If one vertex of a square whose diagonals intersect at the origin is $3\left( {\cos {\rm{\theta }} + i\sin {\rm{\theta }}} \right),$ then the two adjacent vertices are

$ \pm 3\;\left( {\sin {\rm{\theta }} - i\cos {\rm{\theta }}} \right)$

$ \pm \left( {\sin {\rm{\theta }} + i\cos {\rm{\theta }}} \right)$

$ \pm \left( {\cos {\rm{\theta }} - i\sin {\rm{\theta }}} \right)$

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JEE • MATHEMATICS • COMPLEX NUMBERS

Complex numbers practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 901-930 of 979 questions

901 Medium

901 of 979

Understanding +4 | -1

f\left( x \right)

is a polynomial of degree

n

with rational coefficients and

1 + 2i,\;2 - \sqrt 3

and 5 are three roots of

f\left( x \right) = 0

, then the least value of

n

View answer and explanation

Correct option: A

Option A (5) is correct because complex and irrational roots occur in conjugate pairs, so we must include

1-2i

and

2+\sqrt{3}

as roots, along with 5,

1+2i

and

2-\sqrt{3}

, giving a minimum degree of 5.

902 Easy

902 of 979

Application +4 | -1

|z|=1

and

z \neq \pm 1

, then all the values of

\frac{z}{1-z^2}

lie on

View answer and explanation

Correct option: D

the

Y

-axis

Option D (the

Y

-axis) is correct because the expression simplifies to a purely imaginary number, indicating that all values lie on the Y-axis.

903 Easy

903 of 979

Analyzing +4 | -1

(1+i)(1+2 i)(1+3 i) \ldots(1+n i)=\alpha+i \beta

, then

2 \cdot 5 \cdot 10 \ldots\left(1+n^2\right)

is equal to

View answer and explanation

Correct option: C

\alpha^2+\beta^2

Option C (

\alpha^2+\beta^2

) is correct because taking the modulus squared of both sides of the given equation leads to

2 \cdot 5 \cdot 10 \ldots (1+n^2) = \alpha^2 + \beta^2

904 Medium

904 of 979

Analyzing +4 | -1

Let

S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}

. Then

\sum\limits_{{z \in S}}|z|^{2}

is equal to

View answer and explanation

Correct option: B

The correct answer is B (4) because it is found by solving the given complex equation and summing the squared magnitudes of the solutions.

905 Easy

905 of 979

Analyzing +4 | -1

z^2+z+1=0

, where

z

is a complex number, then values of

S=\left(z+\frac{1}{z}\right)^2+\left(z^2+\frac{1}{z^2}\right)^2+\left(z^3+\frac{1}{z^3}\right)^2+\cdots+\left(z^6+\frac{1}{z^6}\right)^2 \text { is }

View answer and explanation

Correct option: A

Option A (12) is correct because by substituting

z = \omega

(a complex cube root of unity) into the expression and using the properties of

\omega

, the sum simplifies to 12.

906 Easy

906 of 979

Application +4 | -1

The value of

{\log _2}{\log _3} \ldots {\log _{100}}{100^{{{99}^{{{98}^{{.^{{.^{{2^1}}}}}}}}}}}

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because the expression simplifies down to log base 2 of 2, which equals 1.

907 Medium

907 of 979

Application +4 | -1

The greatest and the least value of

\left| {{z_1} + {z_2}} \right|,

{z_1} = 24 + 7i

and

\left| {{z_2}} \right| = 6,

are respectively

View answer and explanation

Correct option: A

31,\;19

Option A (31, 19) is correct because the greatest value is the sum of the magnitudes and the least value is the difference of the magnitudes.

908 Easy

908 of 979

Application +4 | -1

Which of the following is a fourth root of

\frac{1}{2} + i\frac{{\sqrt 3 }}{2}

View answer and explanation

Correct option: A

\mathrm{cis}\frac{\pi }{{12}}

Option A (

\mathrm{cis}\frac{\pi }{{12}}

) is correct because it represents the fourth root of the given complex number in polar form.

909 Easy

909 of 979

Application +4 | -1

z=\frac{1}{2}-2 i

is such that

|z+1|=\alpha z+\beta(1+i)

i=\sqrt{-1}

and

\alpha, \beta \in {R}

, then

\alpha+\beta

is equal to

View answer and explanation

Correct option: D

Option D (3) is correct because solving the equation |z+1|=αz+β(1+i) for α and β yields α=1 and β=2, so α+β=3.

