Let y=f(x) be a quadratic polynomial with leading co-efficient 1 . Let α and β be the roots of the equation f(x)=0 and γ and δ be the roots of the equation f(x)=8 x-32. If α, β, γ and δ (in order) are in A.P. Then find the value of f (4).

Question

Let y=f(x) be a quadratic polynomial with leading co-efficient 1 . Let α and β be the roots of the equation f(x)=0 and &gamma; and δ be the roots of the equation f(x)=8 x-32. If α, β, &gamma; and δ (in order) are in A.P. Then find the value of f (4). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let f(x)=x2-P x+q, α=a, β=a+d a + a + d = P a +2 d + a +3 d = P +δ ⇒ d =2 a ( a +2)= q ( a +4)( a +6)= q +32 ⇒ 8 a +24=32 ⇒ a =1 ⇒ q =3 text and P =4 f ( x )= x 2-4 x +3 f (4)=3

Q. Let $y = f (x)$ be a quadratic polynomial with leading co-efficient 1 . Let $α$ and $β$ be the roots of the equation $f (x) = 0$ and $γ$ and $δ$ be the roots of the equation $f (x) = 8 x - 32$ . If $α, β, γ$ and $δ$ (in order) are in A.P. Then find the value of $f (4)$ .

Let $f (x) = x^{2} - P x + q, α = a, β = a + d$
$a + a + d = P$
$a + 2 d + a + 3 d = P + δ$
$\Rightarrow d = 2$
$a (a + 2) = q$
$(a + 4) (a + 6) = q + 32$
$\Rightarrow 8 a + 24 = 32 \Rightarrow a = 1$
$\Rightarrow q = 3 and P = 4$
$f (x) = x^{2} - 4 x + 3$
$f (4) = 3$

Q. Let y=f(x) be a quadratic polynomial with leading co-efficient 1 . Let α and β be the roots of the equation f(x)=0 and γ and δ be the roots of the equation f(x)=8x−32. If α,β,γ and δ (in order) are in A.P. Then find the value of f(4).

Let f(x)=x2−Px+q,α=a,β=a+d a+a+d=P a+2d+a+3d=P+δ ⇒d=2 a(a+2)=q (a+4)(a+6)=q+32 ⇒8a+24=32⇒a=1 ⇒q=3 and P=4 f(x)=x2−4x+3 f(4)=3

Q. Let $y = f (x)$ be a quadratic polynomial with leading co-efficient 1 . Let $α$ and $β$ be the roots of the equation $f (x) = 0$ and $γ$ and $δ$ be the roots of the equation $f (x) = 8 x - 32$ . If $α, β, γ$ and $δ$ (in order) are in A.P. Then find the value of $f (4)$ .

Let $f (x) = x^{2} - P x + q, α = a, β = a + d$
$a + a + d = P$
$a + 2 d + a + 3 d = P + δ$
$\Rightarrow d = 2$
$a (a + 2) = q$
$(a + 4) (a + 6) = q + 32$
$\Rightarrow 8 a + 24 = 32 \Rightarrow a = 1$
$\Rightarrow q = 3 and P = 4$
$f (x) = x^{2} - 4 x + 3$
$f (4) = 3$