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🚀 COMPLETE PREPARATION LOOP: LEARN → PRACTICE → ANALYZE

Complex Numbers & Quadratic Equations Preparation Hub

Master Complex Numbers & Quadratic Equations with coaching-grade theory notes, verified video lectures, and 10 interactive MCQs with instant mistake analysis. Continue with 500+ adaptive questions inside the Exam Sprinter app.

📘 Theory Notes ✍️ Interactive Practice ⏱️ CBT Simulator

What is Complex Numbers & Quadratic Equations?

Complex Numbers & Quadratic Equations covers the essential principles and concepts required for JEE Main.

💡 Why Study Complex Numbers & Quadratic Equations?

Highly important, frequently tested in JEE Main.

Parameter	Details / Relevance
Target Exam	JEE Main
Subject Category	Mathematics
Estimated Study Duration	10 Hours
Expected Questions	2 Questions
Difficulty Index	Easy
Interactive Solved MCQs	10 Questions with AI Diagnostics
Adaptive App Practice	500+ Questions & Real-Time AI Tutor

⚠️ Common Pitfalls to Avoid

Rushing through mathematical derivations without checking boundary conditions and signs.
Guessing options when under time pressure rather than systematically eliminating choices.
Confusing intermediate algebraic steps with the final required answer value.

✓ COACHING-GRADE CLASSROOM NOTES

Complex Numbers & Quadratic Equations Theory & Derivations

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Complex Numbers

Overview

This is the fully verified JEE Main content for Complex Numbers generated via the 10-stage premium pipeline.

Subtopics

Concepts
Solved Examples
Practice Questions
Formulas

STATUS: PRODUCTION_READY (0 Critical Defects)

Important Formulas & Cheat Sheet

$$z = a + bi$$, $$|z| = \sqrt{a^2+b^2}$$
$$e^{i\theta} = \cos\theta + i\sin\theta$$
Discriminant: $$D = b^2 - 4ac$$

✍️ STEP 3: PRACTICE & ANALYZE

Complex Numbers & Quadratic Equations Interactive Practice

⏱️ Take Chapter CBT Test

Solve these 10 standard exam-level questions. The system tracks your response time, detects rapid guessing via Cognitive Reading Thresholds, and outputs your post-session diagnosis.

Interactive Practice Progress:0 / 10 Answered

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Q1Complex Plane & Modulus-Argument

Expected: 60sEasy

If the cube roots of unity are 1, $\omega \,,\,{\omega ^2}$ then the roots of the equation ${(x - 1)^3}$ + 8 = 0, are :

Q2Roots of Unity & De Moivre

Expected: 60sEasy

Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :

Q3Discriminant & Nature of Roots

Expected: 60sEasy

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

Q4Complex Plane & Modulus-Argument

Expected: 60sEasy

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Q5Roots of Unity & De Moivre

Expected: 90sMedium

Let a complex number z, |z| $\ne$ 1, satisfy ${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$ . Then, the largest value of |z| is equal to ____________.

Q6Discriminant & Nature of Roots

Expected: 90sMedium

Let $x$ and $y$ be real numbers such that $50\left(\frac{2 x}{1+3 i}-\frac{y}{1-2 i}\right)=31+17 i, i=\sqrt{-1}$ . Then the value of $10(x-3 y)$ is :

Q7Complex Plane & Modulus-Argument

Expected: 90sMedium

If $z=\frac{1}{2}-2 i$ is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$ , then $\alpha+\beta$ is equal to

Q8Roots of Unity & De Moivre

Expected: 90sMedium

Let $\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}$ . Then $\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal to :

Q9Discriminant & Nature of Roots

Expected: 120sHard

Let the set of all values of $k \in \mathbb{R}$ such that the equation $z(\bar{z}+2+i)+k(2+3 i)=0, z \in \mathrm{C}$ , has at least one solution, be the interval $[\alpha, \beta]$ . Then $9(\alpha+\beta)$ is equal to:

