Question 93

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

Correct Answer: 24

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.

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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

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