How to Use This Page
This page is your complete study companion for Chapter 2 — Complex Numbers. Start with the Key Concepts table to refresh definitions and formulas, then work through the MCQs, fill-in-the-blanks, and true-or-false sections for quick practice. Use the match-the-following exercise to strengthen associations. The brief and detailed answer sections mirror the 2-mark and 5-mark questions you will face in the board exam. Finally, review the Common Mistakes and Exam Checklist before every test to avoid losing marks.
Chapter 2 — Exercise Solutions Guide
Complex number problems become easy when every expression is converted into either a + ib form or polar form. Do not mix both forms in the same step unless required.
Key Concepts (Quick Recap)
| Term | Definition | ||
|---|---|---|---|
| Complex number | A number of the form z = a + ib where a is real part and b is imaginary part | ||
| Modulus | The absolute value of z; given by \ | z\ | = sqrt(a² + b²) |
| Argument | The angle theta made by z with the positive real axis; tan theta = b / a | ||
| Conjugate | The conjugate of z = a + ib is z bar = a - ib | ||
| Polar form | z = r(cos theta + i sin theta) where r = \ | z\ | and theta = arg(z) |
| De Moivre's Theorem | [r(cos theta + i sin theta)]ⁿ = rⁿ(cos n theta + i sin n theta) | ||
| Euler's form | z = r e^(i theta); connects polar form to exponential notation | ||
| Cube roots of unity | 1, omega, omega² where omega = cos(2pi/3) + i sin(2pi/3) | ||
| Properties of omega | omega³ = 1; 1 + omega + omega² = 0; omega² = omega bar | ||
| Square root of i | sqrt(i) = (1 + i) / sqrt2 | ||
| Principal value of argument | The value of arg(z) in the interval (-pi, pi] | ||
| Real part | Re(z) = a for z = a + ib | ||
| Imaginary part | Im(z) = b for z = a + ib | ||
| Argand diagram | A 2D plane with real axis (horizontal) and imaginary axis (vertical) representing complex numbers |
Formula Recap
| Concept | Formula | ||||||
|---|---|---|---|---|---|---|---|
| z | a + ib | ||||||
| Modulus | \ | z\ | = sqrt(a² + b²) | ||||
| Argument | tan theta = b / a, adjusted by quadrant | ||||||
| Conjugate | z bar = a - ib | ||||||
| Polar form | z = r(cos theta + i sin theta) | ||||||
| De Moivre | [r(cos theta + i sin theta)]ⁿ = rⁿ(cos n theta + i sin n theta) | ||||||
| z₁z₂ | \ | z₁z₂\ | = \ | z₁\ | \ | z₂\ | ; arg(z₁z₂) = arg(z₁) + arg(z₂) |
| z₁/z₂ | \ | z₁/z₂\ | = \ | z₁\ | /\ | z₂\ | ; arg(z₁/z₂) = arg(z₁) - arg(z₂) |
I. Choose the Best Answer (MCQs)
7. The modulus of the product z₁z₂ is equal to
| (a) \ | z₁\ | + \ | z₂\ | |
|---|---|---|---|---|
| (b) \ | z₁\ | - \ | z₂\ | |
| (c) \ | z₁\ | \ | z₂\ | |
| (d) \ | z₁\ | / \ | z₂\ | |
| **Answer: (c) \ | z₁\ | \ | z₂\ | ** (The modulus of a product equals the product of the moduli.) |
II. Fill in the Blanks
7. The product of a complex number and its conjugate is always ______.
| **Answer: a non-negative real number (specifically, \ | z\ | ²)** |
|---|
III. True or False
1. The modulus of a complex number is always non-negative. True. (Modulus is defined as sqrt(a² + b²), which is always >= 0.)
2. The conjugate of a purely imaginary number equals the negative of that number. True. (For z = 0 + bi, the conjugate is 0 - bi = -bi = -(bi).)
3. De Moivre's theorem can be applied only for positive integer powers. False. Correction: De Moivre's theorem holds for all rational powers n = p/q when properly interpreted for roots.
4. The argument of z = 0 is 0. False. Correction: The argument of z = 0 is undefined because the zero complex number has no well-defined angle.
5. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. True. (This is the fundamental condition for equality of complex numbers.)
6. The product of a complex number and its conjugate equals the square of its modulus.
| True. (z multiplied by z bar = a² + b² = \ | z\ | ².) |
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7. i raised to any integer power can only take the values 1, -1, i, or -i. True. (Powers of i cycle: i, -1, -i, 1, and then repeat.)
8. The modulus of z₁/z₂ equals the ratio of the moduli of z₁ and z₂.
| True. (\ | z₁/z₂\ | = \ | z₁\ | /\ | z₂\ | , provided z₂ is not zero.) |
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IV. Match the Following
| Column A | Column B | ||
|---|---|---|---|
| 1. Modulus of z = a + ib | (c) sqrt(a² + b²) | ||
| 2. Conjugate of z = a + ib | (f) a - ib | ||
| 3. Argument of z = 1 + i | (a) pi/4 | ||
| 4. De Moivre's theorem | (e) (cos theta + i sin theta)ⁿ = cos n theta + i sin n theta | ||
| 5. Cube roots of unity sum | (b) 0 | ||
| 6. z × z bar | (g) \ | z\ | ² |
| 7. Polar form | (d) r(cos theta + i sin theta) | ||
| 8. i² | (h) -1 |
Standard Solution Pattern
1. Identify real part and imaginary part. 2. Find modulus using sqrt(a² + b²). 3. Find argument using quadrant. 4. Convert to polar form if power or root is asked. 5. Apply De Moivre theorem for powers and roots.
