If $x = \left( {\frac{{1 + i}}{2}} \right),\;\;\left( {{\rm{where}}\,\;i = \sqrt { - 1} } \right)$, then the expression $2{x^4} - 2{x^2} + x + 3$ equals

$\frac{{\left( {3 + i} \right)}}{2}$

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

If $\left| z \right| = 1$ and $w = \frac{{z - 1}}{{z + 1}}$ (where $z \ne - 1$), then ${\rm{Re}}\left( w \right)$ is

$\frac{1}{{{{\left| {z + 1} \right|}^2}}}$

$\left| {\frac{1}{{z + 1}}} \right|.\frac{1}{{{{\left| {z + 1} \right|}^2}}}$

$\frac{{\sqrt 2 }}{{{{\left| {z + 1} \right|}^2}}}$

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JEE • MATHEMATICS • COMPLEX NUMBERS

Complex numbers practice · Medium

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Showing 181-210 of 337 questions

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181 Medium

181 of 337

Analyzing +4 | -1

1, \omega, \omega^2, \omega^3, \ldots, \omega^{n-1}

are the

n^{\text {th }}

roots of unity, then

(1-\omega)\left(1-\omega^2\right)\left(1-\omega^3\right) \ldots\left(1-\omega^{n-1}\right)

View answer and explanation

Correct option: C

n

Option C (

n

) is correct because the given expression is equal to the derivative of

z^n - 1 = 0

evaluated at z=1, which results in n.

182 Medium

182 of 337

Application +4 | -1

Let

z

be the complex number satisfying

|z+16|=4| z+1|

, then:

View answer and explanation

Correct option: A

|z|=4

Option A (

|z|=4

) is correct because the equation represents a circle in the complex plane, and solving for

|z|

gives the radius.

183 Medium

183 of 337

Analyzing +4 | -1

\mathrm{g}\;\left( x \right)

and

h\left( x \right)

are two polynomials such that the polynomial

P\left( x \right) = \mathrm{g}\left( {{x^3}} \right) + x\;h\left( {{x^3}} \right)

is divisible by

{x^2} + x + 1,

then which one of the following is not true?

View answer and explanation

Correct option: B

\mathrm{g}\left( 1 \right) = h\left( 1 \right) \ne 0

Option B (

\mathrm{g}\left( 1 \right) = h\left( 1 \right) \ne 0

) is not true because the correct deduction is that

\mathrm{g}\left( 1 \right) = h\left( 1 \right) = 0

184 Medium

184 of 337

Application +4 | -1

Let

z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5}

. If

R\,(z)

and

I\,(z)

respectively denote the real and imaginary parts of

z

, then

View answer and explanation

Correct option: B

I\left( z \right) = 0

Option B (

I\left( z \right) = 0

) is correct because the imaginary part of the complex number

z

is zero, as

z

simplifies to a real number.

185 Medium

185 of 337

Application +4 | -1

\arg (z) < 0

, then

\arg (-z)-\arg (z)

is equal to

View answer and explanation

Correct option: A

\pi

Option A (

\pi

) is correct because when

\arg(z) < 0

\arg(-z)

\arg(z) + \pi

186 Medium

186 of 337

Application +4 | -1

Let

f(x)=x^4+a x^3+b x^2+c x+d

be a polynomial with real coefficients and real zeroes. If

|f(i)|=1

, (where

i=\sqrt{-1}

) then

a+b+c+d

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because if |f(i)| = 1, all real zeroes must be zero, leading to f(x) = x⁴, and thus a+b+c+d = 0.

187 Medium

187 of 337

Application +4 | -1

The closest distance of the origin from a curve given as

a\bar z + \bar az + {\left| a \right|^2} = 0

View answer and explanation

Correct option: D

\frac{{\left| a \right|}}{2}

Option D (

\frac{{\left| a \right|}}{2}

) is correct because the closest distance of the origin from the given curve is found using the formula for the distance from a point to a line.

188 Medium

188 of 337

Analyzing +4 | -1

w = \alpha + i\beta

, where

\beta \ne 0

and

z \ne 1

, satisfies the condition that

\left( {\frac{{w - \bar wz}}{{1 - z}}} \right)

is purely real, then the set of values of

z

View answer and explanation

Correct option: B

\left| z \right| = 1

and

z \ne 1

Option B (

\left| z \right| = 1

and

z \ne 1

) is correct because the condition that

\left( {\frac{{w - \bar wz}}{{1 - z}}} \right)

is purely real leads to

|z| = 1

, and since

z \ne 1

, the answer is

\left| z \right| = 1

and

z \ne 1

189 Medium

189 of 337

Application +4 | -1

-3+x^2 y i

and

x^2+y+4 i

are conjugate of each other, then

|x|+|y|

View answer and explanation

Correct option: A

Option A (5) is correct because solving the equations derived from the conjugate property gives

|x|+|y|=5

190 Medium

190 of 337

Application +4 | -1

x = \left( {\frac{{1 + i}}{2}} \right),\;\;\left( {\mathrm{where}\,\;i = \sqrt { - 1} } \right)

, then the expression

2{x^4} - 2{x^2} + x + 3

equals

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is obtained by substituting the value of x and simplifying the expression using the given condition.

