Verified Solution • 2025–26 Session moderate

QJEEM23Ma2d29: JEE Main 2023 Mathematics - Class 11 JEEM23Ma2d29 Solution: The number of elements in the set is _________.

The number of elements in the set \{ n \in Z:|{n^2} - 10n + 19| < 6\} is _________.

Step-by-Step Verified Solution

Given, \left|n^2-10 n+19\right|<6

\Rightarrow-6 < n^2-10 n+19 < 6

Take, -6 < n^2-10 n+19 and n^2-10 n+19 < 6

⇒n2−10n+25gt;0amp; and n2−10n+13lt;0⇒(n−5)2gt;0amp; and n=10±100−522lt;0\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}

⇒n∈Z−{5}\Rightarrow n \in \mathbb{Z}-\{5\}

amp;∴n∈[5−23,5+23]amp;∴n∈[13,8.3]amp;∴n=2,3,4,5,6,7,8\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}

Thus, number of element in the set is ' 6 '

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