The number of elements in the set { n : | n 2 - 10 n + 19 | 6 } is ___________ . [2023]

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Solution

Q.
The number of elements in the set ${n \in ℤ : | n^{2} - 10 n + 19 | < 6}$ is ___________ . [2023]

Ans.
(6)

We have, $| n^{2} - 10 n + 19 |$ $< 6, n \in ℤ$

$\Rightarrow$ $| {(n - 5)}^{2} - 6 |$ $< 6$

$\Rightarrow 0 < {(n - 5)}^{2} < 12$

$\Rightarrow {(n - 5)}^{2} = 1, 4, 9$

$\Rightarrow n - 5 = \pm 1, \pm 2, \pm 3$

$∴ 6 values of n exist.$

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