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Q.
The number of integral values of $\alpha$ for which the equation $| x -1|-2| x -3|+| x +2|=\alpha^2-2 \alpha+1$ has atleast one solution, is
Relations and Functions - Part 2
Solution:
$f ( x )=| x -1|-2| x -3|+| x +2|$
$\therefore$ Range of $f ( x )$ is $[-7,7]$
$f(x)=\alpha^2-2 \alpha+1$ will have a solution if $\alpha^2-2 \alpha+1 \leq 7$
$(\alpha-1)^2 \leq 7$
$-\sqrt{7}+1 \leq \alpha \leq \sqrt{7}+1$
$\therefore$ number of integral values of $\alpha$ is 5
