Complex numbers are an important and high-weightage topic that is asked in the JEE Advanced Exam. The topic carries an approximate weightage of 9-10% and candidates can expect 2 to 3 questions from this section in the JEE Advanced Exam 2025.
The complex number section focuses on understanding imaginary numbers, modulus, and properties of complex roots. The most asked topics in the Complex numbers are the Argand Plane, Roots of Unity, Nature of Roots, and Complex conjugates.
Must Read Articles:
- Download JEE Advanced Last 10 year PYQ for Free
- Download JEE 2024 Question Paper with Detailed Solutions
Below is the list of Complex number questions with detailed solutions asked in JEE Advanced Exam.
Question 1:
Question 2:
A value of b for which the equations x2+bx−1=0,x2+x+b=0 have one root in common is
Question 3:
Let z=cosθ+isinθ. Then, the value of
Question 4:
In the quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then, the equation p[p(x)]=0 has
- only purely imaginary roots
- all real roots
- two real and two purely imaginary roots
- neither real nor purely imaginary roots
Question 5:
Let α, β be the roots of the equation −px+r=0 and a2 ,2β be the roots of the equation x2−qx+r=0. Then, the value of r is
Question 6:
A man walks a distance of 3 units from the origin towards the North-East (N 45 degrees E) direction. From there, he walks a distance of 4 units towards the northwest (N 45 degrees W) direction to reach a point P. Then, the position of P in the Arg and plane is
Question 7:
If a, b,c are the sides of a triangle ABC such that x2−2(a+b+c)x+3λ(ab+bc+ca)=0 has real roots, then
Question 8:
The shaded region, where P=(−1,0),Q=(−1+2,2)R=(−1+2,−2),S=(1,0) is represented by
Question 9:
The minimum value of ∣a+bω+cω2∣ , where a,b and c are all not equal integers and ω(≠1) ω(=1) is a cube root of unity, is
Question 10:
If one root is square of the other root of the equation
x2+px+q=0, then the relation between p and q is
- p3−q(3p−1)+q2=0
- p3−q(3p+1)+q2=0
- p3+q(3p−1)+q2=0
- p3+q(3p+1)+q2=0
Question 11:
If ω(≠1) ω(=1) be a cube root of unity and (1+ω2)n=(1+ω4)n, then the least positive value of n is
- 2
- 3
- 5
- 6
JEE Advanced Exam Pattern 2025
- Mode of the exam: The examination was conducted ONLY in the Computer Based Test (CBT) mode.
- Number of papers: Two question papers: Paper 1 and Paper 2 of three hours duration each. Both papers are compulsory.
- Language: English and Hindi
- Total duration: 6 hours
- Number of sections: Each question paper consisted of three separate sections, viz., Physics, Chemistry and Mathematics.
- Marking scheme: Details of the marking scheme were provided in the “Instructions to Candidates” section at the time of examination. The same is also explained further below.
JEE Advanced 2025 Mathematics Syllabus
JEE Advanced Mathematics Syllabus consists of nine units- Sets, Relations and Functions ; Algebra ; Matrices; Probability and Statistics; Trignometry; Analytical Geometry; Differential Calculus ; Integral Calculus and Vectors. Check JEE Advanced 2025 Syllabus
JEE Advanced Mathematics Syllabus according to the seven units is defined below.
| Units | Sub Units |
|---|---|
| Sets, Relations, and Functions | - Sets and their representations, different kinds of sets (empty, finite, and infinite), algebra of sets, intersection, complement, difference, and symmetric difference of sets and their algebraic properties, De-Morgan’s laws on union, intersection, and difference (for finite number of sets), and practical problems based on them.- Cartesian product of finite sets, ordered pair, relations, domain, and codomain of relations, equivalence relation.- Function as a special case of relation, functions as mappings, domain, codomain, range of functions, invertible functions, even and odd functions, into, onto and one-to-one functions, special functions (polynomial, trigonometric, exponential, logarithmic, power, absolute value, greatest integer, etc.), sum, difference, product, and composition of functions. |
| Algebra | - Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.- Statement of fundamental theorem of algebra, quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.- Arithmetic and geometric progressions, arithmetic and geometric means, sums of finite arithmetic and geometric progressions, infinite geometric series, sum of the first n natural numbers, sums of squares and cubes of the first n natural numbers.- Logarithms and their properties, permutations and combinations, binomial theorem for a positive integral index, properties of binomial coefficients. |
| Matrices | - Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, elementary row and column transformations, determinant of a square matrix of order up to three, adjoint of a matrix, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric, and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables. |
| Probability and Statistics | - Random experiment, sample space, different types of events (impossible, simple, compound), addition and multiplication rules of probability, conditional probability, independence of events, total probability, Bayes Theorem, computation of probability of events using permutations and combinations.- Measure of central tendency and dispersion, mean, median, mode, mean deviation, standard deviation, and variance of grouped and ungrouped data, analysis of the frequency distribution with the same mean but different variance, random variable, mean, and variance of the random variable. |
