Class 10 Maths Chapter 1 Exercise 1.1 Solutions 2026 – Complex Numbers Notes PDF

Unit 1: Complex Numbers | Exercise 1.1 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 1.1 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.

Why Do We Need Complex Numbers?

Ordinary real numbers cannot answer every equation. An expression like x² + 1 = 0 has no solution among real numbers, because no real number squared can ever be negative. To close this gap, mathematicians introduced a new number, i, defined so that i² = −1. Once i exists, equations that used to have “no solution” suddenly do — and an entire new branch of numbers, called complex numbers, opens up. Exercise 1.1 is where students meet this idea for the first time.

Key Concepts Covered in Exercise 1.1

The Standard Form z = a + bi

Every complex number is written with two parts: a real part (a) and an imaginary part (b), combined as z = a + bi. Recognizing a and b correctly in any given complex number is the first skill this exercise builds.

Real Part and Imaginary Part

The real part is written Re(z) and the imaginary part Im(z). For example, in z = 5 − 2i, Re(z) = 5 and Im(z) = −2 — note that the imaginary part is the coefficient of i, not the term −2i itself.

The Imaginary Unit i

Defined by i² = −1, this single identity is the foundation of the entire chapter. Every later exercise — operations, conjugates, modulus, polar form — traces back to this one rule.

Equality of Two Complex Numbers

Two complex numbers a + bi and c + di are equal only when their real parts match (a = c) and their imaginary parts match (b = d). This rule is frequently used to solve for unknown variables inside a complex equation.

Step-by-Step Solved Examples

Q. No. 1: State the real and imaginary parts of z = −6 + 9i.

Re(z) = −6

Im(z) = 9

Q. No. 2: Find the values of x and y if (x + 2) + (y − 3)i = 5 − i.

Compare real parts: x + 2 = 5  →  x = 3

Compare imaginary parts: y − 3 = −1  →  y = 2

Q. No. 3: Simplify i² × i³.

i² = −1 and i³ = −i

i² × i³ = (−1) × (−i) = i

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 1.1 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: Objective questions on standard form, i², and identifying real/imaginary parts
• Short Questions: Direct 2–3 line questions such as finding Re(z) and Im(z) for a given number
• Long Questions: Multi-step problems combining equality of complex numbers with basic algebra

Where Complex Numbers Are Actually Used

Complex numbers are not just an abstract exercise — they underpin electrical engineering (AC circuit analysis), signal processing, control systems, computer graphics, and quantum mechanics. Knowing this often helps the concept feel less arbitrary to students meeting it for the first time.

Common Mistakes Students Make in Exercise 1.1

• Writing the imaginary part as −2i instead of just −2 (Im(z) is a real number, not a term with i)
• Forgetting that i² = −1 when simplifying products or powers of i
• Assuming two complex numbers are equal just because they look similar, without checking both parts separately

Why This Exercise Matters for the Board Exam

Exercise 1.1 rarely appears in full on its own in the board paper, but its definitions are assumed knowledge for nearly every other question in the chapter — including MCQs, short questions, and the long questions on operations, conjugates, and modulus. Skipping this exercise usually means struggling with everything that follows it.

Quick Links – Rest of Chapter 1: Complex Numbers

Links for sections other than Exercise 1.1 will be activated as they are published on this site.

Download Exercise 1.1 Notes PDF

Click below to view or download the complete Exercise 1.1 notes in PDF format.

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 1.1 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. Do I need any background knowledge before starting Exercise 1.1?

No — Exercise 1.1 is the very first exercise of the chapter and introduces complex numbers from scratch. Only basic algebra is required.

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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