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  • Let m and n be the numbers of real roots of the quadratic equations <img alt="\mathrm{x}^2-12 \mathrm{x}+[\mathrm{x}]+31=0 \text { and } \mathrm{x}^2-5|\mathrm{x}+2|-4" src="https://entranceco

Let m and n be the numbers of real roots of the quadratic equations \mathrm{x}^2-12 \mathrm{x}+[\mathrm{x}]+31=0 \text { and } \mathrm{x}^2-5|\mathrm{x}+2|-4=0 respectively,where [x] denotes the greatest integer \leq x. Then m2 +mn + n2 is equal to

Option: 1

9


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\begin{aligned} & x^2-12 x+[x]+31=0 \\ & \{x\}=x^2-11 \quad x+31 \\ & 0 \leq x^2-11 x+31<1 \\ & x^2-11 x+30<0 \\ & x \in(5,6) \\ & \text { so } \quad[x]=5 \\ & x^2-12 x+5+31=0 \\ & x^2-12 x+36=0 \\ & x=6 \quad \text { but } x \in(5,6) \\ & \text { so } \quad x \in \phi \\ & m=0 \end{aligned}

Now x^2-5|x+2|-4=0 x \geq-2 \begin{array}{ll} x^2-5 x-14=0 & x^2+5 x+6=0 \\ (x-7)(x+2)=0 & (x+3)(x+2)=0 \\ x=7,-2 & x=-3,-2 \end{array} \begin{aligned} & \mathrm{x}=\{7,-2,-3\} \\ & \mathrm{n}=3 \\ & \mathrm{~m}^2+\mathrm{mn}+\mathrm{n}^2=\mathrm{n}^2=9 \end{aligned}

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