex numbers such that $\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right| = 1,$ then $\left| {{z_1} + {z_2} + {z_3}} \right|$ is
A.
equal to 1
B.
less than 1
C.
greater than 3
D.
equal to 3
2000 Q61 JEE Advanced MCQ
14 Mar 2026
If $\arg \left( z \right) < 0,$ then $\arg \left( { - z} \right) - \arg \left( z \right) = $
A.
$\pi $
B.
$ - \pi $
C.
$ - {\pi \over 2}$
D.
${\pi \over 2}$
1999 Q62 JEE Advanced MCQ
14 Mar 2026
$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$ is equal to
A.
$1 - i\sqrt 3 $
B.
$ - 1 + i\sqrt 3 $
C.
$i\sqrt 3 $
D.
$ - i\sqrt 3 $
1999 Q63 JEE Advanced Numerical
14 Mar 2026
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$.
1998 Q64 JEE Advanced MCQ
14 Mar 2026
If $\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$ , then
A.
x = 3, y = 2
B.
x = 1, y = 3
C.
x = 0, y = 3
D.
x = 0, y = 0
1998 Q65 JEE Advanced MCQ
14 Mar 2026
If ${\omega}$ is an imaginary cube root of unity, then ${(1\, + \omega \, - {\omega ^2})^7}$ equals
A.
$128\omega $
B.
$ - 128\omega $
C.
$128{\omega ^2}$
D.
$ - 128{\omega ^2}$
1998 Q66 JEE Advanced MCQ
14 Mar 2026
The value of the sum $\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$ , where i = $\sqrt { - 1} $, equals
A.
i
B.
i - 1
C.
- i
D.
0
1997 Q67 JEE Advanced Numerical
14 Mar 2026
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 represent ${z_1}$ and ${z_2}$ in the complex plane. If $\angle AOB = \alpha \ne 0\,$ and OA = OB, where O is the origin, prove that ${p^2} = 4q\,{\cos ^2}\left( {{\alpha \over 2}} \right)$.
1996 Q68 JEE Advanced MCQ
14 Mar 2026
For positive integers ${n_1},\,{n_2}$ the value of the expression ${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$
where $i = \sqrt { - 1} $ is real number if and only if
A.
${n_1} = {n_2} + 1$
B.
${n_1} = {n_2} - 1$
C.
${n_1} = {n_2}$
D.
${n_1} > 0,\,{n_2} > 0$
1996 Q69 JEE Advanced Numerical
14 Mar 2026
Find all non-zero complex numbers Z satisfying $\overline Z = i{Z^2}$.
1996 Q70 JEE Advanced Numerical
14 Mar 2026
The value of the expression
$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\left( {n - {\omega ^2}} \right),$

where $\omega $ is an imaginary cube root of unity, is..........

