REVISION (COMPETITIVE)

COMPLEX NUMBERS 

1) α is the real cube root of 2 and β, γ are its imaginary cube roots, 

then (xβ + yγ + zα)/(xγ + yα + zβ) is equal to 

a) ³√2 b) ³√2 ω½ c) ω² d) none 

2) The modulus amplitude form of the complex number -1 - I is

a) -√2(cos(π/4) + i sin(π/4))

b)  √2(cos(3π/4) + i sin(3π/4))

c) √2(cos(-3π/4) + i sin(-3π/4)) d) none 

3) The amplitude of √12 + 6{(1- i)/(1+ i)} is 

a) π/3 b) 2π/3 c) -π/3 d) -2π/3

4) If x√2 = 1+ √-1, then the value of x⁶+ x⁴+ x² +1 is

a) 0 b) 4 c) -4 d) none 

5) The value of √i + √-i is/are

a) √2 b) ±√2 c) ±√2 i d) ±√2, ±√2 i

6) If y= √(z² + 6x +8), then (1- iy)¹⁾² is equal to 

a) ±(1/√2){i √(x +4) - √(x +2)}

b) ±(i/√2){√(x +4) - √(x +2)}

c) ±(1/√2){√(x +4) - i √(x +2)} d) none 

7) The square roots of a² + 1/a² - 4i(a - 1/a) - 6 are

a) ±(a - i/a -2)

b) ±(a - 1/a -2i)

c) ±(a - 1/a +2i) d) none 

8) If ³√(x + iy)= a + ib, where a,b, x, y are real, then x/a + y/b is equal to 

a) 4(b² - a²) b) 4(a² + b²) c) 4(a² - b²) d) none 

9) If n is a +ve integer, not a multiple of 3, the {(1+ √-3)/2}ⁿ +  {(-1- √-3)/2}ⁿ is equal to 

a) -1 b) 2 c) 0 d) none 

10) If x + iy moves on the line 3x + 4y +5=0, then the least value of |x + it| is 

a) 0 unit b) 1/5 unit c) 1 unit d) none 

11) If z₁ = √3 i and z₂= -1+ √3 i, then amp(z₁z₂) is equal to 

a) 7π/6 b) -5π/6 c) 5π/6 d) none 

12) If {(2- i)x + i}/(1+ i)  + {(1+ 2i)y + i}/(1- i) = -1/2 + 5i/2, where x and y are real, then x - y is equal to 

a) 1 b) -1 c) 6 d) 8

13) If {(1+ i)x - 2i}/(3+ i)  + {(2- 3i)y + i}/(3- i)= i, where x and y are real, then 4x + 9y is equal to 

a) 10 b) -10 c) 3 d) -3

14) The modulus of {(1- i)/(3+ i)} + 4i/5 is

a) √5 unit b) √11/5 unit  c) √5/5  unit  d) none

15) The least positive integer n such that {2i/(1+ i)}ⁿ is a positive integer, is

a) 2 b) 4 c) 8 d) 16

16) If (√3 + i)¹⁰⁰ = 2¹⁰¹ (a + ib), then a is equal to 

a) 4 b) -4 c) 1/4 d) -1/4

17) If (a + 1)²/(2a - i) = p + iq, then p² + q² is equal to 

a) (a² +1)²/(4a² -1) b) (a²+1)²/(2a² -1) c) (a² +1)²/(4a² +1) d) none

18) The complex numbers z= x + iy which satisfy the equation|(z - 5i)/(z + 5i)|= 1, lie on 

a) x = -5 b)  y= 6 c) the x-axis d) the y-axis 

19) For any complex number z= x + it, if the imaginary part of (2z +1)/(iz + 1) is -2, then the locus of z is 

a) a straight line 

b) a circle 

c) an ellipse  d) none

20) If the complex number z= x + iy satisfies the condition |(z - k)/(z + ki)|= 1, where k is any real number, then the locus of z is 

a) a straight line b) a circle c) an ellipse d) none

21) The complex number z= x + iy satisfying the condition amp{(z - i)/(z + i)}=π/4 lies on

a) a straight line b) a circle c) an ellipse d) none 

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