910 Easy

910 of 979

Understanding +4 | -1

The points representing complex number

z

for which

\left| {z - 3} \right| = \left| {z - 5} \right|

lie on the locus given by

View answer and explanation

Correct option: C

A straight line

Option D (Perpendicular bisector of line joining z1 and z2) is correct because the given condition implies that the point P is equidistant from z1 and z2, which defines the perpendicular bisector.

911 Easy

911 of 979

Analyzing +4 | -1

\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1

and

z_1+z_2+z_3=0

, then area of the triangle whose vertices are

z_1, z_2, z_3

View answer and explanation

Correct option: A

3 \sqrt{3} / 4

Option A (

3 \sqrt{3} / 4

) is correct because the given conditions imply that the vertices form an equilateral triangle with side

\sqrt{3}

, and thus the area is

3 \sqrt{3} / 4

912 Medium

912 of 979

Application +4 | -1

The modulus of

\frac{(3+2 i)^2}{(4-3 i)}

View answer and explanation

Correct option: A

\frac{13}{5}

Option A (

\frac{13}{5}

) is correct because the modulus of a quotient is the quotient of the moduli, and the modulus of a product is the product of the moduli.

913 Easy

913 of 979

Analyzing +4 | -1

{z_1},\;{z_2},\;{z_3},\;{z_4}

are four complex numbers represented by the vertices of a quadrilateral taken in order such that

{z_1} - {z_4} = {z_2} - {z_3}\;\;\mathrm{and}\;\arg \left( {\frac{{{z_4} - {z_1}}}{{{z_2} - {z_1}}}} \right) = \frac{\pi }{2},\;

then the quadrilateral is

View answer and explanation

Correct option: B

A rectangle

Option B is correct because the quadrilateral is a rectangle, as the diagonals bisect each other and one angle is

\frac{\pi}{2}

914 Easy

914 of 979

Analyzing +4 | -1

The number of complex numbers

z

such that

\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|

equals

View answer and explanation

Correct option: B

Option B (1) is correct because the only complex number satisfying the given conditions is

z = i

915 Hard

915 of 979

Application +4 | -1

1, \omega, \omega^2, \ldots, \omega^{n-1}

are the

n, n

th roots of unity and

z_1, z_2

are any two complex numbers, then

\sum_{k=0}^{n-1}\left|z_1+\omega^k z_2\right|^2

is equal to

View answer and explanation

Correct option: A

n\left[\left|z_1\right|^2+\left|z_2\right|^2\right]

Option A (

n[|z_1|^2+|z_2|^2]

) is correct because it is derived from expanding the sum and using the properties of nth roots of unity.

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916 Easy

916 of 979

Understanding +4 | -1

z

is a complex number, then

View answer and explanation

Correct option: B

\left|z^2\right|=|z|^2

Option B (|z²|=|z|^2) is correct because the magnitude of the square of a complex number is equal to the square of its magnitude.

917 Easy

917 of 979

Application +4 | -1

n

is a positive integer, then

{\left( {1 + i} \right)^n} + {\left( {1 - i} \right)^n}

is equal to

View answer and explanation

Correct option: C

{\left( {\sqrt 2 } \right)^{n + 2}}\cos \left( {\frac{{n\pi }}{4}} \right)

Option C (

{\left( {\sqrt 2 } \right)^{n + 2}}\cos \left( {\frac{{n\pi }}{4}} \right)

) is correct because it is the simplified form of the given expression using De Moivre's theorem.

918 Easy

918 of 979

Application +4 | -1

\alpha

is a complex number satisfying the equation

{\alpha ^2} + \alpha + 1 = 0

, then

{\alpha ^{31}}

is equal to

View answer and explanation

Correct option: A

\alpha

Option A (

\alpha

) is correct because

{\alpha ^{31}}

simplifies to

\alpha

when

\alpha

is a complex number satisfying the given equation.

919 Easy

919 of 979

Analyzing +4 | -1

The locus of the points representing the complex numbers

z

for which

\left| z \right| - 2 = \left| {z - i} \right| - \left| {z + 5\;i} \right| = 0

View answer and explanation

Correct option: C

The single point

\left( {0,\; - 2} \right)

Option C is correct. The locus of the complex number z is the single point (0, -2) because it satisfies both the circle equation and the perpendicular bisector condition.

