If $a = \cos {\rm{\theta }} + i\sin {\rm{\theta }},{\rm{\;then\;}}\frac{{1 + a}}{{1 - a}}$ is equal to

$i\cot \frac{{\rm{\theta }}}{2}$

$\cot \frac{{\rm{\theta }}}{2}$

$i\tan \frac{{\rm{\theta }}}{2}$

If $z_1$ lies on $|z|=r$, then equation of tangent at $z_1$ is

$\frac{z}{\bar{z}_1}+\frac{\bar{z}}{z_1}=r$

$\frac{z}{\bar{z}_1}+\frac{\bar{z}}{z_1}=2$

$\frac{z}{z_1}+\frac{\bar{z}}{\bar{z}_1}=r$

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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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721 Medium

721 of 979

Application +4 | -1

The minimum value of

\left| {a + b\omega + c{\omega ^2}} \right|

, where

a,\;b

and

c

are all not equal integers and

\omega \left( { \ne 1} \right)

is a cube root of unity, is

View answer and explanation

Correct option: C

Option C (1) is correct because the minimum value of the expression is 1 when a, b, and c are distinct integers.

722 Medium

722 of 979

Application +4 | -1

Let

{z_1},{z_2},{z_3}

be three vertices of an equilateral triangle circumscribing the circle

\left| z \right| = \frac{1}{2}.

{z_1} = \frac{1}{2} + i\frac{{\sqrt 3 }}{2}

and

{z_1},{z_2},{z_3}

were in anticlockwise sense, then

{z_2}

View answer and explanation

Correct option: D

- 1

The correct answer is D (

- 1

) because, in an equilateral triangle, the vertices are equally spaced around the circumscribing circle, and using the properties of complex numbers, z2 = -1.

723 Easy

723 of 979

Application +4 | -1

\alpha

and

\beta

are the complex cube roots of unity, then

\alpha^4+\beta^4+\alpha^{-1} \beta^{-1}=

View answer and explanation

Correct option: A

Option A (0) is correct because, after substituting

\alpha = \omega

and

\beta = \omega^2

and simplifying using the properties of cube roots of unity, the expression evaluates to 0.

724 Easy

724 of 979

Application +4 | -1

The argument of the complex number

\frac{{13 - 5i}}{{4 - 9i}}

View answer and explanation

Correct option: B

\frac{\pi }{4}

Option B

\frac{\pi }{4}

is correct because it is the argument of the simplified complex number, found by rationalizing the denominator and calculating the arctangent of the imaginary to real part ratio.

725 Easy

725 of 979

Application +4 | -1

\left|z_1-1\right|<1,\left|z_2-2\right|<2

and

\left|z_3-3\right|<3

, then

\mid z_1+

z_2+z_3 \mid

View answer and explanation

Correct option: C

is less than 12

Option C (is less than 12) is correct because by triangle inequality, the magnitude of the sum is less than the sum of the magnitudes.

726 Easy

726 of 979

Analyzing +4 | -1

z \in \mathbf{C}, z \notin \mathbf{R}

, and

a=z^2+3 z+5

, then

a

cannot take value

View answer and explanation

Correct option: C

11 / 4

Option C (11/4) is correct because the expression a = z² + 3z + 5 cannot take the value 11/4 when z is not a real number.

727 Easy

727 of 979

Analyzing +4 | -1

The value of

\sum\limits_{{k=1}}^{n-1}(n-k) \cos \frac{2 k \pi}{n} \quad(n \geq 3

is an integer)

View answer and explanation

Correct option: A

-\frac{n}{2}

Option A (-n/2) is correct because the sum can be simplified using trigonometric series identities.

728 Medium

728 of 979

Application +4 | -1

Principal Argument of complex number

\frac{(1+i)^5(1+\sqrt{3} i)^2}{-2 i(-\sqrt{3}+i)}

View answer and explanation

Correct option: C

\frac{-5 \pi}{12}

Option C (-5π/12) is correct because it is the principal argument obtained after calculating the argument of the complex number and adjusting to the range (-π, π].

729 Hard

729 of 979

Application +4 | -1

When

f(z)

is divided by

z-i

, the remainder is

i

and if

f(z)

is divided by

z+i

, the remainder is

1+i

. If

f(z)

is divided by

z^2+1

, then the remainder is

View answer and explanation

Correct option: B

\frac{1}{2}(i z+1+2 i)

Option B,

\frac{1}{2}(i z+1+2 i)

, is correct because it represents the remainder when f(z) is divided by

z^2+1

, obtained using the remainder theorem.

