6
Question
Suppose a,b denotes the distinct real roots of the quadratic polynomial x2+20x−2020 and suppose c,d denotes 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
Suppose a,b denotes the distinct real roots of the quadratic polynomial x2+20x−2020 and suppose c,d denotes 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
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Solution
The correct option is D 16000
x2+20x−2020=0 has two roots a,b∈R.
a+b=−20 & a⋅b=−2020
& x2−20x+2020=0 has two roots c,d∈ complex.
c+d=20 & c⋅d=2020
now
=ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)
=a2c−ac2+a2d−ad2+b2c−bc2+b2d−bd2
=a2(c+d)+b2(c+d)−c2(a+b)−d2(a+b)
=(a2+b2)(c+d)−(a+b)(c2+d2)
=[(a+b)2−2ab](c+d)−[(c+d)2−2cd](a+b)
Put value a,b,c & d then,
=[(−20)2−2(−2020)](20)−[(20)2−2(2020)](−20)
=[(400+4040)](20)−(−20)[(20)2−4040]
=20[4440−3640]
=20[800]=16000
x2+20x−2020=0 has two roots a,b∈R.
a+b=−20 & a⋅b=−2020
& x2−20x+2020=0 has two roots c,d∈ complex.
c+d=20 & c⋅d=2020
now
=ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)
=a2c−ac2+a2d−ad2+b2c−bc2+b2d−bd2
=a2(c+d)+b2(c+d)−c2(a+b)−d2(a+b)
=(a2+b2)(c+d)−(a+b)(c2+d2)
=[(a+b)2−2ab](c+d)−[(c+d)2−2cd](a+b)
Put value a,b,c & d then,
=[(−20)2−2(−2020)](20)−[(20)2−2(2020)](−20)
=[(400+4040)](20)−(−20)[(20)2−4040]
=20[4440−3640]
=20[800]=16000
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