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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24 — Easy Set $f_1(x)=f_2(x)$ to get a quadratic. Two distinct real roots require a positive discriminant; this bounds $n$, and the largest integer below the bound is the answer. 1 Form the equation. $$\begin{aligned} &x^2 + 11x + n = x\ &\Rightarrow\ x^2 + 10x + n = 0 \end{aligned}$$ 2 Impose two distinct real roots. Require discriminant $D>0$. $$\begin{aligned} &D = 10^2 - 4(1)(n) > 0\ &\Rightarrow\ 100 - 4n > 0\ &\Rightarrow\ n 3 Largest integer. The greatest integer strictly less than $25$. $$

CAT 2017 Slot 1 — QA Question 27

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