Question

Let $f\left(x\right)=(\begin{array}{c}{x}^2-1,\mathrm{if}0<x<2\\ 2x+3,\mathrm{if}2\le x<3\end{array}$, a quadratic equation whose roots are $\underset{x\to 2^{-}}{\lim }f\left(x\right)\mathrm{and}\underset{x\to 2^{+}}{\lim }f\left(x\right)$ is

• A. $x^2-6x+9=0$
• B. $x^2-7x+8=0$
• C. $x^2-14x+49=0$
• D. $x^2-10x+21=0$

Answer 1

The correct option is D $x^2-10x+21=0$

$\mathrm{Let}\mathrm{us}\mathrm{first}\mathrm{find}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{required}\mathrm{quadratic}\mathrm{equation}i.e.\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }{x}^2-1=4-1=3\underset{x\to 2^{+}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }2x+3=2\left(2\right)+3=7\mathrm{Now},\mathrm{let}\mathrm{us}\mathrm{find}\mathrm{the}\mathrm{qudratic}\mathrm{equation}\mathrm{with}\mathrm{roots}\mathrm{as}x=3\mathrm{and}x=7(x-3)(x-7)=0\Rightarrow x^2-10x+21=0$