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JEE • MATHEMATICS • COMPLEX NUMBERS

Complex numbers practice · Easy

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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 1-30 of 612 questions

1 Easy

1 of 612

Application +4 | -1

The number of non-zero integer solutions of the equation

{\left| {1 - i} \right|^x} = {2^x}

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the equation |1 - i|^x = 2^x has only one solution, x = 0, which is not a non-zero integer.

2 Easy

2 of 612

Application +4 | -1

Let

\left|z_r-r\right| \leq r, \forall r=1,2,3, \ldots, n

, then

\left|\sum\limits_{r=1}^n z_r\right|

is less than

View answer and explanation

Correct option: C

n(n+1)

Option C (

n(n+1)

) is correct because it represents the upper bound of the sum of the complex numbers based on the given inequality.

3 Easy

3 of 612

Application +4 | -1

2x = - 1 + \sqrt 3 i

, then the value of

{\left( {1 - {x^2} + x} \right)^6} - {\left( {1 - x + {x^2}} \right)^6}

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because substituting x = ω (cube root of unity) simplifies the expression to 0.

4 Easy

4 of 612

Application +4 | -1

The amplitude of

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

View answer and explanation

Correct option: D

\frac{\pi }{6}

The correct answer is D (

\frac{\pi}{6}

) because it is the argument of the simplified complex number.

5 Easy

5 of 612

Analyzing +4 | -1

The smallest positive integral value of

n

such that

{\left[ {\frac{{1 + \sin \frac{\pi }{8} + i\cos \frac{\pi }{8}}}{{1 + \sin \frac{\pi }{8} - i\cos \frac{\pi }{8}}}} \right]^n}

is purely imaginary, is equal to

View answer and explanation

Correct option: A

The correct answer is A (4) because when n=4, the expression becomes purely imaginary, satisfying the condition.

6 Easy

6 of 612

Application +4 | -1

\left( {3 + i} \right)\left( {z + \bar z} \right) - \left( {2 + i} \right)\left( {z - \bar z} \right) + 14i = 0

, then

z\bar z

is equal to

View answer and explanation

Correct option: C

Option C (10) is correct because substituting z = x + iy into the given equation and solving the resulting system of equations yields x = -1 and y = 3, thus z

\bar z

= 10.

7 Easy

7 of 612

Analyzing +4 | -1

z_1, z_2, z_3

and

z_4

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

z_1-z_4=z_2-z_3

and

\operatorname{amp}\left(\frac{z_4-z_1}{z_2-z_1}\right)=\frac{\pi}{2}

, then the quadrilateral is a

View answer and explanation

Correct option: C

cyclic quadrilateral

Option C (cyclic quadrilateral) is correct because the given conditions imply that the quadrilateral is a rectangle, which is a cyclic quadrilateral.

8 Easy

8 of 612

Application +4 | -1

The equation

|z-1|^2+|z+1|^2=4

represents on the Argand plane

View answer and explanation

Correct option: D

A circle with centre origin and radius unity

Option D (A circle with centre origin and radius unity) is correct because the equation simplifies to

|z|=1

, which represents a circle centered at the origin with radius 1.

9 Easy

9 of 612

Understanding +4 | -1

The locus represented by the equation

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

View answer and explanation

Correct option: C

A line through the origin

Option C (A line through the origin) is correct because the equation represents the perpendicular bisector of the line segment joining 1 and

i

, which passes through the origin.

10 Easy

10 of 612

Analyzing +4 | -1

The greatest number among

\sqrt[3]{{9,\;}}\sqrt[4]{{11}},\;\sqrt[6]{{17}}

View answer and explanation

Correct option: A

\sqrt[3]{9}

Option A is correct because when expressed with a common exponent,

\sqrt[3]{9}

has the largest base (6561) compared to the other options.

