Class 10 Maths Chapter 1 Exercise 1.2 Solutions 2026 – Complex Numbers Notes PDF

Unit 1: Complex Numbers | Exercise 1.2 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 1.2 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 Definitions to Calculations

Exercise 1.1 taught students what a complex number is. Exercise 1.2 asks a natural follow-up question: if a + bi is a number, can it be added, subtracted, or multiplied like any other number? The answer is yes — complex numbers follow almost all the same rules as real-number algebra, with exactly one extra rule to remember: i² = −1. This exercise is where that rule gets put to work.

Key Concepts Covered in Exercise 1.2

Addition of Complex Numbers

To add (a + bi) + (c + di), add the real parts together and the imaginary parts together separately: the result is (a + c) + (b + d)i. The two parts never mix.

Subtraction of Complex Numbers

Subtraction works the same way as addition, but every sign inside the second bracket must be flipped first. This is where most calculation errors happen, so this exercise gives it extra practice.

Multiplication of Complex Numbers

Multiplying two complex numbers uses the same distributive (FOIL) method as multiplying two algebraic binomials. The only extra step is simplifying any i² term that appears, using i² = −1, before writing the final answer in standard form.

Combining Multiple Operations

Some questions in this exercise chain two or more operations together — for example, add two complex numbers first, then multiply the result by a third. Working through the brackets in the correct order matters here.

Step-by-Step Solved Examples

Q. No. 1: Simplify (4 + 3i) + (2 − 5i).

= (4 + 2) + (3i − 5i)

= 6 − 2i

Q. No. 2: Simplify (7 + 4i) − (3 − 2i).

= (7 − 3) + (4i − (−2i))

= 4 + 6i

Q. No. 3: Simplify (2 + 3i)(1 − 4i).

= 2(1) + 2(−4i) + 3i(1) + 3i(−4i)

= 2 − 8i + 3i − 12i²

= 2 − 5i − 12(−1)   [since i² = −1]

= 2 − 5i + 12

= 14 − 5i

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 1.2 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 objective questions on the sign rules for addition and subtraction
• Short Questions: Direct 2–3 line questions asking to add or subtract two given complex numbers
• Long Questions: Multi-step multiplication problems, sometimes combined with simplifying powers of i

Where This Matters Beyond the Exercise

Algebraic operations on complex numbers are the building blocks for every later topic in this chapter — conjugates, division, and modulus all rely on being able to add, subtract, and multiply complex expressions confidently. Outside the classroom, these same operations appear in circuit analysis and signal processing, wherever real and reactive quantities are combined.

Common Mistakes Students Make in Exercise 1.2

• Forgetting to flip both signs inside the second bracket during subtraction
• Leaving i² unsimplified instead of replacing it with −1
• Adding real and imaginary parts together by mistake, instead of keeping them separate
• Losing a negative sign while expanding brackets in multiplication

Why This Exercise Matters for the Board Exam

Addition, subtraction, and multiplication of complex numbers are among the most frequently repeated question types in the objective and short-question sections of the Punjab Board paper. Because the method is short and formulaic, this exercise is also one of the easiest places to secure guaranteed marks with focused practice.

Quick Links – Rest of Chapter 1: Complex Numbers

Links for sections other than Exercise 1.2 will be activated as they are published on this site.

Download Exercise 1.2 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 1.2 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 1.2?

Students should first complete Exercise 1.1, since Exercise 1.2 assumes familiarity with the standard form a + bi and the identity i² = −1.

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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