JEE Advance - Mathematics (2020 - Paper 1 Offline - No. 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
ac(a $$-$$ c) + ad(a $$-$$ d) + bc(b $$-$$ c) + bd(b $$-$$ d) is
0
8000
8080
16000
Explanation
Given quadratic polynomials, x2 + 20x $$-$$ 2020 and x2 $$-$$ 20x + 2020 having a, b distinct real and c, d distinct complex roots respectively.
So, a + b = $$-$$20, ab = $$-$$2020
and c + d = 20, cd = 2020
Now, ac(a $$-$$c) + ad(a $$-$$ d) + bc(b $$-$$c) + bd(b $$-$$ d)
= a2(c + d) $$-$$ a(c2 + d2) + b2(c + d) $$-$$ b(c2 + d2)
= (c + d) (a2 + b2) $$-$$ (c2 + d2) (a + b)
= (c + d)[(a + b)2 $$-$$ 2ab] $$-$$ (a + b)[(c + d)2 $$-$$ 2cd]
= 20[(20)2 + 4040] + 20[(20)2 $$-$$ 4040]
= 2 $$ \times $$ 20 $$ \times $$ (20)2 = 40 $$ \times $$ 400 = 16000
So, a + b = $$-$$20, ab = $$-$$2020
and c + d = 20, cd = 2020
Now, ac(a $$-$$c) + ad(a $$-$$ d) + bc(b $$-$$c) + bd(b $$-$$ d)
= a2(c + d) $$-$$ a(c2 + d2) + b2(c + d) $$-$$ b(c2 + d2)
= (c + d) (a2 + b2) $$-$$ (c2 + d2) (a + b)
= (c + d)[(a + b)2 $$-$$ 2ab] $$-$$ (a + b)[(c + d)2 $$-$$ 2cd]
= 20[(20)2 + 4040] + 20[(20)2 $$-$$ 4040]
= 2 $$ \times $$ 20 $$ \times $$ (20)2 = 40 $$ \times $$ 400 = 16000
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