11 Easy

11 of 612

Analyzing +4 | -1

\alpha=\cos \left(\frac{8 \pi}{11}\right)+i \sin \left(\frac{8 \pi}{11}\right)

, where

i=\sqrt{-1}

, then

\operatorname{Re}\left(\alpha+\alpha^2+\alpha^3+\alpha^4+\alpha^5\right)

View answer and explanation

Correct option: B

-\frac{1}{2}

Option B (-1/2) is correct because the real part of the sum of the given complex numbers simplifies to -1/2 using properties of roots of unity.

12 Easy

12 of 612

Understanding +4 | -1

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

View answer and explanation

Correct option: D

Can't be determined

Option D (Can't be determined) is correct because the sum depends on the value of n, specifically whether n is even or odd.

13 Easy

13 of 612

Analyzing +4 | -1

z=x+i y

satisfies

|z|-2=0

and

|z-i|-|z+5 i|

=0

, then

View answer and explanation

Correct option: C

x+2 y+4=0

Option C (

x+2 y+4=0

) is correct because the equation represents the perpendicular bisector of the line segment joining (0,1) and (0,-5), and the intersection with |z|=2 gives the solution.

14 Easy

14 of 612

Analyzing +4 | -1

Let

A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta} \text { is purely imaginary}\right\}.

Then the sum of the elements in

A

View answer and explanation

Correct option: A

4 \pi

Option A,

4 \pi

, is the correct sum of the elements in set A, which contains values of

\theta

that make the given expression purely imaginary. The values are

\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}

15 Easy

15 of 612

Application +4 | -1

n

is any positive integer, then the value of

\frac{i^{4 n+1}-i^{4 n-1}}{2}

equals

View answer and explanation

Correct option: C

i

Option C (

i

) is correct because the expression simplifies to

i

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16 Easy

16 of 612

Application +4 | -1

The value of

{\left[ {\frac{{1 + \sin \frac{{2\pi }}{9} + i\cos \frac{{2\pi }}{9}}}{{1 + \sin \frac{{2\pi }}{9} - i\cos \frac{{2\pi }}{9}}}} \right]^3}

View answer and explanation

Correct option: A

- \frac{1}{2}\left( {\sqrt 3 - i} \right)

Option A (

- \frac{1}{2}\left( {\sqrt 3 - i} \right)

) is correct because it is the simplified form of the given complex number expression after applying De Moivre's theorem.

17 Easy

17 of 612

Application +4 | -1

The locus of the centre of the circle which touches the circles

\left| {z - {z_1}} \right| = a

and

\left| {z - {z_2}} \right| = b

externally (

z,{z_1}

and

{z_2}

are complex numbers) will be

View answer and explanation

Correct option: B

A hyperbola

Option B (A hyperbola) is correct because the difference in distances from the center of the variable circle to the centers of the two fixed circles is constant, which is the definition of a hyperbola.

18 Easy

18 of 612

Application +4 | -1

If the points represented by complex numbers

z_1=a+i b, z_2=c+i d

and

z_1-z_2

are collinear, where

i=\sqrt{-1}

, then

View answer and explanation

Correct option: B

a d-b c=0

Option B (

a d-b c=0

) is correct because if three complex numbers are collinear, the imaginary part of the ratio of the differences between them is zero, which leads to the given condition.

19 Easy

19 of 612

Application +4 | -1

The imaginary part of

{\left( {3 + 2\sqrt { - 54} } \right)^{1/2}} - {\left( {3 - 2\sqrt { - 54} } \right)^{1/2}}

can be

View answer and explanation

Correct option: B

- 2\sqrt 6

Option B (

- 2\sqrt 6

) is correct because it is the imaginary part of the simplified expression after taking the square root of the complex numbers.

20 Easy

20 of 612

Application +4 | -1

{\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}\left( {p + iq} \right)

, then

p

and

q

are roots of the equation

View answer and explanation

Correct option: A

{x^2} - \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0

Option A (

{x^2} - \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0

) is correct because it is the quadratic equation whose roots are the real and imaginary parts of the complex number.

