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Question - 1

The value of \(\left[ \frac { -1\quad +\quad i\sqrt { 3 } }{ 2 } \right] ^{ 100 } + \left[ \frac { -1\quad -\quad i\sqrt { 3 } }{ 2 } \right] ^{ 100 } \)is

  • A 2
  • B 0
  • C -1
  • D 1

Question - 2

The modules and amplitude of the complex number \(\left[ e^{ 3-i\frac { \pi }{ 4 } } \right] ^{ 3 }\)are respctively

  • A \(e^{ 9 },\frac { \pi }{ 2 } \)
  • B \(e^{ 9 }\frac { -\pi }{ 2 } \)
  • C \(e^{ 6 }\frac { -3\pi }{ 4 } \)
  • D \(e^{ 9 }\frac { -3\pi }{ 4 } \)

Question - 3

If \(( m - 5 ) + i( n + 4 )\) is the complex conjugate of \(( 2m + 3 ) + i ( 3n - 2 )\) then ( n,m ) are 

  • A \(\left( \frac { -1 }{ 2 } ,-8 \right) \)
  • B \(\left( \frac { -1 }{ 2 } ,8 \right) \)
  • C \(\left( \frac { 1 }{ 2 } ,-8 \right) \)
  • D \(\left( \frac { 1 }{ 2 } ,8 \right) \)

Question - 4

If \(x^{ 2 } +y^{ 2 }\) = 1 then the value of \(\frac { 1+x + iy }{ 1 + (x - iy) } \)  is

  • A \(x - iy \)
  • B \(2x\)
  • C \(-2iy\)
  • D \(x + iy\)

Question - 5

The modulus of the complex number \(2+i\sqrt { 3 } \) is

  • A \(\sqrt { 3 } \)
  • B \(\sqrt { 13 } \)
  • C \(\sqrt { 7 }\)
  • D \(7\)

Question - 6

If \(A+iB=\left( a_{ 1\ }+ ib_{ 1 } \right) \left( a_{ 2 }+ ib_{ 2 } \right) \left( a_{ 3 }+ ib_{ 3 } \right) \) then \(A^{ 2 }\quad +\quad B^{ 2 }\) is

  • A \({ a }_{ 1 }^{ 2 }+{ b }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 }+{ a }_{ 3 }^{ 2 }+{ b }_{ 3 }^{ 2 }\)
  • B \(({ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 })^{ 2 }+({ b }_{ 1 }+{ b }_{ 2 }+{ b }_{ 3 })^{ 2 }\)
  • C \({ (a }_{ 1 }^{ 2 }+{ b }_{ 1 }^{ 2 })({ a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 })({ a }_{ 3 }^{ 2 }+{ b }_{ 3 }^{ 2 })\)
  • D \({ (a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }+{ a }_{ 3 }^{ 2 })^{ 2 }({ b }_{ 1 }^{ 2 }+{ b }_{ 2 }^{ 2 }+{ b }_{ 3 }^{ 2 })^{ 2 }\)

Question - 7

If \( a = 3 + i \) and \(z = 2 - 3i \) then the points on the Argand diagram representing \(az,3az\) and \(-az\) are 

  • A vertices of a right angled triangle 
  • B vertices of an equilateral triangle
  • C vertices of an isosceles triangle
  • D collinear

Question - 8

The points \(z_{ 1 },z_{ 2 },z_{ 3 }\) in the complex plane are the vertices of a parallelogram taken in order if and only if

  • A \(z_{ 1 } +z_{ 4 }=z_{ 2 } +z_{ 3 }\)
  • B \(z_{ 1 } +z_{ 3 }=z_{ 2 }+ z_{ 4 }\)
  • C \(z_{ 1 } + z_{ 2 }=z_{ 3 } + z_{ 4 }\)
  • D \(z_{ 1 } - z_{ 2 }=z_{ 3 } -z_{ 4 }\)

Question - 9

If \(z\) represents a complex number then \(arg(z) +\arg(z)\) is

  • A \(\frac { \pi }{ 4 }\)
  • B \(\frac { \pi }{ 2 } \)
  • C \(0\)
  • D \(\frac { \pi }{ 3 }\)

Question - 10

If the amplitude of a complex number is \(\frac { \pi }{ 2 }\) then the number is

  • A purely imaginary
  • B purely real
  • C \(0\)
  • D neither real nor imaginary
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