JEE Advanced Complex Numbers and Quadratic Equations — previous year questions

45 questions from 2026, 2025, 2024, 2023, 2022, 2021, 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007.

• Match each entry in List-I to the correct entry in List-II and choose the correct option. | List-I | List-II…2026JEE Advanced
• root of unity for which 0 < arg(ω) < π. Let α = arg(∑ₙ₌₁²⁰²⁵ (-ω)ⁿ). Then the value of 3α/π is ______.2025JEE Advanced
• Let R denote the set of all real numbers. Let z₁ = 1 + 2i and z₂ = 3i be two complex numbers, where i =…2025JEE Advanced
• Let S = {a + b√2 : a, b ∈ Z}, T₁ = {(-1+√2)ⁿ : n ∈ N} and T₂ = {(1+√2)ⁿ : n ∈ N}. Then which of the following…2024JEE Advanced
• Let R² denote R × R. Let S = {(a, b, c) : a, b, c ∈ R and ax² + 2bxy + cy² > 0 for all (x, y) ∈ R² - {(0,…2024JEE Advanced
• Let f(x) = x⁴ + ax³ + bx² + c be a polynomial with real coefficients such that f(1) = -9. Suppose that i√3 is…2024JEE Advanced
• Let A = {(1967 + 1686 i sinθ)/(7 - 3 i cosθ) : θ ∈ R}. If A contains exactly one positive integer n, then the…2023JEE Advanced
• Let z be a complex number with non-zero imaginary part. If (2+3z+4z²)/(2-3z+4z²) is a real number, then the…2022JEE Advanced
• Let z̄ denote the complex conjugate of a complex number z and let i = √(-1). In the set of complex numbers,…2022JEE Advanced
• Let z̄ denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both…2022JEE Advanced
• The product of all positive real values of x satisfying the equation x^((16(log₅ x)³ - 68 log₅ x)) = 5⁻¹⁶ is…2022JEE Advanced
• Let θ₁, θ₂, …, θ₁₀ be positive valued angles (in radian) such that θ₁ + θ₂ + … + θ₁₀ = 2π. Define the complex…2021JEE Advanced
• For x ∈ R, then number of real roots of the equation 3x² - 4|x² - 1| + x - 1 = 0 is ______.2021JEE Advanced
• Let S be the set of all complex numbers z satisfying |z²+z+1|=1. Then which of the following statements…2020JEE Advanced
• Let S be the set of all complex numbers z satisfying z⁴-|z|⁴=4 i z², where i=√(-1). Then the minimum possible…2020JEE Advanced
• Suppose a, b denote the distinct real roots of the quadratic polynomial x²+20x-2020 and suppose c, d denote…2020JEE Advanced
• Let S be the set of all complex numbers z satisfying |z - 2 + i| ≥ √5. If the complex number z₀ is such that…2019JEE Advanced
• set {|a + bω + cω²|² : a,b,c distinct non-zero integers} equals underlinehspace2cm.2019JEE Advanced
• Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy (x, y ∈ R, i = √(-1)) of the…2018JEE Advanced
• For a non-zero complex number z, let arg(z) denote the principal argument with -π<arg(z)≤π. Then, which of…2018JEE Advanced
• Let a, b, x and y be real numbers such that a - b = 1 and y ≠ 0. If the complex number z = x + iy satisfies…2017JEE Advanced
• Let a, b ∈ R and a² + b² ≠ 0. Suppose S = { z ∈ C : Z = 1/(a+ibt), ∈ R⁺, t ≠ 0 }, where i = √(-1). If z = x +…2016JEE Advanced
• Let -π/6<θ<-π/12. Suppose θ₁ and B₁ are the roots of the equation x²-2xsecθ+1=0 and α₂ and β₂ are the roots…2016JEE Advanced
• For any integer k, let αₖ = cos(kπ/7) + i sin(kπ/7), where i = √(-1). The value of the expression (∑ₖ₌₁¹²…2015JEE Advanced
• Let S be the set of all non-zero real numbers α such that the quadratic equation α x² - x + α = 0 has two…2015JEE Advanced
• The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x))…2014JEE Advanced
• If a ∈ R and the equation -3(x-[x])²+2(x-[x])+a²=0 (where [x] denotes the greatest integer ≤ x) has no…2014JEE Advanced
• Let complex numbers α and 1/α lie on circles (x - x₀)² + (y - y₀)² = r² and (x - x₀)² + (y - y₀)² = 4r²…2013JEE Advanced
• Let w = (√3+i)/2 and P = {wⁿ : n = 1, 2, 3, …}. Further H₁ = {z ∈ C : Re z > 1/2} and H₂ = {z ∈ C : Re z <…2013JEE Advanced
• If 3ˣ = 4ˣ⁻¹, then x =2013JEE Advanced
• Let z be a complex number such that the imaginary part of z is non-zero and a = z² + z + 1 is real. Then a…2012JEE Advanced
• If z is any complex number satisfying |z - 3 - 2i| ≤ 2, then the minimum value of |2z - 6 + 5i| is2011JEE Advanced
• Let α and β be the roots of x² - 6x - 2 = 0, with α > β. If aₙ = αⁿ - βⁿ for n ≥ 1, then the value of (a₁₀ -…2011JEE Advanced
• Let (x₀, y₀) be the solution of the following equations (2x)^(ln 2) = (3y)^(ln 3) ; 3^(ln x) = 2^(ln y). Then…2011JEE Advanced
• A value of b for which the equations x²+bx-1=0 x²+x+b=0 have one root in common is2011JEE Advanced
• Let z₁ and z₂ be two distinct complex numbers and let z = (1 - t) z₁ + t z₂ for some real number t with 0 < t…2010JEE Advanced
• Let p and q be real numbers such that p ≠ 0, p³ ≠ q and p³ ≠ -q. If α and β are nonzero complex numbers…2010JEE Advanced
• Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices…2009JEE Advanced
• Let z = cosθ + isinθ. Then the value of ∑ₘ₌₁¹⁵ Im(z²ᵐ⁻¹) at θ = 2^∘ is2009JEE Advanced
• The smallest value of k, for which both the roots of the equation x² - 8kx + 16(k² - k + 1) = 0 are real,…2009JEE Advanced
• A particle P starts from the point z₀ = 1 + 2i, where i = √(-1). It moves horizontally away from origin by 5…2008JEE Advanced
• Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x² + 2px + q = 0 and α, 1/β are…2008JEE Advanced
• If |z| = 1 and z ≠ ± 1, then all the values of z/(1-z²) lie on2007JEE Advanced
• A man walks a distance of 3 units from the origin towards the north-east (N 45^∘ E) direction. From there, he…2007JEE Advanced
• Let α, β be the roots of the equation x² - px + r = 0 and α/2, 2β be the roots of the equation x² - qx + r =…2007JEE Advanced

Practise these with video solutions