| Trigonometry | - Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, general solution of trigonometric equations.- Inverse trigonometric functions (principal value only) and their elementary properties. |
| Analytical Geometry | - Topics related to coordinate geometry, straight lines, circles, conic sections, and other geometric properties. |
| Differential Calculus | - Concepts of limits, continuity, differentiation, derivatives of functions, application of derivatives, increasing and decreasing functions, maxima and minima. |
| Integral Calculus | - Indefinite and definite integrals, properties of integrals, fundamental theorem of calculus, methods of integration, applications of integration. |
| Vectors | - Vectors and their properties, operations on vectors, scalar and vector products, applications in geometry and physics. |
JEE Advanced Mathematics Chapter-wise Weightage
However, based on the previous year's analysis, the chapters along with the number of questions and marks tabulated below:
| Chapter Name | Number of Questions | Marks |
|---|---|---|
| Application of Derivatives | 3 | 9 |
| Ellipse | 1 | 3 |
| Circles | 2 | 6 |
| Parabola | 2 | 7 |
| Definite Integral | 2 | 7 |
| Permutations and Combinations | 1 | 3 |
| Sequence and Series | 1 | 3 |
| Matrices | 2 | 7 |
| Limit and Continuity | 1 | 4 |
| Probability | 1 | 4 |
| Complex Numbers | 1 | 4 |
| Hyperbola | 1 | 4 |
JEE Advanced - Marks wise Weightage of Topics
| Topics | Marks |
|---|---|
| Calculus | 40-50 marks |
| Vector and 3D Geometry | 15-20 marks |
| Probability & Permutation & Combination | 15-20 marks |
| Parabola, Hyperbola, Ellipse, Rectangular Hyperbola | 15 marks |
| Complex Numbers | 15 marks |
Important Books for Mathematics:
JEE Advanced 2024 Question Paper PDF with Solutions
| Paper Name | Question Paper Link | Solutions Link |
|---|---|---|
| JEE Advanced 2024 Paper 1 Question Paper (English) | Download PDF | Check Solutions |
| JEE Advanced 2024 Paper 2 Question Paper (English) | Download PDF | Check Solutions |
| JEE Advanced 2024 Paper 1 Question Paper (Hindi) | Download PDF | Check Solutions |
| JEE Advanced 2024 Paper 1 Question Paper (Hindi) | Download PDF | Check Solutions |
JEE Advanced Last 5 Year Mathematics Papers Pdf
Download free JEE Advanced Previous Year Question Paper PDFs from the table provided below:-
| JEE Advanced Paper Name | Question Paper Link |
|---|---|
| JEE Advanced 2024 Question Paper PDF | Click Here |
| JEE Advanced 2023 Question Paper PDF | Click Here |
| JEE Advanced 2022 Question Paper PDF | Click Here |
| JEE Advanced 2021 Question Paper PDF | Click Here |
| JEE Advanced 2020 Question Paper PDF | Click Here |
| JEE Advanced 2019 Question Paper PDF | Click Here |
| JEE Advanced 2018 Question Paper PDF | Click Here |
| JEE Advanced 2017 Question Paper PDF | Click Here |
| JEE Advanced 2016 Question Paper PDF | Click Here |
JEE Advanced 2023 Question Paper PDF Download With Solutions
| Paper | Question Paper PDF | Answer Key PDF |
|---|---|---|
| Question Paper 1 PDF | Download PDF | Check Solutions |
| Question Paper 2 PDF | Download PDF | Check Solutions |
| Question Paper AAT PDF | Download PDF | Check Solutions |
JEE Advanced 2022 Question Paper
| 2022 Question Paper | PDF Download |
|---|---|
| Question Paper 1 PDF | Click here |
| Question Paper 2 PDF | Click here |
JEE Advanced 2021 Question Paper
| Question Paper | PDF Download | ||
|---|---|---|---|
| Question Paper 1 PDF | Physics | Chemistry | Maths |
| Question Paper 2 PDF | Physics | Chemistry | Maths |
JEE Advanced 2020 Question Paper
| Question Paper | PDF Download |
| Question Paper 1 PDF | Click here |
| Question Paper 2 PDF | Click here |
Frequently Asked Questions
Q1. Why is 17 a complex number?
17 is considered as a real number, it can also be expressed as complex number. A complex number can be written in the form of a+bi, where a and b are real numbers, and i is the imaginary number.
In the case of 17, the value of b can be zero and it can be written as 17+ 0i. This shows that all real numbers are complex with their imaginary part equal to zero.
Q2. How are complex numbers related to quadratic equations?
Complex numbers play a major role in solving quadratic equations when the equation has no real roots. In a quadratic equation of the form ax2 +bx+c= 0, the nature of the root depends on the discriminant.
- If the discriminant is positive, the equation has two real roots.
- If the discriminant is zero, the equation has one repeated real root.
- If the discriminant is negative the equation has no real roots.
Q3. What happens when a quadratic has complex zero?
When a quadratic equation has complex zeros, the graph of such an equation does not touch or cross the x-axis. The parabola lies above or below the x-axis.
For example, the graph of y=x2+4 stays above the x-axis because the minimum value of parabola is 4. The roots of this equation are ±2i which is imaginary and thus, has no real x-intercepts.
Q4. What rank is 100 marks in JEE Advanced?
100 marks rank in JEE Advanced 2025 is based on the difficulty of the exam and cutoff trends.
Expected Rank (Based on Last Year Trends) :
100-110 marks → 15,000 - 18,000 rank
120-130 marks → 10,000 - 12,000 rank
150+ marks → Less than 5,000 rank
Note: The rank is updated annually, so use maximum accuracy with no negative marking.
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