If z is a complex number, then the number of common roots of the equation z1985 + z100 + 1 = 0 and z3 + 2z2 + 2z + 1 =…

Year	2026	2025	2024	Total
Questions	11	16	14	41

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

z³-1 = (z-1)(z²+z+1) Cube roots of unity: 1, ω, ω²

ω³ = 1 and 1+ω+ω² = 0

Core Logic

Let's first find the roots of the lower degree polynomial:

z³ + 2z² + 2z + 1 = 0

Group the terms:

(z³ + 1) + 2z(z + 1) = 0

(z + 1)(z² - z + 1) + 2z(z + 1) = 0

(z + 1)(z² - z + 1 + 2z) = 0

(z + 1)(z² + z + 1) = 0

Thus, the roots are $z = -1$ $z = -1$, and the roots of $z² + z + 1 = 0$ $z^2 + z + 1 = 0$ which are $z = ω, ω²$ $z = \omega, \omega^2$.

Step 1: Check z = -1

Substitute $z = -1$ $z = -1$ into the first equation $z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0$ $z^{1985} + z^{100} + 1 = 0$:

(-1)¹⁹⁸⁵ + (-1)¹⁰⁰ + 1 = -1 + 1 + 1 = 1 \neq 0

So, $z = -1$ $z = -1$ is not a common root.

Step 2: Check z = ω and ω²

Substitute $z = ω$ $z = \omega$:

ω¹⁹⁸⁵ + ω¹⁰⁰ + 1

Reduce the powers modulo 3 (since $ω³ = 1$ $\omega^3 = 1$): $1985 = 3 \times 661 + 2 \Rightarrow ω¹⁹⁸⁵ = ω²$ $100 = 3 \times 33 + 1 \Rightarrow ω¹⁰⁰ = ω¹ = ω$ So, $ω¹⁹⁸⁵ + ω¹⁰⁰ + 1 = ω² + ω + 1 = 0$ . $z = ω$ $z = \omega$ is a common root.

Substitute $z = ω²$ $z = \omega^2$:

(ω²)¹⁹⁸⁵ + (ω²)¹⁰⁰ + 1 = ω³⁹⁷⁰ + ω²⁰⁰ + 1

$3970 = 3 \times 1323 + 1 \Rightarrow ω³⁹⁷⁰ = ω$ $200 = 3 \times 66 + 2 \Rightarrow ω²⁰⁰ = ω²$ So, $ω³⁹⁷⁰ + ω²⁰⁰ + 1 = ω + ω² + 1 = 0$ . $z = ω²$ $z = \omega^2$ is also a common root.

Step 3: Conclusion

There are exactly 2 common roots: $ω$ $\omega$ and $ω²$ $\omega^2$.

Pattern Recognition

Whenever $z²+z+1$ $z^2+z+1$ emerges as a factor, its roots $ω$ $\omega$ and $ω²$ $\omega^2$ can be directly tested in massive degree polynomials using exponent reduction modulo 3.

Chapter Mix

Class 11 Maths: Complex Numbers

More Complex Numbers Previous-Year Questions — Page 9

Q26 jee_main_2024_31_jan_morning Properties of Modulus and Argument

α

denotes the number of solutions of

|1 - i|^x = 2^x

and

β = ((|z|)/( (z)))

, where

z = (π)/(4) (1 + i)⁴ ( 1 - \sqrt(π) i\sqrt(π) + i + \sqrt(π) - i1 + \sqrt(π) i), i = \sqrt(-1)

, then the distance of the point

(α, β)

from the line

4x - 3y = 7

Numerical Answer. Answer: 3 to 3

Solution

Core Logic

|1 - i|^x = 2^x (\sqrt(2))^x = 2^x 2x/2 = 2^x

This implies $(x)/(2) = x x = 0$ $\frac{x}{2} = x \implies x = 0$. There is exactly 1 solution, so $α = 1$ $\alpha = 1$.

Step 1: Simplify complex number z

(1+i)⁴ = ((1+i)²)² = (1 + i² + 2i)² = (2i)² = -4

Thus, $z = -π ( (1-\sqrt(π)i)(\sqrt(π)-i)π + 1 + (\sqrt(π)-i)(1-\sqrt(π)i)1 + π )$

Step 2: Simplify Bracket

Let's expand the terms directly: $z = (π)/(4)(-4) [ \sqrt(π) - π i - i - \sqrt(π)π + 1 + \sqrt(π) - i - π i - \sqrt(π)1 + π ]$

= -π [ (-i(π+1))/(π+1) + (-i(π+1))/(π+1) ]

= -π [ -i - i ] = 2π i

Step 3: Find beta

For $z = 2π i$ $z = 2\pi i$: $|z| = 2π$ $|z| = 2\pi$ and $(z) = (π)/(2)$ $\arg(z) = \frac{\pi}{2}$.

β = (|z|)/( (z)) = (2π)/(π/2) = 4

Step 4: Distance from Line

Distance of point $(α, β) = (1, 4)$ $(\alpha, \beta) = (1, 4)$ from the line $4x - 3y - 7 = 0$ $4x - 3y - 7 = 0$:

D = |4(1) - 3(4) - 7|\sqrt(4² + (-3)²) = (|4 - 12 - 7|)/(5) = (|-15|)/(5) = 3

Chapter Mix

Class 11 Maths: Complex Numbers and Quadratic Equations Class 11 Maths: Straight Lines

If $z$ $z$ is a complex number, then the number of common roots of the equation $z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0$ $z^{1985} + z^{100} + 1 = 0$ and $z³ + 2z² + 2z + 1 = 0$ $z^3 + 2z^2 + 2z + 1 = 0$ , is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Check z = -1

Step 2: Check z = ω and ω²

Step 3: Conclusion

Pattern Recognition

Chapter Mix

Reference Study Guides

More Complex Numbers Previous-Year Questions — Page 9

Solution

Core Logic

Step 1: Simplify complex number z

Step 2: Simplify Bracket

Step 3: Find beta

Step 4: Distance from Line

Chapter Mix

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If z$z$ is a complex number, then the number of common roots of the equation z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0$z^{1985} + z^{100} + 1 = 0$ and z³ + 2z² + 2z + 1 = 0$z^3 + 2z^2 + 2z + 1 = 0$ , is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Check z = -1

Step 2: Check z = ω and ω²

Step 3: Conclusion

Pattern Recognition

Chapter Mix

Reference Study Guides

More Complex Numbers Previous-Year Questions — Page 9

Solution

Core Logic

Step 1: Simplify complex number z

Step 2: Simplify Bracket

Step 3: Find beta

Step 4: Distance from Line

Chapter Mix

More Complex Numbers Questions — jee_main_2024_30_january_evening

Select Your Target Exam

Full Mock Test Hub

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Exam Analysis Tools

Chapter Weightage & Trends

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JEE Main Exam Papers

Chapter Weightage Board

Formula Recognition

Exam Questions

Select an Exam

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If $z$ $z$ is a complex number, then the number of common roots of the equation $z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0$ $z^{1985} + z^{100} + 1 = 0$ and $z³ + 2z² + 2z + 1 = 0$ $z^3 + 2z^2 + 2z + 1 = 0$ , is equal to: