Question

If $f\left(0\right)=1,f\left(1\right)=5,f\left(2\right)=11$, then the equation of polynomial of degree two is

• A. $x^2+1=0$
• B. $x^2+3x+1=0$
• C. $x^2-2x+1=0$
• D. None of these

Answer 1

The correct option is B $x^2+3x+1=0$

Given, $f\left(0\right)=1,f\left(1\right)=5,f\left(2\right)=11$
Let the second degree equation be
$f\left(x\right)=a{x}^2+bx+c$
$\therefore f\left(0\right)=0+0+c$
$\Rightarrow c=1$ ..... (i)
$f\left(1\right)=a+b+c$
$\Rightarrow 5=a+b+1$ [from Eq. (i)]
$\Rightarrow a+b=4$ ..... (ii)
and $f\left(2\right)=4a+2b+1$
$\Rightarrow 11=4a+2b+1$
$\Rightarrow 2ab+b=5$ ...... (iii)
On solving Eqs. (ii) and (iii), we get
$a=1,b=3$
$\therefore$ The required equation is
$f\left(x\right)=x^2+3x+1$