920 Medium

920 of 979

Analyzing +4 | -1

|z|=1, z \neq 1

, then value of

\arg \left(\frac{1}{1-z}\right)

cannot exceed

View answer and explanation

Correct option: A

\pi / 2

Option A (

\pi / 2

) is correct because the argument of the given expression is bounded by

-\pi/2

and

\pi/2

, and thus cannot exceed

\pi/2

921 Easy

921 of 979

Application +4 | -1

x+i y=\frac{a+i b}{a-i b}

, then

x^2+y^2=

View answer and explanation

Correct option: A

Option A (1) is correct because when x+iy is multiplied by its conjugate, the result is 1, which means x² + y² = 1.

922 Medium

922 of 979

Application +4 | -1

\alpha

and

\beta

are the roots of

x^2-2 x+2=0

, then

\alpha^n+\beta^n=

View answer and explanation

Correct option: B

2^{\frac{n}{2}+1} \cos \frac{n \pi}{4}

Option B,

2^{\frac{n}{2}+1} \cos \frac{n \pi}{4}

, is correct because it is the simplified form of the expression

\alpha^n + \beta^n

where

\alpha

and

\beta

are the roots of the given quadratic equation.

923 Easy

923 of 979

Application +4 | -1

The value of

\left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right)\left( {2 - {\omega ^{10}}} \right)\left( {2 - {\omega ^{11}}} \right)

, where

\omega

is the complex cube root of unity, is

View answer and explanation

Correct option: A

Option A (49) is correct because by using the property that

\omega^3 = 1

and

1 + \omega + \omega^2 = 0

, the expression simplifies to 49.

924 Easy

924 of 979

Analyzing +4 | -1

Let

z

and

w

be two complex numbers such that

|z|=|w|=1

and

|z+i w|=|z-i \bar{w}|=2

. Then

z

equals

View answer and explanation

Correct option: C

1 or -1

Option C (1 or -1) is correct because the given conditions imply that z must be a real number with magnitude 1.

925 Easy

925 of 979

Application +4 | -1

The number of complex numbers

z

which satisfy

z^2+2|z|^2=2

View answer and explanation

Correct option: D

Option D (4) is correct because by considering both real and imaginary cases for z, we find four complex numbers that satisfy the given equation.

926 Easy

926 of 979

Application +4 | -1

The radius of the circle

\left| {\frac{{z - i}}{{z + i}}} \right| = 5

is given by

View answer and explanation

Correct option: B

\frac{5}{{12}}

Option B (5/12) is correct because the radius of the circle is calculated using the formula derived from the given complex equation.

927 Easy

927 of 979

Analyzing +4 | -1

|z-2-i|=|z| \mid \sin \left(\frac{\pi}{4}-\arg (z)\right)

, then locus of

z

View answer and explanation

Correct option: C

a parabola

Option C (a parabola) is correct because the given equation simplifies to the form

\sqrt{{(x-2)^2 + (y-1)^2}} = \frac{{1}}{{\sqrt{{2}}}} |x-y|

, which represents a parabola.

928 Medium

928 of 979

Application +4 | -1

x=\omega-\omega^2-2

, then the value of

x^4+3 x^3+2 x^2-11 x-6

View answer and explanation

Correct option: A

Option A (1) is correct because by substituting

x = \omega - \omega^2 - 2

and simplifying, the value of the expression is 1.

929 Easy

929 of 979

Application +4 | -1

\omega

is a cube root of unity, then the value of

{\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5}

View answer and explanation

Correct option: B

Option B (32) is correct because using the properties of cube roots of unity (1 + ω + ω² = 0 and ω³ = 1), the expression simplifies to 32.

930 Medium

930 of 979

Application +4 | -1

If one vertex of a square whose diagonals intersect at the origin is

3\left( {\cos \mathrm{\theta } + i\sin \mathrm{\theta }} \right),

then the two adjacent vertices are

View answer and explanation

Correct option: A

\pm 3\;\left( {\sin \mathrm{\theta } - i\cos \mathrm{\theta }} \right)

Option A (

\pm 3\;\left( {\sin \mathrm{\theta } - i\cos \mathrm{\theta }} \right)

) is correct because rotating the given vertex by

\pm \frac{\pi}{2}

gives the adjacent vertices.

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