Class 11 Math Ex 1.2 Unlocked: Argand Diagrams & Polar Form Secrets
Complete Exercise 1.2 Solved Notes + Expert Quiz
Visualizing the Imaginary!
Assalam-o-Alaikum Students! In the last exercise, we learned how to play with Complex Numbers using algebra. But have you ever wondered what a Complex Number actually looks like? Today, we are going to take these numbers out of the equations and put them onto a graph. This is where Math meets Art!
We will be using the Argand Diagram, which is just a fancy name for a graph where the x-axis is "Real" and the y-axis is "Imaginary". We will also learn the Polar Form—a way to describe a number not by its coordinates, but by its distance from the center and the angle it makes. It’s like giving someone a compass and a distance instead of a map address.
The trickiest part of this exercise is finding the correct angle ($\theta$), also known as the Argument. But don't worry! I have created a simple "Quadrant Rule" for you that will make finding the angle as easy as ABC. By the end of this article, you’ll be graphing and converting complex numbers like a pro. Let’s get started!
💡 The Polar Formulas
To convert $z = x + iy$ into Polar Form $z = r(\cos \theta + i\sin \theta)$, use these:
- $r = \sqrt{x^2 + y^2}$
- $\theta = \tan^{-1}(y/x)$
Exercise 1.2 - Step-by-Step Solutions
Q1: Represent $2 + 3i$ on Argand Diagram
Teacher's Note: Here $x=2$ (Real) and $y=3$ (Imaginary).
Step: Move 2 units on the Real axis (right) and 3 units up on the Imaginary axis.
Result: The point $(2, 3)$ in the first quadrant represents $2 + 3i$.
MCQ 1.1: The x-axis in Argand diagram is? (A) Imaginary (B) Real Answer
(B) Real
MCQ 1.2: In which quadrant does $-2-3i$ lie? (A) 2nd (B) 3rd Answer
(B) 3rd
Q2: Find r and $\theta$ for $z = 1 + i\sqrt{3}$
Step 1 (r): $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1+3} = 2$.
Step 2 ($\theta$): $\theta = \tan^{-1}(\sqrt{3}/1) = 60^\circ$ or $\pi/3$.
Result: $r = 2, \theta = \pi/3$.
MCQ 2.1: If $z = i$, its argument is? (A) $0^\circ$ (B) $90^\circ$ Answer
(B) $90^\circ$
MCQ 2.2: Distance of $z$ from origin is called? (A) Argument (B) Modulus Answer
(B) Modulus
Q3: Write $z = -1 + i$ in Polar Form
Step 1: $r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$.
Step 2: Since $x$ is negative and $y$ is positive, it's 2nd Quadrant. $\theta = 180^\circ - 45^\circ = 135^\circ$.
Result: $z = \sqrt{2}(\cos 135^\circ + i\sin 135^\circ)$.
MCQ 3.1: Polar form is also known as? (A) Trigonometric form (B) Vector form Answer
(A)
MCQ 3.2: Value of $\cos(\pi/2)$ is? (A) 1 (B) 0 Answer
(B) 0
Q4: Properties of Arguments
Concept: When we multiply two complex numbers, their
moduli multiply but their
arguments add.
$arg(z_1 \cdot z_2) = arg(z_1) + arg(z_2)$.
MCQ 4.1: $arg(z_1 / z_2) =$ ? (A) $arg(z_1) - arg(z_2)$ (B) $arg(z_1) + arg(z_2)$ Answer
(A)
MCQ 4.2: Modulus of $z_1 \cdot z_2$ is? (A) $|z_1| + |z_2|$ (B) $|z_1| \cdot |z_2|$ Answer
(B)
Q5: Solve $z^2 = -i$
Step: Use De Moivre's Theorem (Introductory level). Convert $-i$ to polar: $1(\cos 270^\circ + i\sin 270^\circ)$.
Result: Taking square root gives two values by dividing the angle by 2.
MCQ 5.1: How many square roots does a complex number have? (A) 1 (B) 2 Answer
(B) 2
MCQ 5.2: De Moivre's theorem involves? (A) Powers (B) Addition Answer
(A) Powers
🎉 Great Job! You can now "See" Math!
Visualizing numbers on a graph makes engineering and physics much easier. You're doing great!
