This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Argand Plane and Polar Representation”.

1. Which axis is known as real axis in argand plane?

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: The plane having a complex number assigned to each of its point is called the

complex plane or the Argand plane. When (x + y i) is plotted in argand plane, x-axis is real axis.

2. Which axis is known as imaginary axis in argand plane?

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: The plane having a complex number assigned to each of its point is called the

complex plane or the Argand plane. When (x + y i) is plotted in argand plane, y-axis is imaginary axis.

3. 2+i0 is point on ______________

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: Since imaginary part of complex number is zero. So, it is plotted on real axis i.e. x-axis.

2+i0 is point on x-axis.

4. 6i is point on ____________________

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: Since real part of complex number is zero. So, it is plotted on imaginary axis i.e. y-axis.

6i is point on y-axis.

5. \(\sqrt{x^2+y^2}\) distance of point representing complex number x+y i from origin.

a) True

b) False

View Answer

Explanation: Since complex number x + y i is represented by (x, y) on argand plane, distance of point (x, y) from origin is \(\sqrt{x^2+y^2}\).

6. Find mirror image of point representing x+i y on real axis.

a) (x, y)

b) (-x, -y)

c) (-x, y)

d) (x, -y)

View Answer

Explanation: Mirror image of point (x, y) on real axis is (x, -y).

Since real axis is acting as mirror x-coordinate remains same whereas y-coordinate gets inverted.

So, (x, -y) is mirror image of (x, y) on real axis.

7. Find mirror image of point representing x+i y on imaginary axis.

a) (x, y)

b) (-x, -y)

c) (-x, y)

d) (x, -y)

View Answer

Explanation: Mirror image of point (x, y) on imaginary axis is (-x, y).

Since imaginary axis is acting as mirror y-coordinate remains same whereas x-coordinate gets inverted. So, (-x, y) is mirror image of (x, y) on imaginary axis.

8. If P and Q are conjugate complex numbers then their points on argand plane are mirror image on __________________

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: Conjugate complex numbers means their real part is same and imaginary part is inverted i.e. same x part and opposite imaginary part. So, they are mirror image on real axis i.e. x-axis.

9. In polar representation of a complex number (r, 2π) lies on ____________

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: To convert polar representation (r, θ) into argand plane (x, y), substitute x = r cos θ and y = r sin θ.

x = r cos 2π = r and y = r sin 2π = 0.

Argand plane representation is (r, 0). Since imaginary part is zero, so it lies on real axis i.e. x-axis.

10. In polar representation of a complex number (r, π/2) lies on ____________

a) x-axis

b) y-axis

c) z-axis

d) any axis

View Answer

Explanation: To convert polar representation (r, θ) into argand plane (x, y), substitute x=r cos θ and y=r sin θ.

x=r cos π/2 = 0 and y=r sin π/2 = r.

Argand plane representation is (0, r). Since real part is zero, so it lies on imaginary axis i.e. y-axis.

11. Convert (8, 2π/3) into Argand plane representation.

a) (-4, 4\(\sqrt{3}\))

b) (4, 4\(\sqrt{3}\))

c) (4\(\sqrt{3}\), 4)

d) (-4\(\sqrt{3}\), 4)

View Answer

Explanation: To convert polar representation (r, θ) into argand plane (x, y), substitute x=r cos θ and y=r sin θ.

x=8cos 2π/3 = 8cos(π-π/3) = 8(-1/2) = -4.

y=8sin 2π/3 = 8sin(π-π/3) = 8(\(\sqrt{3}\)/2) = 4\(\sqrt{3}\).

12. Convert -1+i into polar form.

a) \(\sqrt{2}\), 5π/4

b) \(\sqrt{2}\), 3π/4

c) –\(\sqrt{2}\), π/4

d) \(\sqrt{2}\), π/4

View Answer

Explanation: r=\(\sqrt{x^2+y^2}=\sqrt{(-1)^2+1^2}=\sqrt{1+1}=\sqrt{2}\).

r cos θ = -1 and r sin θ = 1 So, θ is in 2

^{nd}quadrant since sin is positive and cos is negative.

tan θ = -1 => tan θ = -tan π/4

=> tan θ = tan (π-π/4) = tan 3π/4

=> θ=3π/4.

13. Convert -1-i into polar form.

a) \(\sqrt{2}\), 5π/4

b) \(\sqrt{2}\), 3π/4

c) \(\sqrt{2}\), -3π/4

d) \(\sqrt{2}\), π/4

View Answer

Explanation: r=\(\sqrt{x^2+y^2} = \sqrt{(-1)^2+1^2} = \sqrt{1+1} = \sqrt{2}\).

r cos θ = -1 and r sin θ = -1 => θ is in 3

^{rd}quadrant since sin and cos both negative.

tan θ = 1 => θ= -3π/4.

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