## SUBJECT 1.

Let the expression $E(x)=\sqrt{x^{2}+4 x+4}+\sqrt{x^{2}-6 x+9}, x \in \mathbb{R}$.

a) Determine the cardinality of the set $A=\{n \in \mathbb{N} \mid E(\sqrt{n})=5\}$.

b) Let $m \in \mathbb{R}$ be the minimum value of the expression $E(x)$. Determine $m$ and the set $B=\{x \in \mathbb{R} \mid E(x)=m\}$.
## Subject 1.

a) We have $E(x)=\sqrt{(x+2)^{2}}+\sqrt{(x-3)^{2}}=|x+2|+|x-3|$. 1 p

$E(\sqrt{n})=5 \Leftrightarrow|\underbrace{\sqrt{n}+2}_{>0}+| \sqrt{n}-3|=5 \Leftrightarrow| \sqrt{n}-3 \mid=3-\sqrt{n} \Leftrightarrow \sqrt{n} \leq 3 \Leftrightarrow n \leq 9$. 2 p

But $n \in \mathbf{N}$ and thus $A=\{0,1,2, \ldots, 9\} \Rightarrow \operatorname{Card}(A)=10$.

1 p

b) We have $E(x)=|x+2|+|x-3|=|x+2|+|3-x| \geq|x+2+3-x|=5, \forall x \in \mathbf{R}$. (1) .................1p Since $E(0)=5$ and $E(x) \geq 5, \forall x \in \mathbf{R}$, the minimum value of the expression $E(x)$ is $m=5$. .............1p In inequality (1) we have equality $E(x)=5 \Leftrightarrow(x+2)(3-x) \geq 0 \Leftrightarrow x \in[-2 ; 3]$.

Thus, $B=\{x \in \mathbf{R} \mid E(x)=5\} \Leftrightarrow B=[-2 ; 3]$. 1 p