Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Quadratic Equation and Inequalities question

2020 · Shift 1 · Q19
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Advanced
  3. /Mathematics
  4. /Quadratic Equation and Inequalities
  5. /2020 · Shift 1 · Q19

Quadratic Equation and Inequalities question

2020 · Shift 1 · Q19

JEE AdvancedMathematicsQuadratic Equation and InequalitiesMCQ+3 / −1
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
  1. A
    0
  2. B
    8000
  3. C
    8080
  4. D
    16000
View written solutionFree

Correct answer: D

  1. Let P(x)=x2+20x−2020P(x)=x^2+20x-2020P(x)=x2+20x−2020 and its distinct real roots be a,ba,ba,b.

Using Vieta's formulas:

\qquad ab=-2020.$$ 2. Let $$Q(x)=x^2-20x+2020$$ and its distinct complex roots be $c,d$. Again by Vieta's formulas: $$c+d=20, \qquad cd=2020.$$ 3. We need to evaluate $$S=ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d).$$ Group terms with $a$ and $b$: $$S=a\big(c(a-c)+d(a-d)\big)+b\big(c(b-c)+d(b-d)\big).$$ Now simplify inside: $$c(a-c)+d(a-d)=a(c+d)-(c^2+d^2),$$ $$c(b-c)+d(b-d)=b(c+d)-(c^2+d^2).$$ So, $$S=a\big(a(c+d)-(c^2+d^2)\big)+b\big(b(c+d)-(c^2+d^2)\big).$$ Hence, $$S=(a^2+b^2)(c+d)-(a+b)(c^2+d^2).$$ 4. Compute $a^2+b^2$ and $c^2+d^2$. For $a,b$: $$a^2+b^2=(a+b)^2-2ab=(-20)^2-2(-2020)=400+4040=4440.$$ For $c,d$: $$c^2+d^2=(c+d)^2-2cd=20^2-2(2020)=400-4040=-3640.$$ 5. Substitute: $$S=4440(20)-(-20)(-3640).$$ Now, $$4440\cdot 20=88800,$$ $$(-20)(-3640)=72800.$$ Therefore, $$S=88800-72800=16000.$$ 6. So the correct option is $$\boxed{16000}$$ which is option $\boxed{D}$. 7. Comparison with stored answer: Stored correct answer is $D$, which matches our result.
PreviousNext

More from Quadratic Equation and Inequalities

  • Let α and β be the roots of x2−x−1=0, with α>β. For all positive integers n, define an​=α−βαn−βn​,n≥1b1​=1andbn​=an−1​+an+1​,n≥2…2019 · Multiple correct
  • Let a, b, c three non-zero real numbers such that the equation 3​acosx+2bsinx=c,x∈[−2π​,2π​], has two distinct real roots α and β with $\alpha + \beta = {\pi \over…2018 · Numerical
  • Let p, q be integers and let α, β be the roots of the equation, x2 − x − 1 = 0 where αeβ. For n = 0, 1, 2, ........, let an = p α n + q β n. FACT : If a and b are rational numbers and a + b 5​…2017 · MCQ
  • Let p, q be integers and let α, β be the roots of the equation, x2 − x − 1 = 0 where αeβ. For n = 0, 1, 2, ........, let an = p α n + q β n. FACT : If a and b are rational numbers and a + b 5​…2017 · MCQ
  • Let −6π​<θ<−12π​. 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.Ifα1​>β1​…2016 · MCQ
  • Let S be the set of all non-zero real numbers α such that the quadratic equation αx2−x+α=0 has two distinct real roots x1​ and x2​ satisfying the inequality ∣x1​−x2​∣<1.…2015 · Multiple correct
  • The quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x))=0 has2014 · MCQ
  • If 3x=4x−1, then x=2013 · Multiple correct