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JEE Advanced Level Test: Complex Number- 2 - JEE MCQ


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30 Questions MCQ Test Chapter-wise Tests for JEE Main & Advanced - JEE Advanced Level Test: Complex Number- 2

JEE Advanced Level Test: Complex Number- 2 for JEE 2024 is part of Chapter-wise Tests for JEE Main & Advanced preparation. The JEE Advanced Level Test: Complex Number- 2 questions and answers have been prepared according to the JEE exam syllabus.The JEE Advanced Level Test: Complex Number- 2 MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for JEE Advanced Level Test: Complex Number- 2 below.
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JEE Advanced Level Test: Complex Number- 2 - Question 1

If z2/z1 is imaginary, then   is

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 1

Let   

Using Componendo and Dividenode, we get

an imaginary number


= an imaginary number

= an imaginary number

JEE Advanced Level Test: Complex Number- 2 - Question 2

If (a + bi)11 = x + iy, where a, b, x, y ∈ R , then (b + ai)11 equals

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 2





 we get 

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JEE Advanced Level Test: Complex Number- 2 - Question 3

If zk, then z1z2z3z4  is equal to 

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 3

We have, zk = ωk where ω  



(by De Movire's theorem)
= cos π + i sin π = -1

JEE Advanced Level Test: Complex Number- 2 - Question 4

If (a1 +ib1)(a2 +ib2) ....(an + ibn) = A +iB, then  is equal to

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 4


Since, it given that
(a1 +ib1)(a2 +ib2) ....(an + ibn) = A +iB

Taking conjugate throughout, we have

Multiplying (i) and (ii), we have

JEE Advanced Level Test: Complex Number- 2 - Question 5

If ω is a complex cube root of unity, then the value of  is

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 5


Since, ω3 = 1, hence

JEE Advanced Level Test: Complex Number- 2 - Question 6

The complex numbers sin x + i cos 2x and cos x – i sin 2x are conjugate to each to other, for

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 6

Option (a) :- for x = nπ
let n=1
so, x= 1 x π =π
therefore , sinx + icos2x
sinπ + icos 2π
0 + i= i
Now,
cosx - isin2x
cosπ - isin2π
or, -1 -0 = -1
hence no conjugate
Option (b) :- x = 0
sin0 + icos0
0+i= i
now ,
cos0 - isin0
= 1
hence no conjugate.
option (c) :- x = (n+1/2)π
if n= 1
x=3π/2
sin3π/2 + icos3π
-1 - i
Now,
cos3π/2 - isin3π
0-0= 0
hence no conjugate
Thus option D is correct

JEE Advanced Level Test: Complex Number- 2 - Question 7

If z = 1 + i tan a , where  then |z| is equal to 

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 7



Since, α lies i III Quadrant and only tan is positive in this quadrant
⇒ |z| = -sec α

JEE Advanced Level Test: Complex Number- 2 - Question 8

If in polar form z1 = cos α + i sin α, z2 = cos β + i sin β, z3 = cos γ + i sin γ and z1 + z2 + z3 = 0 , then  

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 8

z1 = (1, α) = cosα + isinα = e  ......(i)
⇒ z2 = (1, β) cosβ+isinβ = ei/β  ......(ii)
⇒ z3 =(1, γ). cosγ +isinγ =eiγ   ......(iii)
Also, Z1 +Z2 +Z3 = 0 

⇒ (cosα+cosβ+cosγ)+i(sinα+sinβ+sinγ) = 0
⇒ ccosα+cosβ+cosγ = 0
and sinα+sinβ+sinγ = 0 ... (iv) 



[(Using (i), (ii), (iii)] 
⇒ E = (cosα+cosβ+cosγ - i (sinα+sinβ+sinγ)
   [(Using (i)]

JEE Advanced Level Test: Complex Number- 2 - Question 9

If z = 1 + i√3, then

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 9





JEE Advanced Level Test: Complex Number- 2 - Question 10

If |z1 - 1| < 1, |z2 - 2|, 2, |z3 - 3|< 3, then |z1 + z2 +z3|

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 10


    (Triangle Inequality)
 ....(i)

       ....(ii)

Adding (i), (ii),(iii), we have 


JEE Advanced Level Test: Complex Number- 2 - Question 11

  is equal to

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 11


JEE Advanced Level Test: Complex Number- 2 - Question 12

If a is the nth root of unity, then 1 + 2a + 3a2 + ...... to n terms equal to   

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 12


⇒ αS = α + 2α2 + 3α3 + ...+(n -1)αn-1 +nαn

On subtracting, we get


Since, α is nth root of unity, therfore
αn = 1 

JEE Advanced Level Test: Complex Number- 2 - Question 13

If 1, ω, ω2, ...ωn -1are nth roots of unity, then (1 - ω)(1 - ω2)...(1 - ωn-1) is equal to    

