Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20264 AprilEvening Shift
Exam
JEE Main
Year
2026
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2026 · 4 April · Evening Shift · Mathematics · Complex Number

Let S=\ z ∈ ℂ: z^2+4z+16=0\ . Then ∑_ z ∈ S |z+ 3 i|^2 is equal to:

Let S={z∈C:z2+4z+16=0}S=\{z \in \mathbb{C}: z^2+4z+16=0\}. Then ∑z∈S∣z+3i∣2\sum_{z \in S}|z+\sqrt{3}i|^2 is equal to:

Options

  1. A4242
  2. B2323
  3. C2727
  4. D3838

Correct answer

D. 3838

Step-by-step solution

Given the equation z2+4z+16=0z^2 + 4z + 16 = 0, we find its roots using the quadratic formula:

z=−4±16−642=−4±−482=−2±23iz = \dfrac{-4 \pm \sqrt{16 - 64}}{2} = \dfrac{-4 \pm \sqrt{-48}}{2} = -2 \pm 2\sqrt{3}i

The set SS contains two complex numbers: z1=−2+23iz_1 = -2 + 2\sqrt{3}i and z2=−2−23iz_2 = -2 - 2\sqrt{3}i.

We need to evaluate the sum ∑z∈S∣z+3i∣2\sum_{z \in S} |z + \sqrt{3}i|^2, which is ∣z1+3i∣2+∣z2+3i∣2|z_1 + \sqrt{3}i|^2 + |z_2 + \sqrt{3}i|^2.

For z1=−2+23iz_1 = -2 + 2\sqrt{3}i:

∣z1+3i∣2=∣−2+23i+3i∣2=∣−2+33i∣2|z_1 + \sqrt{3}i|^2 = |-2 + 2\sqrt{3}i + \sqrt{3}i|^2 = |-2 + 3\sqrt{3}i|^2

∣−2+33i∣2=(−2)2+(33)2=4+27=31|-2 + 3\sqrt{3}i|^2 = (-2)^2 + (3\sqrt{3})^2 = 4 + 27 = 31

For z2=−2−23iz_2 = -2 - 2\sqrt{3}i:

∣z2+3i∣2=∣−2−23i+3i∣2=∣−2−3i∣2|z_2 + \sqrt{3}i|^2 = |-2 - 2\sqrt{3}i + \sqrt{3}i|^2 = |-2 - \sqrt{3}i|^2

∣−2−3i∣2=(−2)2+(−3)2=4+3=7|-2 - \sqrt{3}i|^2 = (-2)^2 + (-\sqrt{3})^2 = 4 + 3 = 7

Adding these values together:

∑z∈S∣z+3i∣2=31+7=38\sum_{z \in S} |z + \sqrt{3}i|^2 = 31 + 7 = 38

Answer: 3838
Practice Complex Number on Quantrex Academy →

More from Complex Number

The number of values of z ∈ ℂ , satisfying the equations |z-(4+8i)|= 10 and… JEE Main 2026Let S = \ z ∈ ℂ : z^2 + 6 \,iz - 3 = 0\ . Then ∑ _ z ∈ S z^8 is equal to : JEE Main 2026Let the set of all values of k ∈ ℝ such that the equation z( z + 2 + i) + k(2 + 3i) = 0… JEE Main 2026Let z_1, z_2 ∈ ℂ be the distinct solutions of the equation z^2 + 4z - (1 + 12i) = 0 .… JEE Main 2026Let z be a complex number such that |z+2| = |z-2| and ((z+3)/(z-i)) = (π)/(4) . Then… JEE Main 2026Let the circles C_1 : |z| = r and C_2 : |z - 3 - 4i| = 5 , z ∈ ℂ , be such that C_2 lies… JEE Main 2026Let x and y be real numbers such that 50((2x)/(1+3i) - (y)/(1-2i)) = 31 + 17i , i = -1 .… JEE Main 2026Let A=\ z ∈ ℂ:|z-2| 4\ and B=\ z ∈ ℂ:|z-2|+|z+2|=5\ . Then the max \ |z_ 1 -z_ 2 |: z_ 1… JEE Main 2026Let z be a complex number such that |z-6|=5 and |z+2-6 i|=5 . Then the value of z^ 3 +3… JEE Main 2026Let z=(1+i)(1+2 i)(1+3 i) …(1+n i) , where i= -1 . If |z|^ 2 =44200 , then n is equal to… JEE Main 2026If z= 3 2 +(i)/(2), i= -1 , then (z^ 201 -i)^ 8 is equal to JEE Main 2026Let S=\ z: 3 |2 z-3(1+i)| 7\ be a set of complex numbers. Then _ z ∈ S |(z+(1)/(2)(5+3… JEE Main 2026Let S=\ z ∈ ℂ: 4 z^ 2 + z =0\ . Then ∑_ z ∈ S |z|^ 2 is equal to: JEE Main 2026Let α= -1+i 3 2 and β= -1-i 3 2 , i= -1 . If (7-7 α+9 β)^ 20 +(9+7 α-7 β)^ 20 +(-7+9 α+7… JEE Main 2026Let z be the complex number satisfying |z-5| ≤ 3 and having maximum positive principal… JEE Main 2026If x^ 2 +x+1=0 , then the value of (x+(1)/(x))^ 4 +(x^ 2 + 1 x^ 2 )^ 4 + (x^ 3 + 1 x^ 3… JEE Main 2026 Full Complex Number list Complex Number JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs