- Home
- Standard 12
- Mathematics
माना $f(x)$ एक द्विघाती बहुपद है जिसका मुख्य-गुणांक 1 है तथा $f (0)= p , p \neq 0$ और $f (1)=\frac{1}{3}$ हैं। यदि समीकरणों $f ( x )=0$ तथा $fofofof (x)=0$ का एक उभयनिष्ठ वास्तविक मूल है, तो $f(-3)$ बराबर है
$25$
$24$
$23$
$22$
Solution
Let $f(x)=(x-\alpha)(x-\beta)$
It is given that $f(0)=p \Rightarrow \alpha \beta=p$
and $f(1)=\frac{1}{3} \Rightarrow(1-\alpha)(1-\beta)=\frac{1}{3}$
Now, let us assume that $\alpha$ is the common root of $f(x)=0$ and $f \circ f \circ f o f(x)=0$
$fofofof(x)=0$
$fofof(0) =0$
$f o f(p)=0$
So, $f(p)$ is either $\alpha$ or $\beta$.
$(p-\alpha)(p-\beta)=\alpha$
$(\alpha \beta-\alpha)(\alpha \beta-\beta)=\alpha \Rightarrow(\beta-1)(\alpha-1) \beta=1$
$(\because \alpha \neq 0)$
So, $\beta=3$
$(1-\alpha)(1-3)=\frac{1}{3}$
$\alpha=\frac{7}{6}$
$f(x)=\left(x-\frac{7}{6}\right)(x-3)$
$f(-3)=\left(-3-\frac{7}{6}\right)(3-3)=25$
