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Complex Numbers

523 Questions Start JEE Mains Test
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1987 Q501 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 Q502 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 Q503 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 Q504 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 Q505 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 Q506 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 Q507 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 Q508 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 Q509 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 Q510 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 Q511 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 Q512 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 Q513 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 Q514 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 Q515 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 Q516 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 Q517 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
1980 Q518 JEE Advanced MCQ
14 Mar 2026
The smallest positive integer n for which ${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$ is
A.
$n = 8$
B.
$n = 16$
C.
$n = 12$
D.
none of these
1980 Q519 JEE Advanced Numerical
14 Mar 2026
Find the real values of x and y for which the following equation is satisfied $\,{{(1 + i)x - 2i} \over {3 + i}} + {{(2 + 3i)y + i} \over {3 - i}} = i$
1979 Q520 JEE Advanced MCQ
14 Mar 2026
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
A.
$ - 1,1 + 2\omega ,\,1 + 2{\omega ^2}$
B.
$ - 1,1 - 2\omega ,\,1 - 2{\omega ^2}$
C.
$ - 1, - 1, - 1$
D.
None of these
1979 Q521 JEE Advanced Numerical
14 Mar 2026
If x + iy = $\sqrt {{{a + ib} \over {c + id}}} $, prove that ${({x^2} + {y^2})^2} = {{{a^2} + {b^2}} \over {{c^2} + {d^2}}}$.
1978 Q522 JEE Advanced Numerical
14 Mar 2026
Express ${1 \over {1 - \cos \,\theta + 2i\sin \theta }}$ in the form x + iy.
1978 Q523 JEE Advanced Numerical
14 Mar 2026
If x = a + b, y = a$\gamma $ + b$\beta $ and z = a$\beta $ +b$\gamma $ where $\gamma $ and $\beta $ are the complex cube roots of unity, show that xyz = ${a^3} + {b^3}$.