21 Easy

21 of 612

Application +4 | -1

If the point representing the complex number

{z}_{{k}}

is lying on or inside the circle having centre

(1+0 {i})

and radius is

{k}

, then the maximum value of

\left|z_1+z_2+z_3+z_4+\ldots +z_{20}\right|

View answer and explanation

Correct option: C

230

Option C (230) is correct because the maximum value of the sum of the complex numbers is found by summing the maximum possible magnitudes of each complex number, which forms an arithmetic progression.

22 Easy

22 of 612

Understanding +4 | -1

Let

{z_1}

and

{z_2}

be complex numbers, then

{\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} - {z_2}} \right|^2}

is equal to

View answer and explanation

Correct option: B

2\left( {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right)

Option B is correct because the expression simplifies to

2(\left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2)

using the properties of complex numbers and their conjugates.

23 Easy

23 of 612

Application +4 | -1

z=2-3 i

, then the value of

z^2-4 z+13

View answer and explanation

Correct option: A

Option A (0) is correct because substituting z = 2 - 3i into the expression z² - 4z + 13 results in 0.

24 Easy

24 of 612

Analyzing +4 | -1

Let a complex number

z,\left| z \right| \ne 1,

satisfy

{\log _{\frac{1}{{\sqrt 2 }}}}\left( {\frac{{\left| z \right| + 11}}{{{{\left( {\left| z \right| - 1} \right)}^2}}}} \right) \le 2

. Then, the largest value of

\left| z \right|

is equal to ______.

View answer and explanation

Correct option: D

Option D (7) is correct because it is the largest value of |z| that satisfies the given logarithmic inequality.

25 Easy

25 of 612

Analyzing +4 | -1

Let

{z_1}

and

{z_2}

be two complex numbers such that

\frac{{{z_1}}}{{{z_2}}} + \frac{{{z_2}}}{{{z_1}}} = 1.

Then,

View answer and explanation

Correct option: C

{z_1},{z_2}

and the origin from an equilateral triangle

Option C (

{z_1},{z_2}

and the origin from an equilateral triangle) is correct because the given condition implies the complex numbers form an equilateral triangle with the origin.

26 Easy

26 of 612

Analyzing +4 | -1

Let

{z_1}

and

{z_2}

be the roots of the equation

{z^2} + pz + q = 0

where

p,q

are real. The points represented by

{z_1},\;{z_2}

and the origin form an equilateral triangle, if

View answer and explanation

Correct option: A

{p^2} = 3q

Option A (

{p^2} = 3q

) is correct because it satisfies the condition for the points represented by the roots and the origin to form an equilateral triangle.

27 Easy

27 of 612

Application +4 | -1

The locus of complex number

z=x+

iy where

i=\sqrt{-1}

satisfying relation

\arg (z-a)=\frac{\pi}{4}

where

a \in R

View answer and explanation

Correct option: D

x-y=a, y>0

Option C (

x+y=a, y>0

) is correct because it represents the locus of the complex number satisfying the given argument condition.

28 Easy

28 of 612

Application +4 | -1

(a+b i)^{11}=x+i y

, where

a, b, x, y \in \mathbf{R}

, then

(b+a i)^{11}

equals

View answer and explanation

Correct option: B

-y-i x

Option B (-y - ix) is correct because by using the properties of complex conjugates and manipulating the expression, we find that

(b + ai)^{11} = -y - ix

29 Easy

29 of 612

Analyzing +4 | -1

a, b

and

c

are distinct integers and

\omega(\neq 1)

a

cube root of unity, then the minimum value of

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

, is

View answer and explanation

Correct option: A

2 \sqrt{3}

Option A (

2 \sqrt{3}

) is correct because by using properties of cube root of unity and modulus, the minimum value of the expression can be found.

30 Easy

30 of 612

Application +4 | -1

z_1

and

z_2

be the

n

th root of unity which subtend right angled at the origin. Then,

n

must be of the form

View answer and explanation

Correct option: D

4 k

Option D (

4k

) is correct because the argument difference between two nth roots of unity subtending a right angle at the origin must be

\pm \frac{\pi}{2}

, leading to

n = 4k

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 612 single-correct-answer MCQs on COMPLEX NUMBERS.
✓ Try each question before opening its answer. Read the explanation where one is available.
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