If $z=\cos \theta+i \sin \theta$, then

$z^n+\frac{1}{z^n}=2^n \cos n \theta$

$z^n-\frac{1}{z^n}=2^n i \sin n \theta$

$z^n-\frac{1}{z^n}=(2 i)^n \sin n \theta$

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JEE • MATHEMATICS • COMPLEX NUMBERS

Complex numbers practice · Easy

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Practice format and chapter quiz

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 91-120 of 612 questions

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91 Easy

91 of 612

Understanding +4 | -1

The inequality

|z-i| < |z+i|

represents the region

View answer and explanation

Correct option: C

\operatorname{Im}(z) > 0

Option C (

\operatorname{Im}(z) > 0

) is correct because the inequality

|z-i| < |z+i|

represents the set of all points in the complex plane that are closer to

i

than to

-i

, which is the upper half-plane.

92 Easy

92 of 612

Application +4 | -1

\omega

is a complex cube root of unity, then for positive integral value of

n

, the product of

\omega .\;{\omega ^2}.\;{\omega ^3} \ldots \;{\omega ^n}

will be

View answer and explanation

Correct option: D

Both (2) and (3)

Option D (Both (2) and (3)) is correct because the product simplifies to

\omega^{\frac{{n(n+1)}}{{2}}}

, which can be either 1 or

\omega

depending on the value of n.

93 Easy

93 of 612

Analyzing +4 | -1

The value of

\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}

is equal to

View answer and explanation

Correct option: B

\frac{-1}{2}

Option B (-1/2) is correct because the value is obtained by multiplying and dividing by

2\sin(\frac{2\pi}{7})

and using trigonometric identities.

94 Easy

94 of 612

Analyzing +4 | -1

Let

a = {e^{i\frac{{2\pi }}{3}\;}}

Then, the equation whose roots are

a + {a^{ - 2}}

and

{a^2} + {a^{ - 4}}

View answer and explanation

Correct option: D

{x^2} + 2x + 4 = 0

Option D (

{x^2} + 2x + 4 = 0

) is the correct equation, found by determining the roots

a + a^{-2}

and

a^2 + a^{-4}

and then constructing the quadratic equation.

95 Easy

95 of 612

Application +4 | -1

z=\cos \theta+i \sin \theta

, then

View answer and explanation

Correct option: A

z^n+\frac{1}{z^n}=2 \cos n \theta

Option A (

z^n+\frac{1}{z^n}=2 \cos n \theta

) is correct because it follows directly from De Moivre's theorem and Euler's formula.

96 Easy

96 of 612

Application +4 | -1

Let

\alpha

be a root of the equation

1+x^2+x^4=0

. Then the value of

\alpha^{1011}+\alpha^{2022}-\alpha^{3033}

is equal to

View answer and explanation

Correct option: A

1

Option A (1) is correct because after simplifying the expression

\alpha^{1011}+\alpha^{2022}-\alpha^{3033}

using the properties of roots of unity, the result is 1.

97 Easy

97 of 612

Analyzing +4 | -1

z^3+(3+2 i) z+(-1+i a)=0

, where

i=\sqrt{-1}

, has one real root, the value of

a

lies in the interval

(a \in R)

View answer and explanation

Correct option: B

(-1,0)

Option B ((-1,0)) is correct because the real root condition leads to a cubic equation in 'a', and analyzing the function reveals that 'a' lies in the interval (-1, 0).

98 Easy

98 of 612

Application +4 | -1

The value of

\sqrt i

View answer and explanation

Correct option: D

\pm \frac{1}{{\sqrt 2 }}\left( {1 + i} \right)

Option D (

\pm \frac{1}{{\sqrt 2 }}\left( {1 + i} \right)

) is correct because it is the simplified form of the square root of i.

99 Easy

99 of 612

Application +4 | -1

x = a + b,\;y + a\alpha + b\beta \;\,\mathrm{and\;}\,\mathrm{z} = a\beta + b\alpha ,\;\,\mathrm{where}\,\;\alpha \,\;\mathrm{and}\;\beta

are complex cube roots of unity, then

xyz

is equal to

View answer and explanation

Correct option: B

{a^3} + {b^3}

The correct answer is B (

{a^3} + {b^3}

) because it is obtained by multiplying the given expressions and using the properties of complex cube roots of unity.

