Complex Numbers Basics: JEE Questions & Solutions

If x²+x+1=0, then the value of (x + 1/x)⁴ + (x² + 1/x²)⁴ + (x³ + 1/x³)⁴ + … + (x²⁵ + 1/x²⁵)⁴ is:

Let α = (-1 + i√3)/2 and β = (-1 - i√3)/2, where i = √(-1). If (7 - 7α + 9β)²⁰ + (9 + 7α - 7β)²⁰ + (-7 + 9α + 7β)²⁰ + (14 + 7α + 7β)²⁰ =…

Let S = { z ∈ ℂ: 4z² + z = 0 }. Then Σ_(z ∈ S) |z|² is equal to

If z = √3/2 + i/2, then (z²⁰¹ - i)⁸ is equal to

Let z=(1+i)(1+2i)(1+3i)…(1+ni), where i=√(-1). If |z|²=44200, then n is equal to

Given below are two statements: Statement I: 25¹³+20¹³+8¹³+3¹³ is divisible by 7 Statement II: The integral part of (7+4√3)²⁵ is an odd…

Let z₁, z₂, z₃ be three complex numbers on the circle |z| = 1 with arg(z₁) = -π/4, arg(z₂) = 0 and arg(z₃) = π/4. If |z₁ z₂ + z₂ z₃ + z₃…

If α and β are the roots of the equation 2z² - 3z - 2i = 0, where i = √(-1), then 16 · Re ( (α¹⁹ + β¹⁹ + α¹¹ + β¹¹)/(α¹⁵ + β¹⁵)) · Im (…

The number of real solution(s) of the equation x² + 3x + 2 = min ( |x - 3|, |x + 2|) is:

Let z ∈ ℂ be such that (z²+3i)/(z-2+i) = 2+3i. Then the sum of all possible values of z² is

Let the product of ω₁ = (8 + i) sin θ + (7 + 4i) cos θ and ω₂ = (1 + 8i) sin θ + (4 + 7i) cos θ be α + iβ, where i = √(-1). Let p and q be…

The remainder when ( (64)⁶⁴)⁶⁴ is divided by 7 is equal to:

The sum of the squares of the roots of |x - 2|² + |x - 2| - 2 = 0 and the squares of the roots of x² - 2|x - 3| - 5 = 0, is:

If α satisfies x² + x + 1 = 0 and (1 + α)⁷ = A + Bα + Cα², where A, B, C ≥ 0, then 5(3A - 2B - C) is equal to:

Let the complex numbers α and 1/α lie on the circles |z - z₀| = 2 and |z - z₀| = 4 respectively, where z₀ = 1 + i. Then, the value of…

If z = 1/2 - 2i such that |z + 1| = α z + β(1 + i), where i is the imaginary unit and α, β are real numbers, then α + β is equal to:

Let α, β be the roots of the equation x² - √6x + 3 = 0 such that Im(α) > Im(β). Let a, b be integers not divisible by 3 and n be a natural…

If z = x + iy, xy ≠ 0, satisfies the equation z² + iz = 0, then |z²| is equal to:

If z is a complex number, then the number of common roots of the equations z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0 and z³ + 2z² + 2z + 1 = 0 is equal to:

The number of real solutions of the equation x(x² + 3|x| + 5|x-1| + 6|x-2|) = 0 is:

Let z₁ and z₂ be two complex numbers such that z₁ + z₂ = 5 and z₁³ + z₂³ = 20 + 15i. Then |z₁⁴ + z₂⁴| equals:

The number of real solutions of the equation 3(x² + 1/x²) - 2(x + 1/x) + 5 = 0 is:

The value of ( (1 + sin 2π/9 + i cos 2π/9)/(1 + sin 2π/9 - i cos 2π/9))³ is:

Let S = {α: log₂ (9^(2α-4) + 13) - log₂ (5/2 · 3^(2α-4) + 1) = 2 }. Then the maximum value of β for which the equation x² - 2(Σ_(α ∈ S) α)²…

For two non-zero complex numbers z₁ and z₂, if Re(z₁ z₂) = 0 and Re(z₁ + z₂) = 0, then which of the following are possible? (A) Im(z₁) > 0…

Let x = ( 8√3+13)¹³ and y = ( 7√2+9)⁹. If [t] denotes the greatest integer ≤ t, then

50th root of a number x is 12 and 50th root of another number y is 18. Then the remainder obtained on dividing x + y by 25 is.....

Let a ≠ b be two non-zero real numbers. Then the number of elements in the set X = {z ∈ ℂ: Re(a z² + b z) = a and Re(b z² + a z) = b} is…

For α, β, z ∈ ℂ and λ > 1, if √(λ - 1) is the radius of the circle |z - α|² + |z - β|² = 2λ, then |α - β| is equal to:

If for z = α + iβ, |z + 2| = z + 4(1 + i), then α + β and αβ are the roots of the equation:

Let the complex number z = x + iy be such that (2z - 3i)/(2z + i) is purely imaginary. If x + y² = 0, then y⁴ + y² - y is equal to:

Let S = { z = x + iy: (2z - 3i)/(4z + 2i) is a real number } Then which of the following is NOT correct?

For a ∈ ℂ, let: A = { z ∈ ℂ: Re(a + z) > Im(a + z) }, and B = { z ∈ ℂ: Re(a + z) < Im(a + z) }. Among the two statements: (S1): If Re(a),…

Let S = {z ∈ ℂ - {i, 2i}: (z² + 8iz - 15)/(z² - 3iz - 2) ∈ ℝ }. If α - 13/11i ∈ S, α ∈ ℝ - {0}, then 242α² is equal to.

Let ω = zz + k₁ z + k₂ i z + λ(1+i), where k₁, k₂, λ ∈ ℝ. Let Re(ω) = 0 be the circle C of radius 1 in the first quadrant touching the line…

Let S = { z ∈ ℂ: z = i(z² + Re(z)) }. Then Σ_(z ∈ S) |z|² is equal to: