Class 10 Maths Chapter 1 Exercise 1.4 Solutions 2026 – Complex Numbers Notes PDF

Unit 1: Complex Numbers | Exercise 1.4 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 1.4 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.

Giving a Complex Number an Address

A point on the Argand diagram can be described in two ways: by how far right/left and up/down it sits (its a and b coordinates), or by how far it is from the origin and in which direction. Exercise 1.4 introduces this second description — the distance is called the modulus, and the direction is called the argument. Together, they let a complex number be rewritten in an entirely different but equally valid format called the polar form.

Key Concepts Covered in Exercise 1.4

Modulus of a Complex Number

The modulus of z = a + bi, written |z|, is calculated as |z| = √(a² + b²). Geometrically, it is simply the straight-line distance from the origin to the point (a, b) on the Argand diagram — the same idea as the Pythagorean theorem. The modulus is never negative.

Argument of a Complex Number

The argument, written arg(z) or θ, is the angle formed between the positive real axis and the line joining the origin to the point (a, b). It is generally found using θ = tan⁻¹(b/a), with the sign of a and b checked first to place the angle in the correct quadrant.

Polar Form of a Complex Number

Once the modulus r and argument θ are known, the complex number can be rewritten as z = r(cosθ + i sinθ). This is called the polar form, and it expresses the exact same number using distance and direction instead of a + bi.

Converting Between Standard Form and Polar Form

This exercise gives practice converting in both directions: from a + bi to r(cosθ + i sinθ), and from a given r and θ back into standard a + bi form using a = r cosθ and b = r sinθ.

Step-by-Step Solved Examples

Q. No. 1: Find the modulus and argument of z = 1 + i, and write it in polar form.

|z| = √(1² + 1²) = √2

θ = tan⁻¹(1/1) = tan⁻¹(1) = 45° (since a and b are both positive, angle is in Quadrant I)

Polar form: z = √2(cos45° + i sin45°)

Q. No. 2: Find the modulus of z = −5 − 12i.

|z| = √((−5)² + (−12)²)

|z| = √(25 + 144) = √169

|z| = 13

Q. No. 3: Convert the polar form z = 2(cos60° + i sin60°) back to standard form.

a = r cosθ = 2 cos60° = 2(0.5) = 1

b = r sinθ = 2 sin60° = 2(0.866) ≈ 1.73

Standard form: z ≈ 1 + 1.73i

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 1.4 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 the modulus formula and the range of possible argument values
• Short Questions: Direct 2–3 line questions asking to calculate just the modulus of a given number
• Long Questions: Full conversions between standard form and polar form, with both modulus and argument shown

Where Modulus and Argument Matter Beyond This Exercise

The modulus reappears throughout later mathematics as a measure of a complex number’s “size,” and the polar form becomes especially useful for multiplying, dividing, and finding powers of complex numbers efficiently — topics that return in more depth at the Intermediate (FSc) level. In physics and engineering, modulus and argument describe the amplitude and phase of oscillating signals and AC waveforms.

Common Mistakes Students Make in Exercise 1.4

• Forgetting to check the sign of a and b before finalizing the argument’s quadrant
• Using a calculator in the wrong angle mode (degrees vs. radians) when finding θ
• Writing the modulus as a negative number after an arithmetic slip
• Mixing up the roles of a and b when converting from polar form back to standard form

Why This Exercise Matters for the Board Exam

Modulus questions are short, formula-based, and highly repeatable, making them a reliable source of marks in the short-question section. Polar form conversions appear more often in long questions, where showing the modulus and argument as separate, clearly labeled steps tends to earn more partial credit even if the final answer has a small error.

Quick Links – Rest of Chapter 1: Complex Numbers

Links for sections other than Exercise 1.4 will be activated as they are published on this site.

Download Exercise 1.4 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 1.4 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 1.4?

Students should be comfortable with the Argand diagram from Exercise 1.3, along with basic trigonometric ratios (sine, cosine, tangent) from earlier classes, since Exercise 1.4 uses both to calculate the modulus and argument.

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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