100 Easy

100 of 612

Application +4 | -1

The value of

{\left( {\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}} \right)^6} + {\left( {\frac{{1 - i\sqrt 3 }}{{1 + i\sqrt 3 }}} \right)^6}

View answer and explanation

Correct option: A

Option A (2) is correct because the expression simplifies to

{\omega ^6} + \frac{1}{{{\omega ^6}}}

, which equals 2 using properties of complex cube roots of unity.

101 Easy

101 of 612

Application +4 | -1

The locus of point

z

satisfying

\mathrm{Re}\left( {{z^2}} \right) = 0,

View answer and explanation

Correct option: A

A pair of straight lines

Option A (A pair of straight lines) is correct because the equation Re(z²) = 0 simplifies to y = ±x, representing two straight lines.

102 Easy

102 of 612

Application +4 | -1

Let

{z}_1=6+{i}

and

{z}_2=4-3 {i}

(where

{i}=\sqrt{-1}

). Let

{z}

be a complex number such that

\arg \left(\frac{z-z_1}{z_2-z}\right)=\frac{\pi}{2}

, then

z

satisfies

View answer and explanation

Correct option: B

|z-(5-i)|=\sqrt{5}

Option B,

|z-(5-i)|=\sqrt{5}

, is correct because the locus of

z

is a circle with center at the midpoint of

z_1

and

z_2

and radius equal to half the distance between them.

103 Easy

103 of 612

Application +4 | -1

\frac{{{{\left( {1 + i} \right)}^2}}}{{2 - i}} = x + iy

, then

x + y

is equal to

View answer and explanation

Correct option: C

\frac{2}{5}

Option C (

\frac{2}{5}

) is correct because after simplifying the given expression and equating it to x + iy, the sum x + y equals

\frac{2}{5}

104 Easy

104 of 612

Analyzing +4 | -1

The triangle formed by the points 1,

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

and

i

as vertices in the Argand diagrams is

View answer and explanation

Correct option: C

Isosceles

Option C (Isosceles) is correct because two sides of the triangle formed by the given complex numbers have equal lengths.

105 Easy

105 of 612

Application +4 | -1

\left|\begin{array}{ccc}6 i & -3 i & 1 \\ 4 & 3 i & -1 \\ 20 & 3 & i\end{array}\right|=x+i y

, then

View answer and explanation

Correct option: D

x=0, y=0

Option D (

x=0, y=0

) is correct because the determinant evaluates to 0, resulting in both the real and imaginary parts being zero.

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106 Easy

106 of 612

Understanding +4 | -1

If the complex numbers

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

are in AP, then they lie on

View answer and explanation

Correct option: C

A straight line

Option C (A straight line) is correct because if three complex numbers are in arithmetic progression (AP), they are collinear and lie on a straight line.

107 Easy

107 of 612

Application +4 | -1

{z_1} = 1 + 2i

and

{z_2} = 3 + 5i

, then

{\mathop\mathrm{Re}\nolimits} \left[ {{{\bar z}_2}{z_1}/{z_2}} \right]

is equal to

View answer and explanation

Correct option: D

\frac{{22}}{{17}}

Option D (22/17) is correct because it is the real part of the complex number obtained after simplifying the given expression.

108 Easy

108 of 612

Application +4 | -1

The value of

\left(\frac{1+i}{\sqrt{2}}\right)^8+\left(\frac{1-i}{\sqrt{2}}\right)^8

is equal to

View answer and explanation

Correct option: D

Option D (2) is correct because after converting to polar form and applying De Moivre's theorem, the expression simplifies to 2.

109 Easy

109 of 612

Application +4 | -1

The value of

(1+i)^8+(1-i)^8

2^n

. Value of

n

View answer and explanation

Correct option: A

Option A (5) is correct because

(1+i)^8+(1-i)^8

simplifies to

2^5

, thus n=5.

110 Easy

110 of 612

Application +4 | -1

The locus of the complex number

z

in the Argand plane if

\left|\frac{z-3}{z+3}\right|=2

, is

View answer and explanation

Correct option: C

A circle

Option C (A circle) is correct because the given condition represents a circle in the Argand plane.

