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This mock test of Test: Complex Number (Competition Level) - 1 for JEE helps you for every JEE entrance exam.
This contains 30 Multiple Choice Questions for JEE Test: Complex Number (Competition Level) - 1 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

The locus of the point z = x + iy satisfying the equation is given by

Solution:

i.e. lie on the ⊥ bisector the line joining (1,0) & (-1,0)

i.e. the y axis

∴ 2 lies on x = 0

QUESTION: 2

If Arg then the locus of z is

Solution:

s of z is a circle having (–1, 0) and (1, 0) as its ends of diameter.

QUESTION: 3

In Argand diagram all the complex numbers z satisfying |z - 4i + z + 4i = 10| lie on a

Solution:

|z_{1} - z_{2|} = |4i + 4i| = 8 < 10

∴Then locus of z is ellipse.

QUESTION: 4

If z is a complex number, then represents

Solution:

Let z = x + iy, then

QUESTION: 5

If x_{n} = then x_{1}, x_{2}, x_{3}, .....∞

Solution:

QUESTION: 6

If w is a complex cube root of unity then the value of (1 + ω) (1 + ω)^{2} (1 + ω^{4}) (1 + ω^{8})...2n terms =

Solution:

Put n = 1 and verify

QUESTION: 7

If x^{2} + x + 1 = 0, then the value of is

Solution:

x = ω or ω^{2}

= (1+1+4)9 = 54

QUESTION: 8

f |z - 2 + 2i| = 1, then the least value of |z| is

Solution:

QUESTION: 9

The maximum value of 3z + 9- 7| if |z + 2 - i| = 5 is

Solution:

|3z + 6 - 3i + 3 - 4i| |≤| 3z + 6 - 3i| + |3 - 4i| ≤ 3 |z + 2-i| + |3- 4i|

=15+5 = 20

QUESTION: 10

The value of log is

Solution:

QUESTION: 11

If is purely imaginary then the locus of z is

Solution:

Let z = x + iy

QUESTION: 12

If z_{r} = where r = 1,2,3,4,5 then z_{1}z_{2}z_{3}z_{4}z_{5} =

Solution:

QUESTION: 13

Solution:

4 + 5ω^{334} + 3ω^{365} = 4 + 5ω + 3ω^{2}

=1+ 2ω+3 (1+ω+ω^{2}) =1+ 2ω = i √3

QUESTION: 14

For all complex numbers z_{1}z_{2} such that |z_{1}| = 12 and |z_{2} - 3 - 4i| = 5 the minimum value of |z_{1} -z_{2}| is

Solution:

QUESTION: 15

If a and b are real numbers between 0 and 1 such that the points z_{1} = a + i, z_{2} = 1 + bi and z_{3} =0 form an equilateral triangle then a, b are

Solution:

and simplify

QUESTION: 16

If α,β are the roots of x^{2} - 2x + 4 = 0 then α^{n} + β^{n} =

Solution:

QUESTION: 17

The value of is

Solution:

= i[sum of roots -1] = i (0 - 1) = -i

QUESTION: 18

If the cube roots of unity are 1, ω,ω^{2} , then the roots of the equation ( x - 1)^{3} + 8 = 0 are

Solution:

QUESTION: 19

If |z_{1}| = 2, |z_{2}| = 3, |z_{3}| = 4 and |z_{1} + z_{2} + z_{3}| = 5 then |4z_{2}z_{3} + 9z_{3}z_{1} + 16z_{1}z_{2}| =

Solution:

QUESTION: 20

Let z = x + iy be a complex number where x and y are integers then the area of the rectangle whose vertices are the roots of the equation

Solution:

QUESTION: 21

If a + ib = , then (a, b) equals

Solution:

a + ib = i + (i^{2} + i^{3} + i^{4} + i^{5}) + (i^{6} + i^{7} + i^{8} + i^{9}) +....+ (i^{98} + i^{99} + i^{100} + i^{101})

a + ib = 0 + i ⇒ a = 0, b = 1

QUESTION: 22

If z ∈ C and 2z =| z |+i, then z equals

Solution:

QUESTION: 23

Suppose, a, b, c ∈ C , and | a |=| b |=| c |= 1 and abc = a +b + c, then bc +ca + ab is equal to

Solution:

Now, abc = a + b + c

⇒ bc + ca + ab = 1

QUESTION: 24

The number of complex numbers z which satisfy z^{2} + 2 | z |^{2} = 2 is

Solution:

As z^{2} = 2(1 -|z|^{2}) is real, z is either purely and or purely imaginary.

If z is purely real, then

In this case,

QUESTION: 25

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

Solution:

⇒ sin x - i cos 2x = cos x - i sin 2x

⇒ sin x = cos x and cos 2x = sin 2x

⇒ tan x = 1 and tan 2x = 1

These two equations cannot hold simultaneously.

QUESTION: 26

If z_{1} and z_{2} are two complex numbers are a, b are two real numbers,then |az_{1} - bz_{2}|^{2} + |bz_{1} + az_{2}|^{2} equals

Solution:

We have

QUESTION: 27

If z ≠ 0 is a complex number such that arg(z) = π/4,then

Solution:

As arg(z) =π/4, we can write

QUESTION: 28

Let z and w be two non-zero complex numbers such that |z| = |w| and arg (z) + arg (w) = π. Then z is equal to

Solution:

Let |z| = |w| = r and arg(w) = θ, so that arg (z) = π - θ

We have, z = r [cos(π - θ)]

QUESTION: 29

If z_{1}, z_{2}, z_{3} are complex numbers such that

|z_{1}| = |z_{2}| = |z_{3}| =|1/z_{1} +1/z_{2} + 1/z_{3}| = 1, then find the value of z_{1}+ z_{2} + z_{3} .

Solution:

QUESTION: 30

If (x + iy)^{1/3} = a + ib, then x/a + y/b equals

Solution:

(x + iy)^{1/3} = a + ib

⇒ x + iy = (a^{3} – 3ab^{2}) + (3a^{2}b – b^{3})I

⇒ x/a = a^{2} - b^{2}

Thus x/a + y/b = 4a^{2} - 4b^{2}

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