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Mathematics · Quadratic Equations

JEE Advanced 2020 — Paper 1 — Question 29

Suppose aa, bb denote the distinct real roots of the quadratic polynomial x2+20x−2020x^{2}+20 x-2020 and suppose c,dc, d

denote the distinct complex roots of the quadratic polynomial x2−20x+2020x^{2}-20 x+2020. Then the value of

ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)
  1. Option A:

    0

  2. Option B:

    8000

  3. Option C:

    8080

  4. Option D:

    16000

    Correct

Answer: D

Step-by-step solution

Given a+b=−20,ab=−2020,c+d=20\mathrm{a}+\mathrm{b}=-20, \mathrm{ab}=-2020, \mathrm{c}+\mathrm{d}=20, cd=2020−09−28\mathrm{cd}=2020-09-28

a2c−ac2+a2d−ad2+b2c−b2c−bc2+b2d−bd2a^{2} c-a c^{2}+a^{2} d-a d^{2}+b^{2} c-b^{2} c-b c^{2}+b^{2} d-b d^{2}

=(a2+b2)(c+d)−(a+b)(c2+d2)=\left(a^{2}+b^{2}\right)(c+d)-(a+b)\left(c^{2}+d^{2}\right)

=((a+b)2−2ab)(c+d)−(a+b)((c+d)2−2 cd)=\left((\mathrm{a}+\mathrm{b})^{2}-2 \mathrm{ab}\right)(\mathrm{c}+\mathrm{d})-(\mathrm{a}+\mathrm{b})\left((\mathrm{c}+\mathrm{d})^{2}-2 \mathrm{~cd}\right)

=(400−2×2020)(20)−(−20)(400−2×2020)=(400-2 \times 2020)(20)-(-20)(400-2 \times 2020)

=20×800=16000=20 \times 800=16000

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2020
Paper
Paper 1
Subject
Mathematics
Chapter
Quadratic Equations
Topic
Nature of Roots of Quadratic Equation
Suppose a , b denote the distinct real roots of the quadratic… | JEE Advanced 2020 PYQ with Solution · DhiX AI