191 Medium

191 of 337

Analyzing +4 | -1

Let

S

be the set of all

(\alpha, \beta), \pi<\alpha, \beta<2 \pi

, for which the complex number

\frac{1-i \sin \alpha}{1+2 i \sin \alpha}

is purely imaginary and

\frac{1+i \cos \beta}{1-2 i \cos \beta}

is purely real, Let

Z_{\alpha \beta}=\sin 2 \alpha+i

\cos 2 \beta,(\alpha, \beta) \in S

. Then

\sum\limits_{\left( {\alpha ,\,\beta } \right) \in S} {\left( {i{Z_{\alpha \beta }} + \frac{1}{{i{{\bar Z}_{\alpha \beta }}}}} \right)}

is equal to

View answer and explanation

Correct option: C

Option C (1) is correct because after finding the values of alpha and beta that satisfy the given conditions, substituting them into the expression and simplifying yields 1.

192 Medium

192 of 337

Application +4 | -1

The least positive integer

\mathrm{n}

for which

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

, is

View answer and explanation

Correct option: D

Option D (3) is correct because the least positive integer n that satisfies the given equation is 3, as shown by simplifying the complex number and raising it to the power of 3.

193 Medium

193 of 337

Application +4 | -1

z=\cos 20^{\circ}+i \sin 20^{\circ}

, then

\left|z+2 z^2+3 z^3+\ldots \ldots \ldots+18 z^{18}\right|^{-1}

is:

View answer and explanation

Correct option: B

\frac{1}{9} \sin 10^{\circ}

Option B (

\frac{{1}}{{9}} \sin 10^{\circ}

) is correct because after simplifying the given expression using geometric progression and properties of complex numbers, the magnitude of the resulting complex number is found to be

\frac{{1}}{{9}} \sin 10^{\circ}

194 Medium

194 of 337

Application +4 | -1

The value of

\sum_{k=1}^{10}\left(\sin \frac{2 \pi k}{11}-i \cos \frac{2 \pi k}{11}\right)

View answer and explanation

Correct option: C

i

Option C (

i

) is correct because the summation simplifies to

i

using properties of complex numbers and roots of unity.

195 Medium

195 of 337

Application +4 | -1

i=\sqrt{-1}

, then

4+5\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365}

is equal to

View answer and explanation

Correct option: C

i \sqrt{3}

Option C (

i \sqrt{3}

) is correct because it is the simplified form of the given complex number expression using properties of cube roots of unity.

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196 Medium

196 of 337

Analyzing +4 | -1

If a complex number

z

statisfies the equation

z+\sqrt{2}|z+1|+i=0

, then

|z|

is equal to :

View answer and explanation

Correct option: C

\sqrt{5}

Option C (

\sqrt{5}

) is correct because by substituting

z = x + iy

and equating real and imaginary parts, we can solve for

x

and

y

and then find

|z|

197 Medium

197 of 337

Application +4 | -1

\left| z \right| = 1

and

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

(where

z \ne - 1

), then

\mathrm{Re}\left( w \right)

View answer and explanation

Correct option: A

Option A (0) is correct because the real part of

w

is 0, which is derived using the property that

|z| = 1

and

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

198 Medium

198 of 337

Application +4 | -1

\alpha ,\beta

and

\gamma

are angles such that

\tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma

and

x = \cos \alpha + i\sin \alpha ,\;\;y = \cos \beta + i\sin \beta

and

z = \cos \gamma + i\sin \gamma

, then

xyz

is equal to

View answer and explanation

Correct option: C

+ 1\,\mathrm{or}\, - 1

Option C (

+ 1\;\mathrm{or}\; - 1

) is correct because by using the given condition and trigonometric identities, we can show that

xyz

can be either +1 or -1.

199 Medium

199 of 337

Analyzing +4 | -1

\frac{1}{4}\left( {n + 1} \right)\;n\;\left( {{n^2} + 3n + 4} \right)

View answer and explanation

Correct option: C

From an equilateral triangle

Option C (From an equilateral triangle) is correct because the distance between any two cube roots of unity is the same, forming an equilateral triangle.

