Complex Numbers - Must Do Questions ISI–CMI Maths 2026

Complex numbers are an important part of higher mathematics. They are widely used in exams like ISI, CMI, JEE Advanced, and Olympiads. Many students feel this topic is difficult. This usually happens due to a lack of clear understanding.

In reality, complex numbers become simple when you connect algebra with geometry. Here, we’ll explain the key concepts and important question types. It will help you build clarity and confidence for ISI–CMI Maths 2026.

What are Complex Numbers?

A complex number is written as:

z=x+iy

Here,

• x is the real part

• y is the imaginary part

• ​

Every complex number can be shown on a plane. This is called the complex plane. The horizontal axis represents the real part. The vertical axis represents the imaginary part.

Important Basic Concepts

Before solving advanced problems in complex numbers, it is essential to build a strong foundation with the basic concepts. These ideas form the backbone of almost every ISI and CMI-level question.

1. Conjugate of a Complex Number

If
z=x+iy, then its conjugate is:

zˉ=x−iy

The conjugate reflects the number across the real axis.

2. Modulus of a Complex Number

​

The modulus gives the distance of the point from the origin.

3. Polar Form

z=r(cos⁡θ+isin⁡θ)

This form is useful in trigonometry and geometry-based problems.

Geometry of Complex Numbers

Complex numbers can represent points on a plane. This idea is very useful in exams.

Equilateral Triangle Property

If three complex numbers lie on a unit circle and form an equilateral triangle, then:

z1+z2+z3=0

This is a very important result. It helps in solving many trigonometric problems.

Key Result

From the above relation, we get:

• cos a + cos b + cos c = 0

• sin a + sin b + sin c = 0

These results are useful in simplifying expressions.

Must Do Question Type 1: Trigonometric Identities

Example Concept

Using complex numbers, we can prove values like:

​​

These types of questions test your understanding of identities and transformations.

Strategy

• Convert trigonometric expressions into complex form

• Use Euler’s formula

• Simplify step by step

Must Do Question Type 2: Inequalities

One of the most common results is:

This is true for all real r.

Key Insight

• Equality holds when r=±1

• If the value is less than 2, then rrr is not real

This helps in identifying whether a number is complex or real.

Must Do Question Type 3: Roots of Unity

Roots of unity are very important in ISI–CMI exams.

These are solutions of:

Important Result

This result is used to solve many sum-based problems.

Application

• Sum of cosine series

• Sum of sine series

• Polynomial equations

Must Do Question Type 4: Transformation of Complex Numbers

Consider the function:

This transforms points from one plane to another.

Key Idea

• A straight line can become a curve

• A strip region can become parabolas

This shows the link between algebra and geometry.

Must Do Question Type 5: Maximum and Minimum Value

Consider:

Results

• Maximum value of ∣a∣ is 1

• Minimum value of|∣a∣ is 0

Understanding

• Maximum occurs when all angles are same

• Minimum occurs when points form a regular polygon

Must Do Question Type 6: Inequality in Complex Numbers

For complex numbers in the first quadrant:

Key Idea

• Uses the RMS-AM inequality

• Helps in bounding values

Must Do Question Type 7: Collinearity

If points z, z2, and zq are collinear, then:

• z must be real

Why This Happens

Collinearity restricts the argument of complex numbers.
This forces the number to lie on the real axis.

Must Do Question Type 8: Olympiad-Level Problem

Condition

Result

At least two of the numbers are equal.

Insight

• Convert into trigonometric form

• Use angle relations

• Simplify step by step

Important Tips for ISI–CMI 2026

Preparing for ISI and CMI requires more than just knowing formulas:

1. Focus on Concepts

Do not memorise formulas only. Understand why results work.

2. Practice Geometry

Visualise complex numbers on the plane. It improves clarity.

3. Revise Roots of Unity

Many problems are based on this concept.

4. Solve Mixed Problems

Combine algebra, trigonometry, and geometry.

5. Work on Proofs

ISI and CMI focus on reasoning. Practice proofs regularly.

Complex numbers are not just algebra. They connect geometry and trigonometry. This makes them powerful for solving advanced problems.

For ISI–CMI Maths 2026, focus on understanding patterns. Practice different types of questions. Work on proofs and reasoning.

With regular practice and clear concepts, this topic can become one of your strong areas.