The number of elements in the set $\{ n \in Z:|{n^2} - 10n + 19| < 6\}$ is _________.
Answer (integer)
6
Solution
Given, $\left|n^2-10 n+19\right|<6$
<br/><br/>$\Rightarrow-6 < n^2-10 n+19 < 6$
<br/><br/>Take, $-6 < n^2-10 n+19$ and $n^2-10 n+19 < 6$
<br/><br/>$$
\begin{array}{ll}
\Rightarrow n^2-10 n+25 > 0 & \text { and }\quad n^2-10 n+13 < 0 \\\\
\Rightarrow(n-5)^2 > 0 & \text { and } n=\frac{10 \pm \sqrt{100-52}}{2}<0
\end{array}
$$
<br/><br/>$\Rightarrow n \in \mathbb{Z}-\{5\}$
<br/><br/>$$
\begin{array}{lr}
& \therefore n \in[5-2 \sqrt{3}, 5+2 \sqrt{3}] \\\\
& \therefore n \in[13,8.3] \\\\
& \therefore n=2,3,4,5,6,7,8
\end{array}
$$
<br/><br/>Thus, number of element in the set is ' 6 '
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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