Complex Numbers Questions

MathematicsComplex Numbers105 Questions Found

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 2008Difficult

Comprehension Passage

Let

A, B, C

be three sets of complex numbers as defined below

A = {z : I m z \geq 1}

B = {z : ∣ z - 2 - i ∣ = 3}

C = {z : R e ((1 - i) z) = 2}

Question 1:

The number of elements in the set $A \cap B \cap C$ is

(A)

(B)

(C)

(D)

\infty

Answer:B

Question 2:

Let $z$ be any point in $A \cap B \cap C$ . Then, $∣ z + 1 - i ∣^{2} + ∣ z - 5 - i ∣^{2}$ lies between

(A)

25 and 29

(B)

30 and 34

(C)

35 and 39

(D)

40 and 44

Answer:C

Question 3:

Let $z$ be any point $A \cap B \cap C$ and let $w$ be any point satisfying $∣ w - 2 i ∣ < 3$ . Then, $∣ z ∣ - ∣ w ∣ + 3$ lies between

(A)

-6 and 3

(B)

-3 and 6

(C)

-6 and 6

(D)

3 and 9

Answer:D

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 2013Moderate

View in:English Hindi

Comprehension Passage

Let

S = S_{1} \cap S_{2} \cap S_{3}

, where

S_{1} = {z \in C : ∣ z ∣ < 4}

S_{2} = {z \in C : I m [\frac{z - 1 + 3 i}{1 - 3 i}] > 0}

and

S_{3} = {z \in C : R ez > 0}

Question 1:

Area of $S =$

(A)

f r a c 10 p i 3

(B)

f r a c 20 p i 3

(C)

f r a c 16 p i 3

(D)

f r a c 32 p i 3

Answer:B

Question 2:

$mi n_{z in S} ∣1 - 3 i - z ∣ =$

(A)

f r a c 2 - s q r t 3 2

(B)

f r a c 2 + s q r t 3 2

(C)

f r a c 3 - s q r t 3 2

(D)

f r a c 3 + s q r t 3 2

Answer:C

MathematicsAlgebraic Operations on Complex NumbersJEE Advanced 1992Easy

View in:English Hindi

$z \neq = 0$ is a complex number

List-I

(P)

R ez = 0

(Q)

A r g z = \frac{π}{4}

List-II

(1)

R e z^{2} = 0

(2)

I m z^{2} = 0

(3)

R e z^{2} = I m z^{2}

Answer:P → 2, Q → 1

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 2010Moderate

View in:English Hindi

Match the statements in Column I with those in Column II. [Note : Here $z$ takes values in the complex plane and $I m z$ and $R ez$ denote , respectively, the imaginary part and the real part of $z$ . ]

List-I

(P)

The set of points

z

satisfying

∣ z - i ∣ z ∣∣ = ∣ z + i ∣ z ∣∣

is contained in or equal to

(Q)

The set of points

z

satisfying

∣ z + 4∣ + ∣ z - 4∣ = 10

is contained in or equal to

(R)

∣ w ∣ = 2

, then the set of points

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

is contained in or equal to

(S)

∣ w ∣ = 1

, then the set of points

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

is contained in or equal to

List-II

(1)

an ellipse with eccentricity

\frac{4}{5}

(2)

the set of points

z

satisfying

I m z = 0

(3)

the set of points

z

satisfying

∣ I m z ∣ \leq 1

(4)

the set of points

z

satisfying

∣ R ez ∣ < 2

(5)

the set of points

z

satisfying

∣ z ∣ \leq 3

Answer:Q → 3, Q → 1, R → NaN, S → NaN

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 2011Difficult

View in:English Hindi

Match the statements given in Column-I with the values given in Column-II.

List-I

(P)

a = \hat{j} + 3 \hat{k}

b = - \hat{j} + 3 \hat{k}

and

c = 23 \hat{k}

form a triangle, then the internal angle of the triangle between

a

and

b

(Q)

\int_{a}^{b} (f (x) - 3 x) d x = a^{2} - b^{2}

, then the value of

f (\frac{π}{6})

(R)

The value of

\frac{π ^{2}}{l n 3} \int_{7/6}^{5/6} sec (π x) d x

(S)

The maximum value of

Arg (\frac{1}{1 - z})

for

∣ z ∣ = 1, z \neq = 1

is given by

List-II

(1)

\frac{π}{6}

(2)

\frac{2 π}{3}

(3)

\frac{π}{3}

(4)

π

(5)

\frac{π}{2}

Answer:P → 2, Q → 1, R → 4, S → 5

MathematicsCube Roots and nth Roots of UnityJEE Advanced 2014Moderate

View in:English Hindi

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

List-I

(P)

For each

z_{k}

there exists a

z_{j}

such that

z_{k} \cdot z_{j} = 1

(Q)

There exists a

k \in {1, 2, \dots, 9}

such that

z_{1} \cdot z = z_{k}

has no solution

z

in the set of complex numbers

(R)

\frac{∣1 - z _{1} ∣∣1 - z _{2} ∣ \dots ∣1 - z _{9} ∣}{10}

equals

(S)

1 - \sum_{k = 1}^{9} cos (\frac{2 k π}{10})

equals

List-II

(1)

True

(2)

False

(3)

(4)

Answer:P → 1, Q → 2, R → 3, S → 4

MathematicsCube Roots and nth Roots of UnityJEE Advanced 1978Easy

View in:English Hindi

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

MathematicsConjugate and ModulusJEE Advanced 1979Easy

View in:English Hindi

If $x + i y = \frac{a + ib}{c + i d}$ , prove that $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1981Moderate

View in:English Hindi

Let the complex number $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle. Let $z_{0}$ be the circumcentre of the triangle. Then prove that $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1983Moderate

View in:English Hindi

Prove that the complex numbers $z_{1}, z_{2}$ and the origin form an equilateral triangle only if $z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$ .

...11

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the set A∩B∩C is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be any point in A∩B∩C. Then, ∣z+1−i∣2+∣z−5−i∣2 lies between

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be any point A∩B∩C and let w be any point satisfying ∣w−2i∣<3. Then, ∣z∣−∣w∣+3 lies between

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of S=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } minzinS​∣1−3i−z∣=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=0 is a complex number

List-I

List-II

List-I

List-II

Match the statements given in Column-I with the values given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let zk​=cos(102kπ​)+isin(102kπ​);k=1,2,…,9. Match the statements in List-I with those in List-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=a+b,y=aγ+bβ and z=aβ+bγ where γ and β are the complex cube roots of unity, show that xyz=a3+b3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x+iy=c+ida+ib​​, prove that (x2+y2)2=c2+d2a2+b2​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that the complex numbers z1​,z2​ and the origin form an equilateral triangle only if z12​+z22​−z1​z2​=0.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the set A∩B∩C is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be any point in A∩B∩C. Then, ∣z+1−i∣2+∣z−5−i∣2 lies between

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be any point A∩B∩C and let w be any point satisfying ∣w−2i∣<3. Then, ∣z∣−∣w∣+3 lies between

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of S=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } minzinS​∣1−3i−z∣=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=0 is a complex number

List-I

List-II

List-I

List-II

Match the statements given in Column-I with the values given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let zk​=cos(102kπ​)+isin(102kπ​);k=1,2,…,9. Match the statements in List-I with those in List-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=a+b,y=aγ+bβ and z=aβ+bγ where γ and β are the complex cube roots of unity, show that xyz=a3+b3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x+iy=c+ida+ib​​, prove that (x2+y2)2=c2+d2a2+b2​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that the complex numbers z1​,z2​ and the origin form an equilateral triangle only if z12​+z22​−z1​z2​=0.

The number of elements in the set $A \cap B \cap C$ is

Let $z$ be any point in $A \cap B \cap C$ . Then, $∣ z + 1 - i ∣^{2} + ∣ z - 5 - i ∣^{2}$ lies between

Let $z$ be any point $A \cap B \cap C$ and let $w$ be any point satisfying $∣ w - 2 i ∣ < 3$ . Then, $∣ z ∣ - ∣ w ∣ + 3$ lies between

Area of $S =$

$mi n_{z in S} ∣1 - 3 i - z ∣ =$

$z \neq = 0$ is a complex number

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

If $x + i y = \frac{a + ib}{c + i d}$ , prove that $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$ .

Prove that the complex numbers $z_{1}, z_{2}$ and the origin form an equilateral triangle only if $z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$ .

The number of elements in the set $A \cap B \cap C$ is

Let $z$ be any point in $A \cap B \cap C$ . Then, $∣ z + 1 - i ∣^{2} + ∣ z - 5 - i ∣^{2}$ lies between

Let $z$ be any point $A \cap B \cap C$ and let $w$ be any point satisfying $∣ w - 2 i ∣ < 3$ . Then, $∣ z ∣ - ∣ w ∣ + 3$ lies between

Area of $S =$

$mi n_{z in S} ∣1 - 3 i - z ∣ =$

$z \neq = 0$ is a complex number

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

If $x + i y = \frac{a + ib}{c + i d}$ , prove that $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$ .

Prove that the complex numbers $z_{1}, z_{2}$ and the origin form an equilateral triangle only if $z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$ .