Question

# Let $$f\left ( x \right )=\begin{cases}x^{2}-1 ;0< x< 2 \\ 2x+3; 2\leq x< 3\end{cases}$$The quadratic equation whose roots are $$\displaystyle \lim_{x\to 2^-}f\left ( x \right )$$ and $$\displaystyle \lim_{x\to 2^+}f\left ( x \right )$$ is

A
x26x+9=0
B
x210x+21=0
C
x214x+49=0
D
none of these

Solution

## The correct option is C $$x^{2}-10x+21=0$$$$\lim _{ x\to 2-0 } f\left( x \right) =\lim _{ x\to 2-0 } x^{ 2 }-1=3\\ \lim _{ x\to 2+0 } f\left( x \right) =\lim _{ x\to 2+0 } 2x+3=7$$Then equation whose roots are $$3,7$$$${ x }^{ 2 }-\left( 3+7 \right) x+\left( 3\times 7 \right) =0\\ \Rightarrow { x }^{ 2 }-10x+21=0$$Mathematics

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