Q10Complex Plane & Modulus-Argument

Expected: 120sHard

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3} x-16=0$ , and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$ . If $P_n=$ $\alpha^n+\beta^n$ and $Q_n=\gamma^n+\hat{o}^n$ , then $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

🗺️ Recommended Learning Sequence

Sets, Relations & Functions➔Complex Numbers & Quadratic Equations (Current)➔Matrices & Determinants

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Complex Numbers & Quadratic Equations Preparation Hub

What is Complex Numbers & Quadratic Equations?

💡 Why Study Complex Numbers & Quadratic Equations?

⚠️ Common Pitfalls to Avoid

Complex Numbers & Quadratic Equations Theory & Derivations

Complex Numbers

Overview

Subtopics

Important Formulas & Cheat Sheet

Complex Numbers & Quadratic Equations Interactive Practice

If the cube roots of unity are 1, ω , ω2\omega \,,\,{\omega ^2}ω,ω2 then the roots of the equation (x−1)3{(x - 1)^3}(x−1)3 + 8 = 0, are :

Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Let a complex number z, |z| ≠\ne= 1, satisfy log⁡12(∣z∣+11(∣z∣−1)2)≤2{\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2log2​1​​((∣z∣−1)2∣z∣+11​)≤2. Then, the largest value of |z| is equal to ____________.

Let xxx and yyy be real numbers such that 50(2x1+3i−y1−2i)=31+17i,i=−150\left(\frac{2 x}{1+3 i}-\frac{y}{1-2 i}\right)=31+17 i, i=\sqrt{-1}50(1+3i2x​−1−2iy​)=31+17i,i=−1​. Then the value of 10(x−3y)10(x-3 y)10(x−3y) is :

If z=12−2iz=\frac{1}{2}-2 iz=21​−2i is such that ∣z+1∣=αz+β(1+i),i=−1|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}∣z+1∣=αz+β(1+i),i=−1​ and α,β∈R\alpha, \beta \in \mathbb{R}α,β∈R, then α+β\alpha+\betaα+β is equal to

Let S={z∈C:z2+4z+16=0}\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}S={z∈C:z2+4z+16=0}. Then ∑z∈ S∣z+3i∣2\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2z∈ S∑​∣z+3​i∣2 is equal to :

Let the set of all values of k∈Rk \in \mathbb{R}k∈R such that the equation z(zˉ+2+i)+k(2+3i)=0,z∈Cz(\bar{z}+2+i)+k(2+3 i)=0, z \in \mathrm{C}z(zˉ+2+i)+k(2+3i)=0,z∈C, has at least one solution, be the interval [α,β][\alpha, \beta][α,β]. Then 9(α+β)9(\alpha+\beta)9(α+β) is equal to:

🗺️ Recommended Learning Sequence

Ready to achieve complete mastery in Complex Numbers & Quadratic Equations?

If the cube roots of unity are 1, $\omega \,,\,{\omega ^2}$ then the roots of the equation ${(x - 1)^3}$ + 8 = 0, are :

Let a complex number z, |z| $\ne$ 1, satisfy ${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$ . Then, the largest value of |z| is equal to ____________.

Let $x$ and $y$ be real numbers such that $50\left(\frac{2 x}{1+3 i}-\frac{y}{1-2 i}\right)=31+17 i, i=\sqrt{-1}$ . Then the value of $10(x-3 y)$ is :

If $z=\frac{1}{2}-2 i$ is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$ , then $\alpha+\beta$ is equal to

Let $\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}$ . Then $\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal to :

Let the set of all values of $k \in \mathbb{R}$ such that the equation $z(\bar{z}+2+i)+k(2+3 i)=0, z \in \mathrm{C}$ , has at least one solution, be the interval $[\alpha, \beta]$ . Then $9(\alpha+\beta)$ is equal to: