If $\left|z-\frac{1}{\mathrm{z}}\right|=1$, then:

$|\mathrm{z}|_{\min }=\frac{1+\sqrt{5}}{2}$

If $x+(1 / x)=2 \cos \theta$, then the value of $x$ is

$\cos 2 \theta+i \sin 2 \theta$

$(\cos \theta+i \sin \theta)^2$

$\cos 4 \theta-i \sin 4 \theta$

If $2 \cos \theta=x+\frac{1}{x}$ and $2 \cos \phi=y+\frac{1}{y}$, then

$\frac{x}{y}+\frac{y}{x}=2 \cos (\theta+\phi)$

$\frac{x^m}{y^n}+\frac{y^n}{x^m}=2 \cos (m \theta+n \phi)$

$x y+\frac{1}{x y}=2 \cos (\theta-\phi)$

If $i=\sqrt{-1}$, then $\frac{e^{x i}+e^{-x}}{2}=$

$x+\frac{x^5}{3 !}+\frac{x^3}{5 !}+\ldots x$

$i\left[x-\frac{x^3}{3 !}+\frac{x^3}{5 !}-\ldots \infty\right]$

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JEE • MATHEMATICS • COMPLEX NUMBERS

Complex numbers practice · Mixed difficulty

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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 931-960 of 979 questions

931 Medium

931 of 979

Application +4 | -1

\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{4}

for any complex number

z

, then

|z-2 i|=

View answer and explanation

Correct option: A

2 \sqrt{2}

Option A (

2 \sqrt{2}

) is correct because by letting

z=x+iy

and using the given argument condition, we derive the equation of a circle and find

|z-2i| = 2\sqrt{2}

932 Easy

932 of 979

Analyzing +4 | -1

a, b, c

are integers, not all equal and

\omega(\neq 1)

is a cube root of unity, then minimum value of

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

View answer and explanation

Correct option: B

Option B (1) is correct because the minimum value of the expression

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

is 1 when a, b, and c are integers not all equal, and

\omega

is a cube root of unity.

933 Medium

933 of 979

Application +4 | -1

\left(\frac{1+i}{1-i}\right)^3-\left(\frac{1-i}{1+i}\right)^3=x+i y

, then

(x, y)

is equal to

View answer and explanation

Correct option: C

(-2,0)

Option B (

(0,-2)

) is correct because simplifying the expression using

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

and

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

leads to

i^3 - (-i)^3 = -2i

, so x=0 and y=-2.

934 Easy

934 of 979

Application +4 | -1

The complex number

z = x + iy

which satisfy the equation

\left| {\frac{{z - 5i}}{{z + 5i}}} \right| = 1

lies on

View answer and explanation

Correct option: A

The axis of

x

Option A (The axis of x) is correct because the given condition simplifies to y = 0, which represents the x-axis.

935 Easy

935 of 979

Application +4 | -1

\left|z-\frac{1}{\mathrm{z}}\right|=1

, then:

View answer and explanation

Correct option: A

|z|_{\max }=\frac{1+\sqrt{5}}{2}

Option A (

|z|_{\max } = \frac{1+\sqrt{5}}{2}

) is correct because by using the reverse triangle inequality, we can find the maximum value of

|z|

936 Easy

936 of 979

Application +4 | -1

The value of

{\left( { - \frac{1}{2} + \frac{{\sqrt 3 i}}{2}} \right)^{1000}}

View answer and explanation

Correct option: D

\omega

Option D (

\omega

) is correct because

{\left( { - \frac{{1}}{{2}} + \frac{{\sqrt 3 i}}{{2}}} \right)^{1000}}

simplifies to

\omega

using the properties of complex cube roots of unity.

937 Easy

937 of 979

Analyzing +4 | -1

Let for some real numbers

\alpha

and

\beta ,\,\,a = \alpha - i\beta .

If the system of equations

4ix + \left( {1 + i} \right)y = 0

and

8\left( {\cos \frac{{2\pi }}{3} + i\sin \frac{{2\pi }}{3}} \right)x + \bar ay = 0

has more than one solution, then

\frac{\alpha }{\beta }

is equal to

View answer and explanation

Correct option: B

2 - \sqrt 3

The correct answer is B (

2 - \sqrt 3

) because it is the value of

\frac{\alpha}{\beta}

calculated using the condition for the system of equations to have more than one solution.

938 Easy

938 of 979

Analyzing +4 | -1

\left| {{z_1}} \right| = \left| {{z_2}} \right| = \ldots = \left| {{z_n}} \right| = 1,\;

then the value of

\left| {{z_1} + {z_2} + {z_3} + ... + {z_n}} \right|

View answer and explanation

Correct option: C

\left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + ... + \frac{1}{{{z_n}}}} \right|

Option C (

\left| {\frac{1}{{{{z_1}}}} + \frac{1}{{{{z_2}}}} + ... + \frac{1}{{{{z_n}}}}} \right|

) is correct because it uses the property that if

|z| = 1

, then

\bar{z} = \frac{1}{z}

and

|z| = |\bar{z}|

939 Easy

939 of 979

Analyzing +4 | -1

Number of roots of the equation

z^{10}-z^5-992=0

, where real parts are negative, is

View answer and explanation

Correct option: C

Option C (5) is correct because solving the equation yields two values for z⁵, one of which (z⁵ = -31) has 5 solutions with negative real parts.

940 Easy

940 of 979

Application +4 | -1

The mirror image of the curve

\arg \left(\frac{z-3}{z-i}\right)=\frac{\pi}{6}

i=\sqrt{-1}

in the real axis, is

View answer and explanation

Correct option: D

\arg \left(\frac{z+i}{z-3}\right)=\frac{\pi}{6}

Option D (

\arg \left(\frac{{z + i}}{{z - 3}}\right) = \frac{\pi }{6}

) is correct because the mirror image of a complex number z in the real axis is its conjugate

\bar{z}

, and using the properties of arguments, the equation transforms to the given answer.

941 Easy

941 of 979

Analyzing +4 | -1

|z|=1

and

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

where

z \neq-1)

, then

\operatorname{Re}(\omega)

View answer and explanation

Correct option: A

Option A (0) is correct because the real part of the given expression, where |z|=1 and z is not -1, simplifies to zero, indicating a purely imaginary number.

942 Easy

942 of 979

Remembering +4 | -1

The real part of

\left[ {1 + \cos \left( {\frac{\pi }{5}} \right) + i\sin \left( {\frac{\pi }{5}} \right)} \right]

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (

\frac{1}{2}

) is correct because after rationalizing the denominator, the real part of the expression simplifies to 1/2.

943 Hard

943 of 979

Analyzing +4 | -1

Let

a, b

be two real numbers such that

a b < 0

. If the complex number

\frac{1+a i}{b+i}

is of unit modulus and

a+

i b

lies on the circle

|z-1|=|2 z|

, then a possible value of

\frac{1+[a]}{4 b}

, where

[t]

is greatest integer function is:

View answer and explanation

Correct option: C

-\frac{1}{2}

The correct answer is C (

-\frac{1}{2}

) because it is obtained by using the given conditions to find the values of a and b, and then evaluating the expression.

944 Medium

944 of 979

Application +4 | -1

z = x + iy

and

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

, the locus of

z

View answer and explanation

Correct option: A

x - axis

Option C (

-a

) is correct because by using Vieta's formulas, we can express

{p^2} + {q^2} + pq

in terms of m and a, which simplifies to

-a

945 Easy

945 of 979

Application +4 | -1

x+(1 / x)=2 \cos \theta

, then the value of

x

View answer and explanation

Correct option: A

\cos \theta \pm i \sin \theta

Option A (

\cos \theta \pm i \sin \theta

) is the correct solution because it is derived by solving the quadratic equation resulting from the given equation.

Explore Other Chapters in MATHEMATICS

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946 Easy

946 of 979

Analyzing +4 | -1

\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots+

a_{2 n} x^{2 n}

, then value of

a_0+a_3+a_6+\ldots

View answer and explanation

Correct option: D

3^{n-1}

Option D (

3^{n-1}

) is correct because by substituting

x=1, \omega, \omega^2

into the given polynomial and adding the resulting equations, we can isolate the desired sum.

947 Easy

947 of 979

Application +4 | -1

For a positive integer

n

, the expression

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

equals

View answer and explanation

Correct option: C

2^n

Option C (

2^n

) is correct because after simplification, the expression equals

2^n

948 Easy

948 of 979

Application +4 | -1

\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1

, then

z

lies on

View answer and explanation

Correct option: B

The imaginary axis

Option B (The imaginary axis) is correct because the given condition implies that the argument of

z^2

\pi

, which means the argument of z is

\frac{\pi}{2}

, indicating that z lies on the imaginary axis.

