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If $f_{1}\left ( x \right )=x^{2}+11x+n$ and $f_{2}\left ( x \right )=x$, then the largest positive integer $n$ for which the equation $f_{1}\left ( x \right )=f_{2}\left ( x \right )$ has two distinct real roots, is

1. $24$
2. $23$
3. $19$
4. $10$

Given that,

• $f_{1}(x) = x^{2} + 11x + n$
• $f_{2}(x) = x$

And $f_{1}(x) = f_{2}(x)$

$\Rightarrow x^{2} + 11x + n = x$

$\Rightarrow x^{2} + 10x + n = 0$

If the equation  has two distinct real roots, then

$b^{2}-4ac> 0$

$\Rightarrow 10^{2}-4(1)(n)> 0$

$\Rightarrow 100-4n> 0$

$\Rightarrow 100>4n$

$\Rightarrow 4n<100$

$\Rightarrow n<\frac{100}{4}$

$\Rightarrow n<25$

$\Rightarrow\boxed{n_{\text{max}} = 24}$

$\therefore$ The largest positive integer value of $n$ is $24.$

Correct Answer $:\text{A}$

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