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