JEE Advanced Previous Year Questions (2018 - 2023): Complex Numbers

## 2023

Q1: Let  . If A contains exactly one positive integer n, then the value of n is                    [JEE Advanced 2023 Paper 1]
Ans:
281

For positive integer

Q2: Let z be a complex number satisfying , where  denotes the complex conjugate of z. Let the imaginary part of z be nonzero.
Match each entry in List-I to the correct entries in List-II.

The correct option is:
(a) (P)→(1)(Q)→(3)(R)→(5)(S)→(4)
(b) (P)→(2)(Q)→(1)(R)→(3)(S)→(5)
(c) (P)→(2)(Q)→(4)(R)→(5)(S)→(1)
(d) (P)→(2)(Q)→(3)(R)→(5)(S)→(4)               [JEE Advanced 2023 Paper 1]
Ans:
(b)

Take conjugate both sides

Let

Put in (1)

Also

Now

|z + 1|2 = 4 + 3 = 7
∴ (P)→(2)(Q)→(1)(R)→(3)(S)→(5)
∴ Option (b) is correct.

## 2022

Q1: Let z be a complex number with a non-zero imaginary part. If  is a real number, then the value of |z|2 is _________.            [JEE Advanced 2022 Paper 1]
Ans:
0.49 to 0.51
For a complex number z = x + iy, it's conjugate . Now z is purely real when y = 0.
When y = 0 then z = x + i × (0) = x and
∴   when z is purely real.
Now given,  is real

=

y = 0  not possible as given z is a complex number with non-zero imaginary part.

Q2: Let  denote the complex conjugate of a complex number z and let . In the set of complex numbers, the number of distinct roots of the equation  is _________.     [JEE Advanced 2022 Paper 1]
Ans:
4
Let z = x + iy

Given,

Comparing both sides real part we get,

And comparing both sides imaginary part we get,

Adding equation (1) and (2) we get,

Case 1 : When x = 0 :
Put x =  0  at equation (1), we get

Case 2 : When y = −1/2 :
Put y = −1/2 in equation (1), we get

∴ Number of distinct  z = 4

Q3: Let denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary parts of  are integers, then which of the following is/are possible value(s) of |z| ?
(a)
(b)
(c)
(d)                   [JEE Advanced 2022 Paper 2]
Ans:
(a)
Let, complex number  is a new complex number ω.

Now, Let  where r = |z| and θ = argument

∴ Real part of

Imaginary part of

Given both Re(ω) and Im(ω) are integer.
∴ Let Re(ω) = I1
and Im(ω) = I2

Now,

In option only positive sign is given so ignoring negative sign we get,

From option (A),

Comparing with (1), we get

Putting α = 45 in (1), we get

Option (A) is correct.
We can re-write

Comparing with option (B) we get,

Option (B) is incorrect.
Similarly option (C) and (D) also incorrect.

## 2021

Q1: Let θ1θ2, ........, θ10 = 2π. Define the complex numbers z1 = e1, zk = zk − 1efor k = 2, 3, ......., 10, where i = √−1. Consider the statements P and Q given below :

Then,
(a) P is TRUE and Q is FALSE
(b) Q is TRUE and P is FALSE
(c) both P and Q are TRUE
(d) both P and Q are FALSE                           [JEE Advanced 2021 Paper 1]
Ans:
(c)
Both P and Q are true.
Length of direct distance  length of arc
i.e. | z2  z1 | = length of line AB  length of arc AB.

| z3  z2 | = length of line BC  length of arc BC.
Sum of length of these 10 lines  sum of length of arcs (i.e. 2π) (because θ1 + θ2 + θ3 + .... + θ10 = 2π (given)
| z2  z1 | + | z3  z2 | + ..... + | z1  z10 |  2π  P is true.
And | zk2  zk−12 | = | zk  zk − 1 | | zk + zk − 1 |
As we know that,

4π  Q is true.

Q2: For any complex number w = c + id, let arg⁡(ω)∈(−π, π], where i = √−1. Let α and β be real numbers such that for all complex numbers z = x + iy satisfying  , the ordered pair (x, y) lies on the circle x2 + y2 + 5x − 3y + 4 = 0, Then which of the following statements is (are) TRUE?
(a) α = −1
(b) αβ = 4
(c) αβ = −4
(d) β = 4               [JEE Advanced 2021 Paper 1]
Ans:
(d)
Circle  x2 + y2 + 5x − 3y + 4 = 0 cuts the real axis (X-axis) at (4, 0), (1, 0).

implies z is on arc and (− α, 0) and (− β, 0) subtend π/4 on z.
So, α = 1 and  β = 4
Hence, αβ = 1 × 4 = 4 and β = 4

## 2020

Q1: For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying , where i = √−1. Then the minimum possible value of |z1  z2|2, where z1, z2S with Re(z1) > 0 and Re(z2) < 0 is _____     [JEE Advanced 2020 Paper 2]
Ans: 8
For a complex number z, it is given that,

So, either

Now, Case - I, if z2=0 and z = x + iy
So, x- y2 + 2ixy = 0
⇒ x- y2 = 0
and xy = 0
⇒ x = y  = 0
⇒ z = 0  which is not possible according to given conditions.
Case - II, if  and
z = x + iy
So,

⇒ xy = 1 is an equation of rectangular hyperbola and for minimum value of |z1  z2|2, the z1 and z2 must be vertices of the rectangular hyperbola.
Therefore,  z1 = 1 + i and z2 = -1 - i
∴ Minimum value of   |z1 − z2|2
= (1 + 1)2 + (1 + 1)2
= 4 + 4
= 8