730 Easy

730 of 979

Application +4 | -1

\omega

is a non-real cube root of unity, then the value of

\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}+\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}

is equal to:

View answer and explanation

Correct option: D

-1

Option D (-1) is correct because by multiplying the numerator and denominator of each fraction by appropriate powers of

\omega

, the expression simplifies to

\omega + \frac{{1}}{{\omega}} = -1

731 Easy

731 of 979

Analyzing +4 | -1

Let

S=\left\{z=x+i y: \frac{2 z-3 i}{4 z+2 i}\right.

is a real number

\}

. Then which of the following is NOT correct?

View answer and explanation

Correct option: B

(x, y)=\left(0,-\frac{1}{2}\right)

Option B

((x, y) = (0, -\frac{{1}}{{2}}))

is correct because when

x = 0

and

y = -\frac{{1}}{{2}}

, the denominator of the expression becomes zero, making the expression undefined, thus it is not a real number.

732 Medium

732 of 979

Application +4 | -1

a = \cos \mathrm{\theta } + i\sin \mathrm{\theta },\mathrm{\;then\;}\frac{{1 + a}}{{1 - a}}

is equal to

View answer and explanation

Correct option: C

i\cot \frac{\mathrm{\theta }}{2}

Option C (

i\cot \frac{\mathrm{\theta }}{2}

) is correct because it is the simplified form of the given expression using Euler's formula and trigonometric identities.

733 Easy

733 of 979

Application +4 | -1

z=(3+7 i)(\lambda+i \mu)

, when

\lambda, \mu \in I \sim\{0\}

and

i=\sqrt{-1}

, is purely imaginary then minimum value of

|z|^2

View answer and explanation

Correct option: D

3364

Option D (3364) is correct because it is the minimum value of

|z|^2

when z is purely imaginary, calculated using the given conditions.

734 Medium

734 of 979

Application +4 | -1

Value of

\sum \limits_{k = 1}^6 \left[ {\sin \left( {\frac{{2k\pi }}{7}} \right) - i\cos \left( {\frac{{2k\pi }}{7}} \right)} \right]

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the summation simplifies to 'i', which is not among the given options.

735 Easy

735 of 979

Understanding +4 | -1

The value of

1 + {i^2} + {i^4} + {i^6} + \ldots + {i^{2n}}

View answer and explanation

Correct option: D

Cannot be determined

Option D (Cannot be determined) is correct because the sum depends on the value of

n

, which is not specified.

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736 Easy

736 of 979

Application +4 | -1

The smallest positive integer

n

for which

[(1+i) /

(1-i)]^n=1

View answer and explanation

Correct option: B

Option B (4) is correct because the expression simplifies to i, and i raised to the power of 4 equals 1, which is the smallest positive integer that satisfies the equation.

737 Medium

737 of 979

Application +4 | -1

The principal value of the

\arg (z)

and

|z|

of the complex number

z=\frac{(1+\cos \theta+i \sin \theta)^5}{(\cos \theta+i \sin \theta)^3}

are

View answer and explanation

Correct option: A

-\frac{\theta}{2}, 32 \cos ^5 \frac{\theta}{2}

Option A (-

\frac{{\theta}}{{2}}

32 \cos ^5 \frac{{\theta}}{{2}}

) is correct because after simplifying using trigonometric identities and De Moivre's theorem, the argument and modulus are -

\frac{{\theta}}{{2}}

and

32 \cos ^5 \frac{{\theta}}{{2}}

, respectively.

738 Medium

738 of 979

Application +4 | -1

z=x+i y

is a complex number with

x, y \in Q

and

|z|=1

, then

\left|z^{2 n}-1\right|

is a _____ for every

n \in N

View answer and explanation

Correct option: B

Rational number

Option B (Rational number) is correct because given the conditions, |z⁽²ⁿ⁾ - 1| can be shown to be a rational number using trigonometric identities and properties of rational numbers.

