If f(x)=|x-3 2 x2-18 3 x3-81 x-5 2 x2-50 4 x3-500 1 2 3| then f(1) f(3)+f(3) f(5)+f(5) f(1) is equal to

Question

If f(x)=|x-3 2 x2-18 3 x3-81 x-5 2 x2-50 4 x3-500 1 2 3| then f(1) f(3)+f(3) f(5)+f(5) f(1) is equal to (A) f(1)+f(5) (B) f(3)+f(5) (C) f(1)+f(3) (D) f(1)+f(3)+f(5)

Tardigrade Admin · Accepted Answer

Clearly f(3)=|0 0 0 -2 -32 -392 1 2 3|=0 and f(5)=|2 32 294 0 0 0 1 2 3|=0, where f(1) ≠ 0 ∴ f (1) f (3)+ f (3) f (5)+ f (5) f (1)=0= f (3)+ f (5)

Q. If $f (x) = ∣ ∣ x - 3 x - 5 1 2 x^{2} - 18 2 x^{2} - 50 2 3 x^{3} - 81 4 x^{3} - 500 3 ∣ ∣$ then $f (1) f (3) + f (3) f (5) + f (5) f (1)$ is equal to

$f (1) + f (5)$

$f (3) + f (5)$

$f (1) + f (3)$

$f (1) + f (3) + f (5)$

Clearly $f (3) = ∣ ∣ 0 - 2 1 0 - 32 2 0 - 392 3 ∣ ∣ = 0$ and $f (5) = ∣ ∣ 201320229403 ∣ ∣ = 0$ , where $f (1) \neq = 0$
$∴ f (1) f (3) + f (3) f (5) + f (5) f (1) = 0 = f (3) + f (5)$

Q. If f(x)=∣∣​x−3x−51​2x2−182x2−502​3x3−814x3−5003​∣∣​ then f(1)f(3)+f(3)f(5)+f(5)f(1) is equal to

f(1)+f(5)

f(3)+f(5)

f(1)+f(3)

f(1)+f(3)+f(5)