QUADRATIC EQUATION JEE ADVANCED PYQ (1989-2025)

Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x – 2020 and suppose c,d denote the distinct complex roots of the quadratic polynomial x2 – 20x + 2020. Then the value of ac(a - c) + ad(a - d) + bc(b - c) + bd(b - d) is (A) 0 (B) 8000 (C) 8080 (D) 16000

Let p, q be integers and let α, β be the roots of the equation, x^2-x-1=0 where α≠β. For n=0,1,2, …, let a_n=p α^n+q β^n. Fact: If a and b are rational numbers and a+b √(5)=0, then a=0=b. If a_4=28, then p+2q= (1) 21 (2) 14 (3) 7 (4) 12 (JEE Advanced 2017)

Let p, q be integers and let α, β be the roots of the equation, x^2-x-1=0 where α≠β. For n=0,1,2, …, let a_n=p α^n+q β^n. Fact: If a and b are rational numbers and a+b √(5)=0, then a=0=b. a12 = (A) a11 – a10 (B) a11 + a10 (C) 2a11 + a10 (D) a11 + 2a10 2017 IIT JEE ADVANCED PYQ

Let − π/6 θ π/12. Suppose α1 and β1 are the roots of the equation x2 − 2xsecθ + 1 = 0 and α2 and β2 are the roots of the equation x2 + 2xtanθ − 1 = 0. If α1 β1 and α2 β2, then α1 + β2 equals (a) 2(secθ − tanθ ) (b) 2secθ (c) −2tanθ (D) 0 2016 IIT JEE ADVANCED PYQ

Let S be the set of all non-zero real numbers α such that the quadratic equation αx^2-x+α=0 has two distinct real roots x_1 and x_2 satisfying the inequality |x_1-x_2|1. Which of the following intervals is(are) a subset(s) of S?(A) (-1/2,-1/√(5)) (B) (-1/√(5), 0) (C) (0, 1/√(5)) (D) (1/√(5), 1/2) [JEE (Advanced) 2015]

Let a, b, c be three non-zero real numbers such that the equation √(3)acosx+2 bsinx=c, x∈[-π/2, π/2], has two distinct real roots α and β with α+β=π/3. Then, the value of b/a is… [IIT JEE ADVANCED 2018]

The product of all positive real values of x satisfying the equation x ^( 16 ( l o g 5 x ) 3 − 68 l o g 5 x ) = 5^(-16) is ___ [IIT JEE ADVANCED 2022]

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p[p(x)] = 0 has (A) only purely imaginary roots (B) all real roots (C) two real and two purely imaginary roots (D) neither real nor purely imaginary roots 2014 IIT JEE ADVANCED PYQ

Let α(a) and β(a) be the roots of the equation (3√1 + a -1)x2 + (√1 + a - 1)x + (6√1 + a - 1) = 0 where a − 1. Then lim α a → 0+ α(a) and lim a → 0+ β(a) are (a) - 5/2 and 1 (b) -1/2 and -1 (c) -7/2 and 2 (d) -9/2 and 3 2012 IIT JEE ADVANCED PYQ