### Mathematics - Complex Numbers

#### Question - 1

For positive integers n1, n2 the value of the expression (1+i)n1+(1+i3)n1+(1+i5)n2+(1+i7)n2, where i=$\sqrt { -1 }$, is a real number,if and only if,

• A n1=n2+1
• B n1=n2-1
• C n1=n2
• D n1>0,n2>0

#### Question - 2

If iz3+z2-z+i=0, then $\left| z \right|$equals

• A 1
• B 2
• C $\cfrac { 1 }{ 2 }$
• D $\cfrac { 1 }{ 4 }$

#### Question - 3

Suppose Z1,Z2,Z3 are the vertices of an equilatural triangle incrribed in the circle $\left| z \right|$=2. If ${ Z }_{ 1 }=1+i\sqrt { 3 }$  then Z2 and Z3 are given by

• A  $-1,1-i\sqrt { 3 }$
• B $1-i\sqrt { 3 } , -2$
• C $1-i\sqrt { 3 } ,2$
• D $1-i\sqrt { 3 } ,1$

#### Question - 4

The value of sum, $\sum _{ n=1 }^{ 13 }{ ({ i }^{ n }+{ i }^{ n+1 }) }$where $i=\sqrt { -1 }$, equals

• A $i$
• B $i-1$
• C $-i$
• D 0

#### Question - 5

If $i=\sqrt { -1 }$, then ${ 4+5\left( -\cfrac { 1 }{ 2 } +\cfrac { i\sqrt { 3 } }{ 2 } \right) }^{ 334 }+3{ \left( -\cfrac { 1 }{ 2 } +\cfrac { i\sqrt { 3 } }{ 2 } \right) }^{ 365 }$  is equal to

• A $1-i\sqrt { 3 }$
• B $-1+i\sqrt { 3 }$
• C $i\sqrt { 3 }$
• D $-i\sqrt { 3 }$

#### Question - 6

If $\alpha ,\beta$are different complex numbers with $\left| \beta \right| =1$ , then $\left| \cfrac { \beta -\alpha }{ 1-\overline { \alpha } \beta } \right|$, is equal to

• A 0
• B $\cfrac { 1 }{ 2 }$
• C 1
• D 2

#### Question - 7

The complex number z=x+iy, satisfying the equation.

$\left| \cfrac { z -5i }{ z+ { 5 } i } \right| =1$lies on

• A the x-axis
• B the straight line y=5
• C a circle passing through the origin
• D None of these

#### Question - 8

The relation a+bi < c + id is meaningful  only when

• A a=0, c=0
• B a=0, d=0
• C b=0, c=0
• D b=0, d=0

#### Question - 9

Number of non-zero integral solutions of the equation $(1-i)x=2x$ is

• A 1
• B 2
• C infinite
• D NONE OF THESE

#### Question - 10

If $\left| \cfrac { z-1 }{ z+1 } \right|$ is a purely imaginary number $(z\neq -1)$, then $\left| z \right|$ is equal to

• A 1
• B 2
• C 3
• D 5