JEE Main Maths Complex Numbers & Quadratic Equations 2027: Argand Plane, Modulus & Roots

Complex numbers look exotic until you see them as points and arrows on a plane. Once you do, modulus becomes distance, multiplication becomes rotation, and locus problems become geometry you already know. Paired with quadratic equations, this chapter is one of the most dependable scoring units in JEE Main Maths.

A complex number is a point. Its modulus is the distance from the origin, and its argument is the angle.

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Chapter at a Glance

Priority reflects past-paper trends, not an official NTA weightage.

What the NTA Syllabus Covers

• Complex numbers as ordered pairs of reals, the form $a + ib$ and representation in a plane, Argand diagram
• Algebra of complex numbers, modulus and argument (amplitude)
• Quadratic equations in the real and complex number systems, relations between roots and coefficients, nature of roots, forming equations with given roots

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Master These Topics

1. Modulus, Argument and Polar Form

For $z = x + iy$, the modulus is $\lvert z \rvert = \sqrt{x^2 + y^2}$ and the argument $\theta$ is the angle with the positive real axis. In polar form:

$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$

Key properties: $z\bar{z} = \lvert z \rvert^2$, $\lvert z_1 z_2 \rvert = \lvert z_1 \rvert \lvert z_2 \rvert$, and $\arg(z_1 z_2) = \arg z_1 + \arg z_2$.

Worked example (division): $\dfrac{3 + 4i}{1 - 2i} = \dfrac{(3 + 4i)(1 + 2i)}{5} = \dfrac{3 - 8 + 10i}{5} = -1 + 2i$, so the modulus is $\sqrt{5}$.

Worked example (powers): $1 + i = \sqrt{2}\,e^{i\pi/4}$, so $(1 + i)^8 = (\sqrt{2})^8 e^{2\pi i} = 16$.

Trap: Always place the argument in the correct quadrant. $\tan^{-1}(y/x)$ alone gives the wrong angle when $x \lt 0$.

2. Loci in the Argand Plane

• $\lvert z - z_0 \rvert = r$ is a circle with centre $z_0$ and radius $r$.
• $\lvert z - z_1 \rvert = \lvert z - z_2 \rvert$ is the perpendicular bisector of the segment joining $z_1$ and $z_2$.
• The triangle inequality states $\lvert z_1 + z_2 \rvert \le \lvert z_1 \rvert + \lvert z_2 \rvert$.

Worked example: For $\lvert z - 1 \rvert = \lvert z + i \rvert$ with $z = x + iy$, we get $(x-1)^2 + y^2 = x^2 + (y+1)^2$, which simplifies to $y = -x$, a straight line.

3. Quadratic Equations and the Nature of Roots

For $ax^2 + bx + c = 0$ with discriminant $D = b^2 - 4ac$:

The relations between roots $\alpha, \beta$ and coefficients are $\alpha + \beta = -\dfrac{b}{a}$ and $\alpha\beta = \dfrac{c}{a}$. To form an equation with given roots, use $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.

Worked example (forming): Roots $2 + \sqrt{3}$ and $2 - \sqrt{3}$ have sum $4$ and product $4 - 3 = 1$, so the equation is $x^2 - 4x + 1 = 0$.

Worked example (symmetric functions): If $\alpha, \beta$ are roots of $x^2 - 3x + 1 = 0$, then $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 9 - 2 = 7$ and $\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta) = 27 - 9 = 18$.

Trap: Complex roots come in conjugate pairs only when all coefficients are real.

4. A Handy Extra: Cube Roots of Unity

The unit cube roots $1, \omega, \omega^2$ satisfy $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$. The NTA Main text does not name them explicitly, but they are a common tool in problems on complex numbers.

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Common Traps to Avoid

• Placing the argument in the wrong quadrant.
• Forgetting that conjugate-pair roots need real coefficients.
• Using $\lvert z_1 + z_2 \rvert = \lvert z_1 \rvert + \lvert z_2 \rvert$ as if it were always an equality.
• Dropping the sign of $b$ in the sum of roots $-b/a$.

60-Second Revision Sheet

• $z\bar{z} = \lvert z \rvert^2$; $\lvert z_1 z_2 \rvert = \lvert z_1 \rvert \lvert z_2 \rvert$; $\arg(z_1z_2) = \arg z_1 + \arg z_2$
• Circle: $\lvert z - z_0 \rvert = r$; perpendicular bisector: $\lvert z - z_1 \rvert = \lvert z - z_2 \rvert$
• $\alpha + \beta = -b/a$, $\alpha\beta = c/a$, $D = b^2 - 4ac$
• $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$

Your Study Plan

1. Day 1: algebra of complex numbers, conjugates and division.
2. Day 2: polar form, powers and arguments in all four quadrants.
3. Day 3: loci and geometry in the Argand plane.
4. Day 4: quadratic roots, symmetric functions and a timed mixed set.

Practice Complex Numbers & Quadratic Equations Questions Free → (opens in a new tab)

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Continue Your Maths Journey

• Previous chapter: Sets, Relations & Functions
• Next chapter: Matrices & Determinants
• All 14 JEE Main Maths chapters
• Complete JEE Main Syllabus 2027 guide

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