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# If $\mid\overrightarrow{a}\mid=4$ and $-3\leq\:\lambda\leq 2$,then the range of $\mid\lambda\overrightarrow{a}\mid$ is

$(A)\;[0,8]\quad(B)\;[-12,8]\quad(C)\;[0,12]\quad(D)\;[8,12]$

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A)
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• $|\lambda\:\overrightarrow a|=|\lambda||\overrightarrow a|$
Given $|\overrightarrow a|=4 \: and\:-3\leq\lambda\leq\:2$
Since it is given $-3 \leq \lambda \leq 2$
This implies $0\leq|\lambda|\leq\:3$
Multiply this in equation by $|\overrightarrow a|$ we get,
$=>0\leq|\lambda|\:|\overrightarrow a|\leq\:3|\overrightarrow a|$
But $|\lambda|\:|\overrightarrow a|=|\lambda\overrightarrow a|$ and $|\overrightarrow a|=4$
$=>0\leq\:|\lambda\overrightarrow a|\leq\:3\times4$
$=>0\leq\:|\lambda\overrightarrow a|\leq 12$
Hence the range of $\lambda \: is\: [0,12]$
Hence $C$ is the correct option