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
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