949 Easy

949 of 979

Analyzing +4 | -1

\omega(\neq 1)

is cube root of unity satisfying

\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}=2 \omega^2

and

\frac{1}{a+\omega^2}+\frac{1}{b+\omega^2}+\frac{1}{c+\omega^2}=2 \omega

, then the value of

\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because

\omega

and

\omega^2

are roots of the equation

\frac{{1}}{{a+x}} + \frac{{1}}{{b+x}} + \frac{{1}}{{c+x}} = \frac{{2}}{{x}}

, so substituting

x=1

gives

\frac{{1}}{{a+1}} + \frac{{1}}{{b+1}} + \frac{{1}}{{c+1}} = 2

950 Hard

950 of 979

Application +4 | -1

2 \cos \theta=x+\frac{1}{x}

and

2 \cos \phi=y+\frac{1}{y}

, then

View answer and explanation

Correct option: B

x^m y^n+\frac{1}{x^m y^n}=2 \cos (m \theta+n \phi)

Option B (

x^m y^n+\frac{{1}}{{x^m y^n}}=2 \cos (m \theta+n \phi)

) is correct because it follows from De Moivre's theorem.

951 Easy

951 of 979

Application +4 | -1

The conjugate of the complex number

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

View answer and explanation

Correct option: D

- 1 - i

Option D (

- 1 - i

) is correct because after simplifying the complex number and taking its conjugate, we get -1-i.

952 Easy

952 of 979

Application +4 | -1

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

be a cube root of unity and

{\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n}

, then the least positive value of

n

View answer and explanation

Correct option: B

Option B (3) is correct because it is the smallest positive integer n that satisfies the equation

{\omega}^n = 1

, where

\omega

is a cube root of unity.

953 Easy

953 of 979

Application +4 | -1

z

and

w

are two non-zero complex numbers such that

\left| {zw} \right| = 1

and

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

, then

\bar{z}w

is equal to

View answer and explanation

Correct option: D

- i

Option D (

-i

) is correct because by expressing z and w in exponential form and using the given conditions,

\bar{z}w

simplifies to

-i

954 Medium

954 of 979

Understanding +4 | -1

z_{1}

and

z_{2}

are two non- zero 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

View answer and explanation

Correct option: C

Option C is correct because if |z1+z2| = |z1| + |z2|, then z1 and z2 are collinear and point in the same direction, meaning their arguments are equal, so arg(z1) - arg(z2) = 0.

955 Medium

955 of 979

Application +4 | -1

\arg \left( {{z_1}{z_2}} \right) = 0

and

\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1,

then

View answer and explanation

Correct option: C

{z_1} = {\bar z_2}

Option C (

{z_1} = {\bar z_2}

) is correct because if the argument of the product of two complex numbers is zero and their magnitudes are equal to 1, then the two numbers are complex conjugates.

956 Medium

956 of 979

Application +4 | -1

\mathrm{Re\;}\left( {\frac{{z + 4}}{{2z - i}}} \right) = \frac{1}{2},

then

z

is represented by a point lying on

View answer and explanation

Correct option: C

A straight line

The correct answer is A (A straight line) because the real part condition leads to a linear equation in x and y, representing a straight line.

957 Medium

957 of 979

Analyzing +4 | -1

a,b,c

and

u,\upsilon ,\omega

are complex numbers representing the vertices of two triangles such that

c = \left( {1 - r} \right)\;a + rb

and

\omega = \left( {1 - r} \right)u + r\upsilon ,

where

r

is a complex number, then the two triangles

View answer and explanation

Correct option: B

Are similar

Option B (Are similar) is correct because the given conditions imply that the two triangles have proportional sides and equal angles, thus they are similar.

958 Medium

958 of 979

Application +4 | -1

If the points

z,-i z

and 1 are collinear then

z

lies on

View answer and explanation

Correct option: B

a circle

Option B (a circle) is correct because the collinearity condition leads to an equation of the form |z - c| = r, which represents a circle.

959 Easy

959 of 979

Analyzing +4 | -1

Let

z

and

\omega

be complex numbers. If

\operatorname{Re}(z)=|z-2|

\operatorname{Re}(\omega)=|\omega-2|

and

\arg (z-\omega)=\frac{\pi}{3}

, the value of

\operatorname{Im}(z+\omega)

View answer and explanation

Correct option: D

\frac{4}{\sqrt{3}}

Option D (

\frac{4}{\sqrt{3}}

) is correct because by using parametric form of z and omega, and given argument, the imaginary part of z+omega can be found.

960 Medium

960 of 979

Understanding +4 | -1

i=\sqrt{-1}

, then

\frac{e^{x i}+e^{-x}}{2}=

View answer and explanation

Correct option: B

1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\ldots \infty

Option A (

1+\frac{{x^2}}{{2 !}}+\frac{{x^4}}{{4 !}}+\ldots \infty

) is incorrect. The correct answer should be

1-\frac{{x^2}}{{2 !}}+\frac{{x^4}}{{4 !}}-\ldots \infty

which represents

\cos x

. However, the provided option A is incorrect. The correct option should be B.

Revision notes and key takeaways

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