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 13

Since, 1, ω, ω2 ..... ωn-1 are the nth roots of unity, therefore
xn - 1 = (x -1)(x - ω)(x - ω2)...(x - ωn-1)




Putting x = 1 in above equation, we have

JEE Advanced Level Test: Complex Number- 2 - Question 14

W is cube root of unity (ω ≠ 1)  then the value of ωn + ωn+1 + ωn+2 is (n ∈ N)

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 14

n,n + 1, n +2  are consecutive integers their sum is muttiple of 3, so 



JEE Advanced Level Test: Complex Number- 2 - Question 15

The value of   is

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 15



JEE Advanced Level Test: Complex Number- 2 - Question 16

If α and β are different complex numbers with , then  is equal to    

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 16








JEE Advanced Level Test: Complex Number- 2 - Question 17

For any two non-zero complex numbers z1 and z2 if then the difference of amplitudes of z1 and z2 is      

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 17







JEE Advanced Level Test: Complex Number- 2 - Question 18

If |z| > 5 then the least value of   is       

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 18

According to Triangle Inequality, we have

Now,  i is an increasing function of |z| and |z| > 5
 will take its minimum value at z = 5
⇒ minimum value of 

JEE Advanced Level Test: Complex Number- 2 - Question 19

Let z be a purely imaginary number such that Im(z) < 0. Then arg(z) is equal to      

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 19

Let z = 0 + ib, where b < 0. Then z is represented by a point on the negative direction of y-axis. 

JEE Advanced Level Test: Complex Number- 2 - Question 20

For any complex number z, maximum value of |z| – |z – 1| is        

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 20

According to Trianglr Inequality
 (given)

JEE Advanced Level Test: Complex Number- 2 - Question 21

If a, b are the roots of x2 + px + q = 0, and w is a cube root of unity, then value of  (ωα + ω2β)(ω2α + ωβ) is

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 21

We have, α + β = -p,αβ = q
Now (ωα + ω2β)(ω2α + ωβ) = ω3α24αβ+ω2αβ + ω3β2
= α22 +(ω+ω2)αβ = α22 -αβ = (α + β)2 -3αβ = p2 -3q

JEE Advanced Level Test: Complex Number- 2 - Question 22

If |z| =  z + 3 - 2i, then z equals 

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 22

z =| z | -3+ 2i ⇒ | z |2= (| z | -3)2 + 4 = | z |2 -6 | z | +9 + 4Þ | z | 13/6
=
Thus,  z = - 5/6 + 2i

JEE Advanced Level Test: Complex Number- 2 - Question 23

If ω (≠1) is a cube root of unity and (1 + ω2)11 = a + bω + cω2, then (a, b, c) equals

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 23




JEE Advanced Level Test: Complex Number- 2 - Question 24

If   then

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 24


⇒ z is purely real.
∴ Im (z) = 0

JEE Advanced Level Test: Complex Number- 2 - Question 25

If (ω ≠1) is a complex cube root of unity and (1 + ω4)n = (1 + ω8)n , then the least possible integral value of n is

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 25




JEE Advanced Level Test: Complex Number- 2 - Question 26

then |z| equals

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 26



⇒ |z| = 1

JEE Advanced Level Test: Complex Number- 2 - Question 27

If α (≠ 1) is a fifth root of unity and b (≠ 1) is a fourth root of unity, then z = (1 + α) (1 + β) (1 + α2) (1 + β2) (1 + α3) (1 + β3) equals

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 27

As β ≠ 1 is afourth root of unity,
β4 = 1 ⇒ (1 - β) (1 + β + β2 + β3) = 0


∴ z = 0

JEE Advanced Level Test: Complex Number- 2 - Question 28

The number of complex numbers satisfying  = iz2 is

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 28





so that the equation becomes 1/z = 1z2

where ω (≠1) is a cube root of unity.

JEE Advanced Level Test: Complex Number- 2 - Question 29

If z2 + z + 1 = 0, where z is a complex number, then value of   

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 29





JEE Advanced Level Test: Complex Number- 2 - Question 30

If z1 + z2 + z3 = 0 and |z1| = |z2| = |z3| = 1, then value of equals

Detailed Solution for JEE Advanced Level Test: Complex Number- 2 - Question 30





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