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Complex Numbers

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 7 times: 27 Jun 2022, 28 Jun 2022, 27 Jul 2022, 13 Apr 2023, 30 Jan 2024, 4 Apr 2024, 22 Jan 2026 S2. This one: 22 Jan 2026 S2 See all 7 of this type

    54
    Let S={z∈C:4z2+z‾=0}S =\left\{ z\in C:4z^{2}+\overline{z}= 0 \right\}. Then ∑z∈S∣z∣2\sum_{z \in S} |z|^{2} is equal to :

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  2. Question 2

    This type asked 6 times: 26 Jul 2022, 29 Jan 2023, 1 Feb 2024, 4 Apr 2025, 24 Jan 2026 S1, 28 Jan 2026 S1. This one: 28 Jan 2026 S1 See all 6 of this type

    55
    Let zz be a complex number such that ∣z−6∣=5|z- 6| = 5 and ∣z+2−6i∣=5|z+ 2 - 6i| = 5. Then the value of z3+3z2−15z+141z^{3}+ 3z^{2}- 15z+ 141 is equal to

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  3. Question 3

    This type asked 5 times: 2021 Paper 2, 2021 Paper 19, 29 Jul 2022, 24 Jan 2023, 23 Jan 2026 S2. This one: 23 Jan 2026 S2 See all 5 of this type

    67
    If z=32+i2,i=−1z =\frac{\sqrt{3}}{2}+\frac{i}{2},i =\sqrt{- 1}, then (z201−i)8\left( z^{201}- i \right)^{8} is equal to

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  4. Question 4

    This type asked 5 times: 2021 Paper 25, 29 Jun 2022, 27 Jan 2024, 30 Jan 2024, 22 Jan 2026 S1. This one: 22 Jan 2026 S1 See all 5 of this type

    71
    Let α=−1+i32\alpha = \frac{- 1 + i\sqrt{3}}{2} and β=−1−i32,i=−1\beta = \frac{- 1 - i\sqrt{3}}{2},i = \sqrt{- 1}. If (7−7α+9β)20+(9+7α−7β)20+(−7+9α+7β)20+(14+7α+7β)20=m10(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + ( - 7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, then m is ____\_\_\_\_ .

  5. Question 5

    This type asked 5 times: 2021 Paper 20, 2021 Paper 23, 10 Apr 2023, 9 Apr 2024, 7 Apr 2025. This one: 7 Apr 2025 See all 5 of this type

    61
    Among the statements (S1): The set {z∈C−{−i}:∣z∣=1 and z−iz+i is purely real}\{z\in\mathbb{C}-\{-i\}:|z|=1\text{ and }\frac{z-i}{z+i}\text{ is purely real}\} contains exactly two elements, and (S2): The set {z∈C−{−1}:∣z∣=1 and z−1z+1 is purely imaginary}\{z\in\mathbb{C}-\{-1\}:|z|=1\text{ and }\frac{z-1}{z+1}\text{ is purely imaginary}\} contains infinitely many elements.

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  6. Question 6

    This type asked 5 times: 2021 Paper 7, 2021 Paper 10, 26 Jul 2022, 11 Apr 2023, 28 Jan 2025. This one: 28 Jan 2025 See all 5 of this type

    68
    Let OO be the origin, the point AA be z1=3+22iz_{1}=\sqrt{3}+ 2\sqrt{2}i, the point B(z2)B\left( z_{2} \right) be such that 3∣z2∣=∣z1∣\sqrt{3}\left| z_{2} \right|=\left| z_{1} \right| and arg(z2)=arg(z1)+π6arg\left( z_{2} \right)= arg\left( z_{1} \right)+\frac{\pi}{6}. Then

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  7. Question 7

    This type asked 5 times: 2021 Paper 12, 2021 Paper 17, 27 Jul 2022, 8 Apr 2023, 8 Apr 2024. This one: 8 Apr 2024 See all 5 of this type

    156
    The sum of all possible values of θ∈[−π,2π]\theta \in \lbrack - \pi,2\pi\rbrack, for which 1+icosθ1−2icosθ\frac{1 + icos\theta}{1 - 2icos\theta} is purely imaginary, is equal to

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  8. Question 8

    This type asked 5 times: 2021 Paper 17, 2021 Paper 26, 14 Jun 2022, 29 Jun 2022, 1 Feb 2024. This one: 1 Feb 2024 See all 5 of this type

    87
    Let P={z∈C:∣z+2−3i∣≤1}\mathbf{P = \{ z \in C:|z + 2 - 3i| \leq 1\}} and Q={z∈C:z(1+i)+z‾(1−i)≤−8}\mathbf{Q = \{ z \in C:z(1 + i) +}\overline{z}(1 - i) \leq - 8\}. Let in P∩Q,∣z−3+2i∣P \cap Q,|z - 3 + 2i| be maximum and minimum at z1z_{1} and z2z_{2} respectively. If ∣z1∣2+2∣z2∣2=α+β2\left| z_{1} \right|^{2}+ 2\left| z_{2} \right|^{2}= \alpha + \beta\sqrt{2}, where α,β\alpha,\beta are integers, then α+β\alpha + \beta equals

  9. Question 9

    This type asked 4 times: 29 Jun 2022, 13 Apr 2023, 29 Jan 2024, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 4 of this type

    53
    Let S={z∈C:z2+6iz−3=0}S=\left\{ z\in C:z^{2}+\sqrt{6}iz- 3 = 0 \right\}. Then ∑z∈Sz8\sum_{z\in S} z^{8} is equal to :

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  10. Question 10

    This type asked 4 times: 1 Feb 2024, 29 Jan 2025, 23 Jan 2026 S1, 2 Apr 2026 S2. This one: 2 Apr 2026 S2 See all 4 of this type

    52
    Let the circle C1:∣z∣=rC_{1}:|z| = r and C2:∣z−3−4i∣=5,z∈CC_{2}:|z - 3 - 4i| = 5,z \in\mathbb{C}, be such that C2C_{2} lies within C1C_{1}. If z1z_{1} moves on C1,z2C_{1},z_{2} moves on C2C_{2} and min⁡∣z1−z2∣=2\min\left| z_{1}-z_{2} \right|= 2, then max ∣z1−z2∣\left| z_{1}-z_{2} \right| is equal to :-

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