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JEE Advanced 2020 · Paper 1, Mathematics question 1

Suppose a,ba, b denote the distinct real roots of the quadratic polynomial x2+20x−2020x^2 + 20x - 2020 and suppose c,dc, d denote the distinct complex roots of the quadratic polynomial x2−20x+2020x^2 - 20x + 2020. Then the value of

ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d)

is

Choose one option
Answer and worked solution

Official answer: D

Regroup by aa and bb.
∑=a2(c+d)−a(c2+d2)+b2(c+d)−b(c2+d2)=(c+d)(a2+b2)−(a+b)(c2+d2).\sum = a^2(c+d)-a(c^2+d^2)+b^2(c+d)-b(c^2+d^2)=(c+d)(a^2+b^2)-(a+b)(c^2+d^2).

Roots.

  • a+b=−20a+b=-20, ab=−2020ab=-2020, so a2+b2=400+4040=4440a^2+b^2=400+4040=4440.
  • c+d=20c+d=20, cd=2020cd=2020, so c2+d2=400−4040=−3640c^2+d^2=400-4040=-3640.

Value.
20(4440)−(−20)(−3640)=88 800−72 800=16 000.20(4440)-(-20)(-3640)=88\,800-72\,800=16\,000.
(D)

Topics:Roots and coefficientsAlgebra of complex numbers
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