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practice question on complex number for iitJee , NDA and airforce Page no. 2

COMPLEX NUMBER

Question no. 16
what is the square root of i, where i=√-1
looks_one(1+i)/2
looks_two (1-i)/2
looks_3 (1+i)/√2
looks_4 none of these
Option looks_one(1+i)/2
Solution :
complex_problem_solution16

Question no. 17
If z=,then what is the argument of z is
looks_one 3π/4
looks_two π/4
looks_3 5π/6
looks_4 -3π/4
Option looks_4 -3π/4
Solution :
complex_problem_solution17

Question no. 18
what is ?
looks_one 1
looks_two -1
looks_3 i
looks_4 -i
option looks_3 i
Solution :
complex_problem_solution18

Question no. 19
If A+iB= (4+2i)/(1-2i), then what is the value of A
looks_one -8
looks_two 0
looks_3 4
looks_4 8
Option looks_two 0
Solution :
complex_problem_solution19

Question no. 20
what is the argument of (1-sinΘ) + icosΘ
looks_oneπ/2 -Θ/2
looks_twoπ/2 +Θ/2
looks_3π/4 -Θ/2
looks_4π/4 +Θ/2
Option looks_4π/4 +Θ/2
Solution :
complex_problem_solution20

Question no. 21
if ω is the imaginary cube root of unity, then what is (2-ω +2ω2)27
looks_one 327ω
looks_two -327ω2
looks_3 327
looks_4 -327
Option looks_4 -327
ω , cube root of unity
(2-ω +2ω2)27 = (2-ω +2(-1-ω))27
= (2-ω -2-2ω)27
= (-3ω)27
= (-3)27ω27 = (-3)27(ω3)9
= - 327

Question no. 22
If α and β are the complex cube roots of unity then what is the value of (1+α)(1+α2)(1+β)(1+β2)
looks_one 1
looks_two -1
looks_3 0
looks_4 4
Option looks_one 1
α , β are cube root of unity
α = ω , β = ω2
= ( 1 + α )(1 + α2)( 1 + β)( 1 + β2)
=(1 + ω)(1 + ω2)(1 + ω2)(1 + (ω2)2)
(1 + ω)(1 + ω2)2(1 + ω3.ω)
= ( 1 + ω)(1 + ω2)2( 1 + ω)
=(1 + ω)2(1 + ω2)2
= (-ω2)2(-ω)2 = (-ω2)2(-ω)2
= ω4 . ω2 = ω6 = 1

Question no. 23
If a=, what is the value of |a|2 + aā
looks_one 0
looks_two -1
looks_3 1
looks_4 8
Option looks_4 8
Solution :
complex_problem_solution23

Question no. 24
What is the modulus of ?
looks_one 3/5
looks_two 9/25
looks_3 3/25
looks_4 5/3
Option looks_one 3/5
Solution :
complex_problem_solution24

Question no. 25
If z= 1+ cos(π/5) + i sin(π/5), then what is |z|?
looks_one 2cos(π/5)
looks_two 2sin(π/5)
looks_3 2cos(π/10)
looks_4 2sin(π/10)
Option looks_3 2cos(π/10)
Solution :
complex_problem_solution25

Question no. 26
What is the value of 1+i2 +i4 ........ +i100 ?
looks_one 1
looks_two 0
looks_3 -1
looks_4 none of these
Option looks_one 0
z = 1+i2 +i4 ........ +i100
z =1 + (i2 + i4) + (i6 + i8) + ........(i98 + i100)
z =1 + (1 -1) + (1 - 1) + (1 -1) + .........(1 - 1)
z = 0

Question no. 27
If ω is the complex root unity, then what is the value of ω10 + ω-10 ?
looks_one 2
looks_two -2
looks_3 -1
looks_4 1
Option looks_3 -1
ω , cube root of unity
= ω10 + ω-10
=(ω3)3.ω + 1 /(ω3)3.ω
= ω + 1/ω
=(ω2 + 1)/ω
= -ω/ω
=-1

Question no. 28
If x2+y2=1 , then what is equal to?
looks_one x-iy
looks_two x+iy
looks_3 2x
looks_4-2iy
Option looks_two x+iy
Solution :
complex-number-problem_solution28

Question no. 29
what is the value of (-i)4n+3 + (i41 + i-257)9
looks_one 0
looks_two 1
looks_3 i
looks_4 -1
Option looks_3 i
(-i)4n+3 + (i41 + i -257 )9
(-i)4n.(-i)3 + ( (i4)10.i + (i4)64i)-1)9
1.(-i)3 + (i + i-1)9
-(i)2.i + (i - i)9
i + 0 = i

Question no. 30
let c be the set of complex number and z1 , z2 are in C
1. agr(z1)=arg(z2) ⇒ z1=z2
2. |z1| =|z2| ⇒ z1=z2
which of these statements above is /are correct?
looks_one 1 only
looks_two 2 only
looks_3 both 1 and 2
looks_4 neither 1 and 2
Option looks_4 neither 1 and 2
Statement 1 :
arg(z1) = arg(z2)
assume z1 = 3 + 4i : arg (z1) = tan-1(4/3)
z2 = 6+ i8 : arg(z2) = tan-1(4/3)
if arg(z1) = arg(z2) , it does not imply z1 = z2
Similarly
For statement 2 :
z1 = 1+ 2i : |z1|= √5
z2 = 2 + i : |z2| = √5
|z1| = |z2| but z1 ≠ z2
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