1995 Q71 JEE Advanced MCQ
14 Mar 2026
Let $z$ and $\omega $ be two complex numbers such that
$\left| z \right| \le 1,$ $\left| \omega \right| \le 1$ and $\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2$ then $z$ equals
A.
$1$ or $i$
B.
$i$ or $-i$
C.
$1$ or $ - 1$
D.
$i$ or $ - 1$
1995 Q72 JEE Advanced MCQ
14 Mar 2026
Let $z$ and $\omega $ be two non zero complex numbers such that
$\left| z \right| = \left| \omega \right|$ and ${\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,$ then $z$ equals
A.
$\omega $
B.
$ - \omega $
C.
$\overline \omega $
D.
$ - \overline \omega $
1995 Q73 JEE Advanced MCQ
14 Mar 2026
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 respectively
A.
0, 1
B.
1, 1
C.
1, 0
D.
-1, 1
1995 Q74 JEE Advanced Numerical
14 Mar 2026
If $\left| {Z - W} \right| \le 1,\left| W \right| \le 1$, show that ${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}$
1995 Q75 JEE Advanced Numerical
14 Mar 2026
If $i{z^3} + {z^2} - z + i = 0$ , then show that $\left| z \right| = 1$.
1994 Q76 JEE Advanced Numerical
14 Mar 2026
Suppose Z1, Z2, Z3 are the vertices of an equilateral triangle inscribed in the circle $\left| Z \right| = 2.$ If Z1 = $1 + i\sqrt 3 $ then Z2 = ......., Z3 =..............
1993 Q77 JEE Advanced Numerical
14 Mar 2026
$ABCD$ is a rhombus. Its diagonals $AC$ and $BD$ intersect at the point $M$ and satisfy $BD$ = 2$AC$. If the points $D$ and $M$ represent the complex numbers $1 + i$ and $2 - i$ respectively, then A represents the comp[lex number ..........or..........
1992 Q78 JEE Advanced MCQ
14 Mar 2026
${\rm{z }} \ne {\rm{0}}$ 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${z^2}$ = 0
(r) Re${z^2}$ = Im${z^2}$
A.
(A) - q, (B) - p
B.
(A) - p, (B) - q
C.
(A) - r, (B) - p
D.
(A) - p, (B) - r
1990 Q79 JEE Advanced Numerical
14 Mar 2026
Let ${z_1}$ = 10 + 6i and ${z_2}$ = 4 + 6i. If Z is any complex number such that the argument of ${{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}$ , then prove that $\left| {z - 7 - 9i} \right| = 3\sqrt 2 $.
1989 Q80 JEE Advanced Numerical
14 Mar 2026
If $a,\,b,\,c,$ are the numbers between 0 and 1 such that the ponts ${z_1} = a + i,{z_2} = 1 + bi$ and ${z_3} = 0$ form an equilateral triangle,
then a= .......and b=..........
1988 Q81 JEE Advanced Numerical
14 Mar 2026
For any two complex numbers ${z_1},{z_2}$ and any real number a and b.

$\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2} = .........$

1988 Q82 JEE Advanced MCQ
14 Mar 2026
The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.
A.
TRUE
B.
FALSE
1987 Q83 JEE Advanced MCQ
14 Mar 2026
If ${{{z_1}}}$ and ${{{z_2}}}$ are two nonzero complex numbers such that $\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{z_2}} \right|\,$, then Arg ${z_1}$ - Arg ${z_2}$ is equal to
A.
$ - \pi $
B.
$ - {\pi \over 2}$
C.
0
D.
${\pi \over 2}$
1987 Q84 JEE Advanced MCQ
14 Mar 2026
The value of $\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$ is
A.
- 1
B.
0
C.
- i
D.
i
1987 Q85 JEE Advanced Numerical
14 Mar 2026
If the expression $${{\left[ {\sin \left( {{x \over 2}} \right) + \cos {x \over 2} + i\,\tan \left( x \right)} \right]} \over {\left[ {1 + 2\,i\,\sin \left( {{x \over 2}} \right)} \right]}}$$

is real, then the set of all possible values of $x$ is ..............

