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Complex Numbers

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

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

2003 JEE Advanced MCQ
IIT-JEE 2003 Screening
If $\,\left| z \right| = 1$ and $\omega = {{z - 1} \over {z + 1}}$ (where $z \ne - 1$), then ${\mathop{\rm Re}\nolimits} \left( \omega \right)$ 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 JEE Advanced Numerical
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 $\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1$.
2003 JEE Advanced Numerical
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 $\left| {{a_r}} \right| < 2$.
2002 JEE Advanced MCQ
IIT-JEE 2002
Let $\omega $ $ = - {1 \over 2} + i{{\sqrt 3 } \over 2},$ then the value of the det.
$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|$ is
A.
$3\omega $
B.
$3\omega \left( {\omega - 1} \right)$
C.
$3{\omega ^2}$
D.
$3\omega \left( {1 - \omega } \right)$
2002 JEE Advanced MCQ
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 of $\left| {{z_1} - {z_2}} \right|$ is
A.
0
B.
2
C.
7
D.
17
2002 JEE Advanced Numerical
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. 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 JEE Advanced MCQ
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 vertices of a triangle which is
A.
of area zero
B.
right-angled isosceles
C.
equilateral
D.
obtuse-angled isosceles
2001 JEE Advanced MCQ
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
A.
$4k + 1$
B.
$4k + 2$
C.
$4k + 3$
D.
$4k$
2000 JEE Advanced MCQ
IIT-JEE 2000 Screening
If ${z_1},\,{z_2}$ and ${z_3}$ are complex numbers such that $\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right| = 1,$ 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 JEE Advanced MCQ
IIT-JEE 2000 Screening
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 JEE Advanced MCQ
IIT-JEE 1999
$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$ is equal to
A.
$1 - i\sqrt 3 $
B.
$ - 1 + i\sqrt 3 $
C.
$i\sqrt 3 $
D.
$ - i\sqrt 3 $
1999 JEE Advanced Numerical
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$.
1998 JEE Advanced MCQ
IIT-JEE 1998
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 JEE Advanced MCQ
IIT-JEE 1998
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 JEE Advanced MCQ
IIT-JEE 1998
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 JEE Advanced Numerical
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 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 JEE Advanced MCQ
IIT-JEE 1996
For positive integers ${n_1},\,{n_2}$ the value of the expression ${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$
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 JEE Advanced Numerical
IIT-JEE 1996
Find all non-zero complex numbers Z satisfying $\overline Z = i{Z^2}$.
1996 JEE Advanced Numerical
IIT-JEE 1996
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 JEE Advanced MCQ
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 $\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2$ then $z$ equals
A.
$1$ or $i$
B.
$i$ or $-i$
C.
$1$ or $ - 1$
D.
$i$ or $ - 1$
1995 JEE Advanced MCQ
IIT-JEE 1995 Screening
Let $z$ and $\omega $ be two non zero complex numbers such that
$\left| z \right| = \left| \omega \right|$ and ${\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,$ then $z$ equals
A.
$\omega $
B.
$ - \omega $
C.
$\overline \omega $
D.
$ - \overline \omega $
1995 JEE Advanced MCQ
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 respectively
A.
0, 1
B.
1, 1
C.
1, 0
D.
-1, 1
1995 JEE Advanced Numerical
IIT-JEE 1995
If $\left| {Z - W} \right| \le 1,\left| W \right| \le 1$, show that ${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}$
1995 JEE Advanced Numerical
IIT-JEE 1995
If $i{z^3} + {z^2} - z + i = 0$ , then show that $\left| z \right| = 1$.
1994 JEE Advanced Numerical
IIT-JEE 1994
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 JEE Advanced Numerical
IIT-JEE 1993
$ABCD$ 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 JEE Advanced MCQ
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${z^2}$ = 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 JEE Advanced Numerical
IIT-JEE 1990
Let ${z_1}$ = 10 + 6i and ${z_2}$ = 4 + 6i. If Z is any complex number such that the argument of ${{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}$ , then prove that $\left| {z - 7 - 9i} \right| = 3\sqrt 2 $.
1989 JEE Advanced Numerical
IIT-JEE 1989
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 JEE Advanced Numerical
IIT-JEE 1988
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 JEE Advanced MCQ
IIT-JEE 1988
The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.
A.
TRUE
B.
FALSE
1987 JEE Advanced MCQ
IIT-JEE 1987
If ${{{z_1}}}$ and ${{{z_2}}}$ are two nonzero complex numbers such that $\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{z_2}} \right|\,$, then Arg ${z_1}$ - Arg ${z_2}$ is equal to
A.
$ - \pi $
B.
$ - {\pi \over 2}$
C.
0
D.
${\pi \over 2}$
1987 JEE Advanced MCQ
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
A.
- 1
B.
0
C.
- i
D.
i
1987 JEE Advanced Numerical
IIT-JEE 1987
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 JEE Advanced MSQ
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 ${z_1}$ 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 JEE Advanced Numerical
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 ${1 \over 2}\,{\left| z \right|^2}$ .
1985 JEE Advanced MCQ
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 $w = \left( {1 - r} \right)u + rv,$ 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 JEE Advanced MSQ
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 ${\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0$, 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 JEE Advanced MCQ
IIT-JEE 1985
If three complex numbers are in A.P. then they lie on a circle in the complex plane.
A.
TRUE
B.
FALSE
1984 JEE Advanced Numerical
IIT-JEE 1984
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 JEE Advanced MCQ
IIT-JEE 1984
If the complex numbers, ${Z_1},{Z_2}$ and ${Z_3}$ represent the vertics of an equilateral triangle such that
$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$ then ${Z_1} + {Z_2} + {Z_3} = 0.$
A.
TRUE
B.
FALSE
1983 JEE Advanced MCQ
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 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 JEE Advanced MCQ
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
A.
z1 + z4 = z2 + z3
B.
z1 + z3 = z2 + z4
C.
z1 + z2 = z3 + z4
D.
None of these
1983 JEE Advanced Numerical
IIT-JEE 1983
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 JEE Advanced MCQ
IIT-JEE 1982
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 JEE Advanced MCQ
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
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 JEE Advanced MCQ
IIT-JEE 1981
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 JEE Advanced Numerical
IIT-JEE 1981
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 JEE Advanced MCQ
IIT-JEE 1981
For complex number ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2},$ we write ${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$
Then for all complex numbers $z\,\,with\,\,1 \cap z,$ we have ${{1 - z} \over {1 + z}} \cap 0.$
A.
TRUE
B.
FALSE