Class 10 Maths Chapter 1 Exercise 1.3 Solutions 2026 – Complex Numbers Notes PDF
Unit 1: Complex Numbers | Exercise 1.3 | Punjab Board New Syllabus 2026–27
Updated August 2026: Exercise 1.3 solutions for Complex Numbers have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.
From Algebra to Geometry
So far, complex numbers have only existed as algebraic expressions. Exercise 1.3 gives them a home on paper: the Argand diagram, a plane where every complex number becomes a point. Alongside this graphical picture, the exercise introduces the conjugate — a simple partner number that makes division possible and unlocks a shortcut for finding a complex number’s distance from the origin.
Key Concepts Covered in Exercise 1.3
The Argand Diagram
A complex number z = a + bi is plotted as the point (a, b) on a plane where the horizontal axis is the real axis and the vertical axis is the imaginary axis. This turns an algebraic object into something students can actually see and compare visually.
Conjugate of a Complex Number
The conjugate of z = a + bi is written z̄ = a − bi — the imaginary part simply changes sign while the real part stays the same. This one small change has a big effect: it is the key that makes division of complex numbers possible.
Properties of the Conjugate
A few properties are worth remembering: the conjugate of a conjugate returns the original number; the conjugate of a sum equals the sum of the conjugates; and adding z to its own conjugate always cancels the imaginary part, leaving a purely real number (2a).
Division Using the Conjugate
To divide one complex number by another, multiply both the numerator and denominator by the conjugate of the denominator. Since (a + bi)(a − bi) = a² + b², this removes i from the denominator entirely and leaves the final answer in standard form.
Powers of i Revisited
This exercise continues practicing the repeating 4-term cycle of i (i, −1, −i, 1), now combined with division and conjugate questions rather than treated as a topic on its own.
Step-by-Step Solved Examples
Q. No. 1: Find the conjugate of z = −4 + 6i, and plot both z and its conjugate on an Argand diagram (describe their coordinates).
z̄ = −4 − 6i
z is plotted at (−4, 6); z̄ is plotted at (−4, −6)
Note: z and z̄ are mirror images of each other across the real axis.
Q. No. 2: Simplify (5 + 2i) ÷ (3 − i).
Multiply numerator and denominator by the conjugate (3 + i):
(5 + 2i)(3 + i) / (3 − i)(3 + i)
= (15 + 5i + 6i + 2i²) / (9 − i²)
= (15 + 11i − 2) / (9 + 1) [since i² = −1]
= (13 + 11i) / 10
= 1.3 + 1.1i
Q. No. 3: If z = 3 + 4i, verify that z + z̄ is a real number.
z̄ = 3 − 4i
z + z̄ = (3 + 4i) + (3 − 4i) = 6 + 0i = 6
6 is a real number, so the property is verified.
These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 1.3 in the same numbered, step-by-step format.
MCQs, Short Questions & Long Questions from This Exercise
Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:
- MCQs: Quick questions on finding a conjugate or reading coordinates off an Argand diagram
- Short Questions: Direct 2–3 line questions asking for the conjugate of a given complex number
- Long Questions: Full division problems requiring the conjugate method with all working shown
Why the Argand Diagram and Conjugate Matter Beyond This Exercise
The Argand diagram reappears in Exercise 1.4 when finding the modulus and argument, since both are read directly off this same graphical picture. The conjugate, meanwhile, is the standard tool engineers use when working with impedance in AC circuits, where dividing complex quantities is a routine calculation.
Common Mistakes Students Make in Exercise 1.3
- Forgetting to change the sign of the imaginary part when writing a conjugate
- Multiplying only the numerator by the conjugate and forgetting the denominator (or vice versa)
- Plotting a + bi as (b, a) instead of (a, b) on the Argand diagram
- Leaving i² unsimplified in the denominator after multiplying by the conjugate
Why This Exercise Matters for the Board Exam
Finding a conjugate and dividing complex numbers are two of the most reliably repeated question types in this chapter’s short-question and long-question sections. Because the conjugate method always follows the exact same steps, this exercise rewards students who practice the pattern rather than trying to memorize individual answers.
Quick Links – Rest of Chapter 1: Complex Numbers
| Section | Covers |
| Exercise 1.1 | Standard form, real & imaginary parts, i |
| Exercise 1.2 | Addition, subtraction & multiplication |
| Exercise 1.3 | You are here |
| Exercise 1.4 | Modulus, argument & polar form |
| Review Exercise 1 | Mixed revision, all topics |
| Chapter 1 MCQs | Objective-type practice |
Links for sections other than Exercise 1.3 will be activated as they are published on this site.
Download Exercise 1.3 Notes PDF
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Frequently Asked Questions (FAQs)
Q1. Are these Exercise 1.3 notes free to download?
Yes, all notes on this page are completely free to view and download in PDF format.
Q2. Which board are these notes for?
These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).
Q3. What should I know before starting Exercise 1.3?
Students should be comfortable with Exercises 1.1 and 1.2, since Exercise 1.3 builds on the standard form and basic operations to introduce the conjugate, division, and the Argand diagram.
Q4. Does this page include MCQs and short questions too?
Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.
Q5. How can I download the PDF?
Click the “Download PDF” button above and the notes will open or download directly to your device.
Q6. Are these notes updated for the new 2026–27 syllabus?
Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.
Comments & Feedback
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