If a, b, c are the roots of cubic equation x3-2 x+3=0 and f(x)=ax+bx+cx, then the value of ( f (10)+3 f (7)/ f (8)) is equal to

Question

If a, b, c are the roots of cubic equation x3-2 x+3=0 and f(x)=ax+bx+cx, then the value of ( f (10)+3 f (7)/ f (8)) is equal to (A) 1 (B) 2 (C) 3 (D) 6

Tardigrade Admin · Accepted Answer

Θ f (10) + f (7)= a 10+ b 10+ c 10+3 a 7+3 b 7+3 c 7 = a 7( a 3+3)+ b 7( b 3+3)+ c 7( c 3+3) = a 7 ⋅ 2 a + b 7 ⋅ 2 b + c 7 ⋅ 2 c =2( a 8+ b 8+ c 8) ⇒ ( f (10)+3 f (7)/ f (8))=2

Q. If $a, b, c$ are the roots of cubic equation $x^{3} - 2 x + 3 = 0$ and $f (x) = a^{x} + b^{x} + c^{x}$ , then the value of $\frac{f ( 10 ) + 3 f ( 7 )}{f ( 8 )}$ is equal to

1

2

3

6

$Θ f (10) + f (7) = a^{10} + b^{10} + c^{10} + 3 a^{7} + 3 b^{7} + 3 c^{7}$
$= a^{7} (a^{3} + 3) + b^{7} (b^{3} + 3) + c^{7} (c^{3} + 3)$
$= a^{7} \cdot 2 a + b^{7} \cdot 2 b + c^{7} \cdot 2 c$
$= 2 (a^{8} + b^{8} + c^{8})$
$\Rightarrow \frac{f ( 10 ) + 3 f ( 7 )}{f ( 8 )} = 2$

Q. If a,b,c are the roots of cubic equation x3−2x+3=0 and f(x)=ax+bx+cx, then the value of f(8)f(10)+3f(7)​ is equal to

1

2

3

6

Θf(10)+f(7)=a10+b10+c10+3a7+3b7+3c7 =a7(a3+3)+b7(b3+3)+c7(c3+3) =a7⋅2a+b7⋅2b+c7⋅2c =2(a8+b8+c8) ⇒f(8)f(10)+3f(7)​=2

Q. If $a, b, c$ are the roots of cubic equation $x^{3} - 2 x + 3 = 0$ and $f (x) = a^{x} + b^{x} + c^{x}$ , then the value of $\frac{f ( 10 ) + 3 f ( 7 )}{f ( 8 )}$ is equal to

$Θ f (10) + f (7) = a^{10} + b^{10} + c^{10} + 3 a^{7} + 3 b^{7} + 3 c^{7}$
$= a^{7} (a^{3} + 3) + b^{7} (b^{3} + 3) + c^{7} (c^{3} + 3)$
$= a^{7} \cdot 2 a + b^{7} \cdot 2 b + c^{7} \cdot 2 c$
$= 2 (a^{8} + b^{8} + c^{8})$
$\Rightarrow \frac{f ( 10 ) + 3 f ( 7 )}{f ( 8 )} = 2$