1986 Q86 JEE Advanced MSQ
14 Mar 2026
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 ${z_1}$ has positive real and ${z_2}$ has negative imaginary part, then ${{{z_1}\, + \,{z_2}} \over {{z_1}\, - \,{z_2}}}$ may be
A.
zero
B.
real and positive
C.
real and negative
D.
purely imaginary
1986 Q87 JEE Advanced Numerical
14 Mar 2026
Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is ${1 \over 2}\,{\left| z \right|^2}$ .
1985 Q88 JEE Advanced MCQ
14 Mar 2026
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 $w = \left( {1 - r} \right)u + rv,$ where $w = \left( {1 - r} \right)u + rv,$ is a complex number, then the two triangles
A.
have the same area
B.
are similar
C.
are congruent
D.
none of these
1985 Q89 JEE Advanced MSQ
14 Mar 2026
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 ${\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0$, then the pair of complex numbers ${w_1}$ = a + ic and ${w_2}$ = b+ id satisfies -
A.
$\left| {{w_1}} \right| = 1\,$
B.
$\left| {{w_2}} \right| = 1\,$
C.
${\mathop{\rm Re}\nolimits} ({w_1}\,{\overline w _2}) = 0$
D.
none of these
1985 Q90 JEE Advanced MCQ
14 Mar 2026
If three complex numbers are in A.P. then they lie on a circle in the complex plane.
A.
TRUE
B.
FALSE
1984 Q91 JEE Advanced Numerical
14 Mar 2026
If 1, ${{a_1}}$, ${{a_2}}$......,${a_{n - 1}}$ are the n roots of unity, then show that (1- ${{a_1}}$) (1- ${{a_2}}$) (1- ${{a_3}}$) ....$(1 - \,a{ - _{n - 1}}) = n$
1984 Q92 JEE Advanced MCQ
14 Mar 2026
If the complex numbers, ${Z_1},{Z_2}$ and ${Z_3}$ represent the vertics of an equilateral triangle such that
$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$ then ${Z_1} + {Z_2} + {Z_3} = 0.$
A.
TRUE
B.
FALSE
1983 Q93 JEE Advanced MCQ
14 Mar 2026
If $z = x + iy$ and $\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$ then $\,\left| \omega \right| = 1$ implies that, in the complex plane,
A.
$z$ lies on the imaginary axis
B.
$z$ lies on the real axis
C.
$z$ lies on the unit circle
D.
none of these
1983 Q94 JEE Advanced MCQ
14 Mar 2026
The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order if and only if
A.
z1 + z4 = z2 + z3
B.
z1 + z3 = z2 + z4
C.
z1 + z2 = z3 + z4
D.
None of these
1983 Q95 JEE Advanced Numerical
14 Mar 2026
Prove that the complex numbers ${{z_1}}$, ${{z_2}}$ and the origin form an equilateral triangle only if $z_1^2 + z_2^2 - {z_1}\,{z_2} = 0$.
1982 Q96 JEE Advanced MCQ
14 Mar 2026
The inequality |z-4| < |z-2| represents the region given by
A.
${\mathop{\rm Re}\nolimits} \left( z \right) \ge 0\,\,$
B.
${\mathop{\rm Re}\nolimits} \left( z \right) < 0$
C.
${\mathop{\rm Re}\nolimits} \left( z \right) > 0$
D.
none of these
1982 Q97 JEE Advanced MCQ
14 Mar 2026
If $z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},$ then
A.
${\mathop{\rm Re}\nolimits} \left( z \right) = 0$
B.
${\rm I}m\left( z \right) = 0$
C.
${\mathop{\rm Re}\nolimits} \left( z \right) > 0,\,{\rm I}m\left( z \right) > 0\,$
D.
${\mathop{\rm Re}\nolimits} \left( z \right) > 0,\,{\rm I}m\left( z \right) < 0$
1981 Q98 JEE Advanced MCQ
14 Mar 2026
The complex numbers $z = x + iy$ which satisfy the equation $\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$ lie on
A.
the x-axis
B.
the straight line y=5
C.
a circle passing through the origin
D.
none of these
1981 Q99 JEE Advanced Numerical
14 Mar 2026
Let the complex number ${{z_1}}$, ${{z_2}}$ and ${{z_3}}$ be the vertices of an equilateral triangle. Let ${{z_0}}$ be the circumcentre of the triangle. Then prove that $z_1^2 + z_2^2 + z_3^2 = 3z_0^2$.
1981 Q100 JEE Advanced MCQ
14 Mar 2026
For complex number ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2},$ we write ${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$
Then for all complex numbers $z\,\,with\,\,1 \cap z,$ we have ${{1 - z} \over {1 + z}} \cap 0.$
A.
TRUE
B.
FALSE