739 Easy

739 of 979

Application +4 | -1

z_1

lies on

|z|=r

, then equation of tangent at

z_1

View answer and explanation

Correct option: A

\frac{z}{z_1}+\frac{\bar{z}}{\bar{z}_1}=2

Option A (

\frac{{z}}{{z_1}}+\frac{{\bar{z}}}{{\bar{z}_1}}=2

) is correct because it represents the equation of the tangent to the circle

|z|=r

at the point

z_1

740 Easy

740 of 979

Application +4 | -1

a+i b=\sum\limits_{{k=1}}^{101} i^k

, then

(a, b)

equals

View answer and explanation

Correct option: A

(0,1)

Option A (

(0,1)

) is correct because the sum of the series

\sum_{k=1}^{101} i^k

simplifies to

i

, which corresponds to the complex number

0 + 1i

741 Easy

741 of 979

Analyzing +4 | -1

The points

A, B

and

C

represent the complex numbers

z_1, z_2,(1-i) z_1+i z_2

respectively, on the complex plane (where,

i=\sqrt{-1}

). The

\triangle A B C

, is

View answer and explanation

Correct option: C

isosceles and right angled

Option C (isosceles and right angled) is correct because the triangle satisfies both the conditions for being isosceles (AB=CA) and right-angled ((AB)²⁺(CA)²=(BC)²).

742 Easy

742 of 979

Analyzing +4 | -1

b

and

c

are odd integers, then the equation

{x^2} + bx + c = 0

has

View answer and explanation

Correct option: C

No integer roots

Option C (No integer roots) is correct because if b and c are odd, the discriminant is not a perfect square, thus the roots are irrational.

743 Medium

743 of 979

Application +4 | -1

z\left( {\overline {z + \alpha } } \right) + \bar z\left( {z + \alpha } \right) = 0

where

\alpha

is a complex constant, then

z

is represented by a point on

View answer and explanation

Correct option: A

A circle

Option A (A circle) is correct because the given equation can be rearranged into the standard form of a circle's equation in the complex plane.

744 Medium

744 of 979

Application +4 | -1

z=-2+2 \sqrt{3} i

, then

z^{2 n}+2^{2 n} z^n+2^{4 n}

may be equal to

View answer and explanation

Correct option: C

3 \cdot 4^{2 n}, n

is multiple of 3

Option C (

3 \cdot 4^{2n}, n

is multiple of 3) is correct because when n is a multiple of 3, the expression simplifies to this value.

745 Easy

745 of 979

Application +4 | -1

z=\frac{1+\cos \theta+i \sin \theta}{\sin \theta+i(1+\cos \theta)}(0 < \theta < \pi / 2)

then

|z|

equals

View answer and explanation

Correct option: C

Option C (1) is correct because by using the property |z1/z2| = |z1|/|z2|, the magnitude of the numerator and denominator are equal, resulting in |z| = 1.

746 Easy

746 of 979

Understanding +4 | -1

z_1, z_2

and

z_3

are three complex numbers in AP, then they lie on

View answer and explanation

Correct option: B

a straight line

Option B (a straight line) is correct because if complex numbers are in AP, the middle term is the average of the other two, implying collinearity.

747 Easy

747 of 979

Analyzing +4 | -1

If at least one value of the complex number

z = x + i\;y

satisfy the condition

\left| {z + \sqrt 2 } \right| = \sqrt {{a^2} - 3a + 2}

and the inequality

\left| {z + i\sqrt 2 } \right| < a,

then

View answer and explanation

Correct option: A

a > 2

Option A is correct. The condition for the intersection of the two circles leads to the inequality

a > 2

748 Medium

748 of 979

Application +4 | -1

The value of

\frac{\left(\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}\right)^8}{\left(\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}\right)^8}

View answer and explanation

Correct option: C

Option C (1) is correct because after simplification using Euler's formula, the expression evaluates to 1.

749 Easy

749 of 979

Analyzing +4 | -1

A complex number

z

is said to be unimodular if

|z|=1

.
Suppose

z_1

and

z_2

are complex numbers such that

\frac{z_1-2 z_2}{2-z_1 \bar{z}_2}

is unimodular and

z_2

is not unimodular. Then the point

z_1

lies on a

View answer and explanation

Correct option: A

circle of radius

z

The point

z_1

lies on a circle of radius 2 because the unimodular condition simplifies to

|z_1| = 2

750 Easy

750 of 979

Analyzing +4 | -1

\alpha(\neq 1)

is a fifth root of unity and

\beta(\neq 1)

is a fourth root of unity then

z=(1+\alpha)(1+\beta)\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\alpha^3\right)\left(1+\beta^3\right)

equals

View answer and explanation

Correct option: D

Option D (0) is correct because the expression (1+

\beta

)(1+

\beta^2

) equals zero since

\beta

is a fourth root of unity (

\neq

1).

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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