Worked Pattern
For z = 1 + i:
For z²:
Worked Examples
Example 1 — Division of complex numbers
Simplify (1 + i) / (1 − i).
So the result is i (modulus 1, argument pi/2).
Example 2 — Modulus of a product
| **Find | z₁z₂ | if z₁ = 3 + 4i and z₂ = 5 − 12i.** |
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Example 3 — Polar form
Express z = −1 + √3 i in polar form.
Example 4 — De Moivre's theorem
Find (1 + i)⁵ using De Moivre's theorem.
Roots of Unity
Cube roots of unity are:
Important relations:
V. Answer Briefly (2-Mark Questions)
1. Define a complex number and write its standard form.
A complex number is a number of the form z = a + ib, where a and b are real numbers and i = sqrt(-1). Here a is called the real part (denoted Re(z)) and b is called the imaginary part (denoted Im(z)). Complex numbers extend the real number system to allow solutions to equations like x² + 1 = 0 that have no real solutions.
2. What is the modulus of a complex number? Find the modulus of z = 5 - 12i.
| The modulus of a complex number z = a + ib is defined as | z | = sqrt(a² + b²). It represents the distance of the point (a, b) from the origin in the Argand diagram. For z = 5 - 12i, | z | = sqrt(25 + 144) = sqrt(169) = 13. The modulus is always a non-negative real number. |
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3. State De Moivre's theorem.
De Moivre's theorem states that for any positive integer n, (cos theta + i sin theta)ⁿ = cos(n theta) + i sin(n theta). In polar form, if z = r(cos theta + i sin theta), then zⁿ = rⁿ(cos n theta + i sin n theta). This theorem is extremely useful for computing powers and roots of complex numbers without expanding using the binomial theorem.
4. Write the properties of cube roots of unity.
The cube roots of unity are 1, omega, and omega², where omega = cos(2pi/3) + i sin(2pi/3) = (-1 + i sqrt(3))/2. Their properties are: (i) omega³ = 1, (ii) 1 + omega + omega² = 0, (iii) omega² = omega bar (conjugate of omega), and (iv) omega and omega² are complex conjugates of each other. These properties simplify many algebraic computations.
5. Distinguish between the real part and imaginary part of a complex number.
For a complex number z = a + ib, the real part Re(z) = a is the coefficient of 1 (the real component), and the imaginary part Im(z) = b is the coefficient of i (the imaginary component). Both a and b are real numbers. The real part determines the horizontal position on the Argand diagram, while the imaginary part determines the vertical position.
6. What is the principal value of the argument?
The principal value of the argument of a complex number z is the unique value of arg(z) that lies in the interval (-pi, pi]. It is denoted Arg(z) with a capital A. For a complex number in the first quadrant, Arg(z) = tan⁻¹(b/a). For other quadrants, the angle is adjusted by adding or subtracting pi to bring it into the principal range.
VI. Answer in Detail (5-Mark Questions)
1. Explain the polar form of a complex number and convert z = -1 - i into polar form step by step.
The polar form of a complex number represents z in terms of its distance from the origin (modulus r) and the angle it makes with the positive real axis (argument theta). The general form is z = r(cos theta + i sin theta), also written as z = r e^(i theta) using Euler's formula. To convert z = -1 - i into polar form, first find the modulus: r = sqrt((-1)² + (-1)²) = sqrt(2). Then find the argument: since the point (-1, -1) lies in the third quadrant, tan alpha = 1/1 = 1, so alpha = pi/4. Therefore theta = -(pi - pi/4) = -3pi/4, or equivalently theta = pi + pi/4 = 5pi/4 in [0, 2pi). The principal argument is Arg(z) = -3pi/4. Hence, z = sqrt(2)[cos(-3pi/4) + i sin(-3pi/4)]. Verify: sqrt(2) cos(-3pi/4) = sqrt(2)(-1/sqrt(2)) = -1 and sqrt(2) sin(-3pi/4) = sqrt(2)(-1/sqrt(2)) = -1. This confirms the conversion is correct.
2. State and prove De Moivre's theorem. Use it to find (cos theta + i sin theta)⁶.
De Moivre's theorem states: (cos theta + i sin theta)ⁿ = cos(n theta) + i sin(n theta) for any positive integer n. Proof by mathematical induction: Base case n = 1 is trivially true. Assume the result holds for n = k, i.e., (cos theta + i sin theta)^k = cos(k theta) + i sin(k theta). Multiply both sides by (cos theta + i sin theta): (cos theta + i sin theta)^(k+1) = [cos(k theta) + i sin(k theta)][cos theta + i sin theta] = cos(k theta)cos theta - sin(k theta)sin theta + i[sin(k theta)cos theta + cos(k theta)sin theta] = cos((k+1)theta) + i sin((k+1)theta), using the angle addition formulas. Hence, by induction, the theorem holds for all positive integers n. Now applying the theorem: (cos theta + i sin theta)⁶ = cos(6 theta) + i sin(6 theta). This avoids the tedious binomial expansion and directly gives the result.
