Complex Numbers and Quadratic Equation Test

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Complex Numbers and Quadratic Equation Test - 3

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Question 1
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For a complex number x + iy, if x >0 and y <0 then the number lies in the.

A
Second quadrant

B
Third quadrant

C
First quadrant

D
Fourth quadrant

Solution

When x >0 &y <0, the quadrant contains positive X-axis and negative Y-axis
Therefore, the point lies in the fourth quadrant.
Question 2
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A
0

B
2

C
-1

D
1

Solution
Question 3
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If |z₁ | = 4, |z₂ | = 4, then |z₁ + z₂ + 3 + 4i| is less than

A
2

B
5

C
10

D
13

Solution
Question 4
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If z is a complex number such that is purely imaginary, then what is |z| equal to ?

A
1/2

B
2/3

C
1

D
More than one of the above
Solution
Concept:
If a number ω= a + ib is purely imaginary, then
a = 0
and ω=
Calculation:
Given, z is a complex number such that is purely imaginary,
Let = a + ib, then a = 0
And
Question 5
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If A = then the locus of the point P(x, y) in the cartesian plane is

A
a pair of lines passing through (-1, -1)

B
0

C
a pair of lines passing through (-1, -2)

D
More than one of the above

Solution

z ̅ = x - iy
Real Part of = Real Part of = Real Part of
⇒Real Part of

⇒x² - x - y² + y = 2x² + 2y² - 4y + 2
⇒x² + 3y² + x - 5y + 2 = 0
The above equation does not represent a circle or Straight line.
So No Option is Correct.
Option (2) is correct only when we use Z in place of Z ̅in the numerator of
Question 6
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A point z = P(x,y) lies on the negative direction of y axis, so amp(z) =

A
π

B
-π

C
3(π/2)

D
π/2

Solution

P(-x,-y) lies on the 3rd or 4th quadrant
So, according to the question, points lies in 3rd quadrant
i.e. 3(π/2)
Question 7
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The argument of the complex number -i

A
π/2

B
3 π/4

C
-π/3

D
-π/4

Solution

z = -i
Comlex number z is of the form x + iy
x = 0, y = -1
arg(z) = π−tan −1|y/x|
⇒π−tan −1|-1/0|
= π- (π/2)
⇒ π- π/2
⇒π/2
Question 8
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The modulas of the complex number -1 + √3i is

A
2

B
4

C
1

D
3

Solution

√(-1)² + (√3)²
= 2
Question 9
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The argument of the complex number -1 –√3

A
π/3

B
4 π/3

C
-π/3

D
-2 π/3

Solution

z = a + ib
a = -1, b = -√3
(-1, -√3) lies in third quadrant.
Arg(z) = -π+ tan^-1 (b/a)
= - π+ tan^-1 (√3)
= - π+ tan^-1 (tan π/3)
= - π+ π/3
= -2 π/3
Question 10
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The amplitude of a complex number is called the principal value amplitude if it lies between.

A
-2 π/3, 2 π/3

B
π, 2 π/3

C
-π, π

D
-π/2, π/2

Solution

x = |z| cos θand y = |z| sin θsatisfies infinite values of θand for any infinite values of θis the value of Arg z. Thus, for any unique value of θthat lies in the interval - π<θ≤πand satisfies the above equations x = |z| cos θand y = |z| sin θis known as the principal value of Arg z or Amp z and it is denoted as arg z or amp z.

We know that, cos (2n π+ θ) = cos θand sin (2n π+ θ) = sin θ(where n = 0, ±1, ±2, ±3, .............), then we get,

Amp z = 2n π+ amp z where - π