200 Medium

200 of 337

Analyzing +4 | -1

Given that the equation

{z^2} + \left( {p + iq} \right)z + r + \,\,is = 0

where

p,\;q,\;r,\;s\;

are real and non-zero roots, then

View answer and explanation

Correct option: D

pqs = {s^2} + {q^2}r

Option D (

pqs = {s^2} + {q^2}r

) is correct because substituting a real root into the equation and separating real and imaginary parts leads to this relationship.

201 Medium

201 of 337

Application +4 | -1

p, q

and

r

are positive integers and

\omega

is an imaginary cube root of unity and

f(x)=x^{3 p}+x^{3 q+1}+x^{3 r+2}

, then

f(\omega)

is equal to

View answer and explanation

Correct option: C

Option A (0) is correct because when ω is substituted into the function, the terms simplify to 1 + ω + ω^2, which equals 0.

202 Medium

202 of 337

Application +4 | -1

z

is a complex number of unit modulus and argument

\theta

, then

\arg \left(\frac{1+z}{1+\bar{z}}\right)

equals to

View answer and explanation

Correct option: B

\theta

Option B (

\theta

) is correct because using the property

|z|^2 = z \bar{z} = 1

, we can simplify the expression and find that the argument of the simplified expression is equal to

\theta

203 Medium

203 of 337

Application +4 | -1

Let

A,B

and

C

represent the complex numbers

{z_1},{z_2},{z_3}

respectively on the complex plane. If the circumcenter of the triangle

ABC

lies at the origin, then the orthocentre is represented by the complex number

View answer and explanation

Correct option: D

{z_1} + {z_2} + {z_3}

Option D (

{z_1} + {z_2} + {z_3}

) is correct because in a triangle with the circumcenter at the origin, the orthocenter is the sum of the vertices' complex numbers.

204 Medium

204 of 337

Application +4 | -1

z

and

w

are two non zero complex numbers such that

|z|=|w|

and

\operatorname{Arg} \mathrm{z}+\operatorname{Arg} \mathrm{w}=\pi

then

\mathrm{z}

equals

View answer and explanation

Correct option: B

-\bar{\omega}

Option B (

-\bar{\omega}

) is the correct answer because given the conditions

|z|=|w|

and

\operatorname{Arg} z + \operatorname{Arg} w = \pi

, z can be expressed as

-\bar{\omega}

205 Medium

205 of 337

Application +4 | -1

{z_1},{z_2},{z_3}

are vertices of an equilateral triangle inscribed in the circle

\left| z \right| = 2

and if

{z_1} = 1 + i\sqrt 3 ,

then

View answer and explanation

Correct option: A

{z_1} = - 2,{z_3} = 1 - i\sqrt 3

Option B (

{z_2} = 2,{z_3} = 1 - i\sqrt 3

) is the correct answer because it accurately calculates the other two vertices of the equilateral triangle by rotating the given vertex by 2π/3 and 4π/3.

206 Medium

206 of 337

Application +4 | -1

The multiplicative inverse of

\frac{3+4 i}{4-5 i}

View answer and explanation

Correct option: C

-\frac{8}{25}-\frac{31}{25} i

Option C (

-\frac{8}{25}-\frac{31}{25} i

) is correct because simplifying the complex fraction and then finding its multiplicative inverse yields this result.

207 Medium

207 of 337

Application +4 | -1

Which of the following are correct?

I. Modulus of

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

is 1.

II. Argument of

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

\frac{\pi}{2}

.

III. Modulus of

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

\sqrt{2}

.

IV. Argument of

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

\frac{\pi}{4}

View answer and explanation

Correct option: A

I and II are correct

Option A (I and II are correct) is correct because the modulus of (1+i)/(1-i) is 1 and its argument is π/2, while statements III and IV are incorrect.

208 Medium

208 of 337

Application +4 | -1

The value of

\frac{1}{50}\left(2^{100} \sin \left(\frac{\pi}{100}\right) \sin \left(\frac{2 \pi}{100}\right) \cdots \sin \left(\frac{99}{100} \pi\right)\right)

View answer and explanation

Correct option: A

Option A (2) is correct because by using the properties of roots of unity and trigonometric identities, the value simplifies to 2.

209 Medium

209 of 337

Application +4 | -1

z

is a complex number such that

|z| \geq 1

, then the minimum value of

\left|z+\frac{1}{2}(3+4 i)\right|

View answer and explanation

Correct option: C

\frac{3}{2}

Option C (

\frac{3}{2}

) is correct because it represents the minimum distance between the complex number z (with |z| >= 1) and the given complex number, found using the triangle inequality.

210 Medium

210 of 337

Application +4 | -1

The continued product of all the four values of the complex number

(1+i)^{3 / 4}

View answer and explanation

Correct option: B

2(1-i)

Option C (2(1+i)) is incorrect. Option B (2(1-i)) is correct because by converting to polar form and using De Moivre's theorem, the product of the four roots simplifies to 2(1-i).

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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