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CAT 2017 Slot 1 QA Question & Solution

AlgebraMedium

Question

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

Solution

$f_{1}(x)=x^{2}+11x+n$ and $f_{2}(x) = x$
$f_{1}(x)=f_{2}(x)$
=> $x^{2}+11x+n = x$
=> $x^2 + 10x + n = 0$
=> For this equation to have distinct real roots, b$^2$-4ac>0
$10^2 > 4n$
=> n < 100/4 
=> n < 25
Thus, largest integral value that n can take is 24.