Q2: Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE?
(a) |z + 1/2| ≤ 1/2 for all z ∈ S
(b) |z| ≤ 2 for all z ∈ S
(c) |z + 1/2| ≥ 1/2 for all z ∈ S
(d) The set S has exactly four elements           [JEE Advanced 2020 Paper 1]
Ans: (
b) & (c)
It is given that the complex number satisfying

from Eqs. (i) and (ii), we get

## 2019

Q1: Let ω ≠ 1 be a cube root of unity. Then the maximum of the set  distinct non-zero integers} equals _____   [JEE Advanced 2019 Paper 1]
Ans:
3
Given, ω ≠ 1 be a cube root of unity, then

[as ω3 = 1)

a, b and c are distinct non-zero integers. For minimum value a= 1, b = 2 and c = 3

Q2: Let S be the set of all complex numbers z satisfying . If the complex number z0 is such that  is the maximum of the set , then the principal argument of  is
(a) π / 4
(b) 3π / 4
(c) - π / 2
(d) π / 2                            [JEE Advanced 2019 Paper 1]
Ans:
(c)
The complex number z satisfying , which represents the region outside the circle (including the circumference) having centre (2, −1) and radius √5 units.

Now, for  is maximum.
When |z0 − 1| is minimum. And for this it is required that z0 ∈ S, such that z0 is collinear with the points (2, 1) and (1, 0) and lies on the circumference of the circle |z − 2 + i| = √5.
So let z0 = x + iy, and from the figure 0 < x < 1 and y >0.
So,

is a positive real number, so  is purely negative imaginary number.

## 2018

Q1: Let s, t, r be non-zero complex numbers and L be the set of solutions  of the equation   where  = x  iy. Then, which of the following statement(s) is(are) TRUE?     [JEE Advanced 2018 Paper 2]
(a) If L has exactly one element, then |s| ≠ |t|
(b) If |s| = |t|, then L has infinitely many elements
(c) The number of elements in L ∩ {z:|z − 1 + i|=5} is at most 2
(d) If L has more than one element, then L has infinitely many elements             [JEE Advanced 2018 Paper 2]
Ans:
(a), (c) & (d)
We have,

On taking conjugate,

On solving Eqs. (i) and (ii), we get

(a) For unique solutions of z
It is true
(b) If |s| = |t|, then  may or may not be zero. So, z may have no solutions.∴ L may be an empty set.
It is false.
(c) If elements of set L represents line, then this line and given circle intersect at maximum two point. Hence, it is true.
(d) In this case locus of z is a line, so L has infinite elements. Hence, it is true.

The document JEE Advanced Previous Year Questions (2018 - 2023): Complex Numbers | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on JEE Advanced Previous Year Questions (2018 - 2023): Complex Numbers - Mathematics (Maths) for JEE Main & Advanced

 1. What is the importance of studying complex numbers in JEE Advanced?
Ans. Complex numbers play a crucial role in JEE Advanced as they are used to solve a wide range of mathematical problems. They provide a powerful tool for representing and manipulating quantities that involve both real and imaginary components. In JEE Advanced, complex numbers are extensively used in topics such as algebra, calculus, and coordinate geometry. Understanding complex numbers is essential for solving complex equations, analyzing circuits, and studying the behavior of waves, among other applications.
 2. How are complex numbers represented in the complex plane?
Ans. Complex numbers are represented in the complex plane, also known as the Argand plane, using a two-dimensional coordinate system. In this system, the real part of a complex number is plotted along the x-axis, while the imaginary part is plotted along the y-axis. The complex number z = a + bi, where a and b are real numbers, is represented by the point (a, b) in the complex plane. The distance from the origin to the point represents the magnitude of the complex number, and the angle formed by the positive x-axis and the line connecting the origin and the point represents the argument or phase of the complex number.
 3. How can complex numbers be expressed in polar form?
Ans. Complex numbers can be expressed in polar form using their magnitude and argument. The magnitude of a complex number z = a + bi is given by |z| = √(a^2 + b^2), and the argument θ is the angle formed by the positive x-axis and the line connecting the origin and the point representing the complex number in the complex plane. The polar form of the complex number is then given by z = |z| * e^(iθ), where e represents Euler's number and i is the imaginary unit.
 4. How are complex numbers added and multiplied?
Ans. Complex numbers are added and multiplied by separately adding or multiplying their real and imaginary parts. For addition, the real parts and imaginary parts are added separately. For example, if z1 = a + bi and z2 = c + di, the sum z1 + z2 is equal to (a + c) + (b + d)i. For multiplication, the distributive property is used. The real part of the product is obtained by multiplying the real parts of the complex numbers and subtracting the product of their imaginary parts. The imaginary part of the product is obtained by multiplying the real part of one complex number by the imaginary part of the other and adding it to the product of their imaginary parts. For example, if z1 = a + bi and z2 = c + di, their product z1 * z2 is equal to (ac - bd) + (ad + bc)i.
 5. How can complex numbers be used to solve equations?
Ans. Complex numbers can be used to solve equations by expanding the number system beyond real numbers. They allow us to find solutions to equations that would otherwise have no real solutions. For example, the equation x^2 + 1 = 0 has no real solutions, but by introducing the imaginary unit i, we can solve it by setting x = ±i. Complex numbers can also be used to solve higher degree equations, such as quadratic, cubic, and quartic equations. Additionally, complex numbers can be used to simplify calculations involving trigonometric functions and exponential functions, making them a valuable tool in solving various mathematical problems.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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