Let m and n be numbers of real roots of quadratic equations x^2-12 x+[x]+31=0 and x ^2-5| x +2|-4=0

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4-2.Quadratic Equations and Inequations

hard

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

$9$

$8$

$7$

$6$

(JEE MAIN-2023)

Solution

$x ^2-12 x +[ x ]+31=0$

$\{ x \}= x ^2-11 x +31$

$0 \leq x ^2-11 x +31 < 1$

$x ^2-11 x +30 < 0$

$x \in(5,6)$

$\text { so } \quad[ x ]=5$

$x ^2-12 x +5+31=0$

$x ^2-12 x +36=0$

$x =6 \quad \text { but } x \in(5,6)$

$\text { so } \quad x \in \phi$

$x =\{7,-2,-3\}$

$n =3$

$m ^2+ mn + n ^2= n ^2=9$

Standard 11

Mathematics

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hard

(JEE MAIN-2024)

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Solution

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Consider the following two statements

$I$. Any pair of consistent liner equations in two variables must have a unique solution.

$II$. There do not exist two consecutive integers, the sum of whose squares is $365$.Then,