3. Explain the method of finding the nth roots of a complex number using De Moivre's theorem and find the cube roots of unity.
To find the nth roots of a complex number z = r(cos theta + i sin theta), we use De Moivre's theorem. The nth roots are given by z_k = r^(1/n)[cos((theta + 2k pi)/n) + i sin((theta + 2k pi)/n)] for k = 0, 1, 2, ..., n-1. Each root has the same modulus r^(1/n) but different arguments that are equally spaced by 2pi/n on the Argand diagram, forming a regular n-gon. For the cube roots of unity, we need z such that z³ = 1. Write 1 in polar form: 1 = cos 0 + i sin 0. So r = 1, theta = 0. The cube roots are: for k = 0: z₀ = cos(0) + i sin(0) = 1; for k = 1: z₁ = cos(2pi/3) + i sin(2pi/3) = -1/2 + i sqrt(3)/2 = omega; for k = 2: z₂ = cos(4pi/3) + i sin(4pi/3) = -1/2 - i sqrt(3)/2 = omega². These three roots satisfy 1 + omega + omega² = 0 and omega³ = 1.
4. Derive the properties of the modulus and argument of the product and quotient of two complex numbers.
| For two complex numbers z₁ = r₁(cos theta₁ + i sin theta₁) and z₂ = r₂(cos theta₂ + i sin theta₂), the product z₁z₂ = r₁r₂[cos(theta₁ + theta₂) + i sin(theta₁ + theta₂)]. This shows that | z₁z₂ | = | z₁ | z₂ | and arg(z₁z₂) = arg(z₁) + arg(z₂). For the quotient, z₁/z₂ = (r₁/r₂)[cos(theta₁ - theta₂) + i sin(theta₁ - theta₂)], giving | z₁/z₂ | = | z₁ | / | z₂ | and arg(z₁/z₂) = arg(z₁) - arg(z₂). In algebraic form, for z = a + ib, the product z × z bar = (a + ib)(a - ib) = a² + b² = | z | ², which confirms that the product of a number and its conjugate equals the square of its modulus. These properties are fundamental for simplifying complex arithmetic and for geometric interpretations on the Argand diagram. |
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Common Mistakes to Avoid
1. ❌ Taking the argument without checking which quadrant the point lies in. ✅ Always plot the point (a, b) on the Argand diagram first, identify the quadrant, then adjust the angle accordingly. In Q2, add pi/4 to the angle; in Q3, subtract from pi.
2. ❌ Forgetting that i² = -1 when expanding powers of complex numbers. ✅ Always replace i² with -1 immediately after expansion. For example, (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i, not 1 + 2i + 1.
3. ❌ Expanding (cos theta + i sin theta)ⁿ using the binomial theorem for large n. ✅ Use De Moivre's theorem instead: (cos theta + i sin theta)ⁿ = cos(n theta) + i sin(n theta). This is much faster and avoids errors.
4. ❌ Writing the modulus as a negative number.
| ✅ The modulus | z | = sqrt(a² + b²) is always non-negative. It represents a distance, which cannot be negative. |
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5. ❌ Confusing the conjugate with the negative of a complex number. ✅ The conjugate of z = a + ib is z bar = a - ib (only the sign of the imaginary part changes). The negative is -z = -a - ib (both parts change sign).
6. ❌ Applying arg(z₁z₂) = arg(z₁) + arg(z₂) without adjusting for the principal value range. ✅ After adding the arguments, check if the result lies in (-pi, pi]. If not, add or subtract 2pi to bring it into the principal range.
7. ❌ Assuming that 1 + omega + omega² = 1 for cube roots of unity. ✅ The correct property is 1 + omega + omega² = 0. This is one of the most frequently tested properties.
8. ❌ Computing the argument of z = 0 as 0 or pi. ✅ The argument of the zero complex number is undefined because there is no well-defined angle from the origin to itself.
Exam Checklist
- Identify the real part (a) and imaginary part (b) of every complex number before starting.
- Compute the modulus using the formula sqrt(a² + b²) and always write it as a positive value.
- Determine the quadrant of the point (a, b) before calculating the argument.
- Use the principal value of the argument in the range (-pi, pi].
- Convert to polar form r(cos theta + i sin theta) when powers or roots are asked.
- Apply De Moivre's theorem for computing powers and roots of complex numbers.
- Remember the cube root properties: omega³ = 1 and 1 + omega + omega² = 0.
- Multiply numerator and denominator by the conjugate when dividing complex numbers.
- Verify answers by substituting back into the original expression when possible.
- Write the final answer in the form requested by the question (a + ib form or polar form).
- Show all intermediate steps clearly, especially quadrant adjustments for the argument.