111 Easy

111 of 612

Application +4 | -1

z = \frac{{\sqrt 3 }}{2} + \frac{i}{2}(i = \sqrt { - 1} ),

then

{(1 + iz + {z^5} + i{z^8})^9}

is eqaul to

View answer and explanation

Correct option: C

-1

Option C (-1) is correct because after simplifying the expression using Euler's formula and trigonometric identities, the result is -1.

112 Easy

112 of 612

Understanding +4 | -1

\arg \overline{{z}}+\arg {z} ; {z} \neq 0

is equal to:

View answer and explanation

Correct option: A

Option A (0) is correct because the argument of the conjugate of a complex number is the negative of the argument of the original complex number.

113 Easy

113 of 612

Application +4 | -1

In the argand plane, all the complex numbers satisfying

|z-4 i|+|z+4 i|=10

lie on

View answer and explanation

Correct option: C

an ellipse

Option C (an ellipse) is correct because the given equation represents the locus of a point where the sum of its distances from two fixed points is constant, which is the definition of an ellipse.

114 Easy

114 of 612

Application +4 | -1

\frac{{{{\left( {\cos \theta + i\sin \theta } \right)}^4}}}{{{{\left( {\sin \theta + i\cos \theta } \right)}^5}}}

is equal to

View answer and explanation

Correct option: D

\sin 9\theta - i\cos 9\theta

Option D (

\sin 9\theta - i\cos 9\theta

) is correct because by applying De Moivre's theorem and simplifying, the expression reduces to the given form.

115 Easy

115 of 612

Analyzing +4 | -1

z\left( { \ne - 1} \right)

is a complex number such that

\frac{{z - 1}}{{z + 1}}

is purely imaginary, then

\left| z \right| =

View answer and explanation

Correct option: A

Option A (1) is correct because the given condition implies that the magnitude of z is 1.

116 Easy

116 of 612

Application +4 | -1

If the centre of a regular hexagon is at the origin and one of its vertices on argand diagram is

1 + 2i,

then its perimeter is

View answer and explanation

Correct option: D

6\sqrt 5

Option D (

6\sqrt 5

) is correct because the perimeter of a regular hexagon is six times the length of one of its sides, which is equal to the distance from the origin to a vertex.

117 Easy

117 of 612

Application +4 | -1

The value of

{\left[ {\sqrt 2 \left\{ {\cos \left( {{{56}^0}15'} \right) + i\sin \left( {{{56}^0}\;15'} \right)} \right\}} \right]^8}

View answer and explanation

Correct option: C

16i

Option C (

16i

) is correct because, using De-Moivre's Theorem, the expression simplifies to

16i

118 Easy

118 of 612

Analyzing +4 | -1

a,\;b,\;c

are distinct positive numbers each being different from 1 such that

\left( {{{\log }_b}a.{{\log }_c}a - {{\log }_a}a} \right) + \left( {{{\log }_a}b.{{\log }_c}b - {{\log }_b}b} \right) + \left( {{{\log }_a}c.{{\log }_b}c - {{\log }_c}c} \right) = 0

, then

abc

View answer and explanation

Correct option: C

Option C (1) is correct because simplifying the given logarithmic expression leads to log(abc) = 0, which implies abc = 1.

119 Easy

119 of 612

Application +4 | -1

The conjugate of a complex number is

\frac{1}{{i - 1}}

. Then, that complex number is

View answer and explanation

Correct option: D

- \frac{1}{{i + 1}}

Option D (

- \frac{1}{{i + 1}}

) is correct because it's the complex number whose conjugate is the given expression.

120 Easy

120 of 612

Application +4 | -1

x=a+b, y=a \omega+b \omega^2, z=a \omega^2+b \omega

, then

x y z

is equal to

View answer and explanation

Correct option: B

a^3+b^3

Option B (

a^3+b^3

) is correct because the expression simplifies to

a^3 + b^3

using the properties of cube roots of unity.

Revision notes and key takeaways

Review COMPLEX NUMBERS in your textbook or class notes, then use this set to check your understanding.

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