Question

solve this please!!

Answer 1

Hope u like my process
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=> Displacement (x) of particle varies with time (t) is given as :-

$= > \it \: x = a {e}^{ - \alpha t} + b {e}^{ \beta t}$
So,

=> It's velocity can be given as,

$= > \bf \: v \: = \frac{dx}{dt} = \frac{d}{dt} (a {e}^{ - \alpha t} + b {e}^{ \beta t} ) \\ \\ = > \: v = \bf \underline { \: b \beta {e}^{ \beta t} - a \alpha {e}^{ - \alpha t} \: }$

Now with increase in time,

$\bf - a \alpha {e}^{ - \alpha t} \: \: \: being \: \: negative \: \: decreases\\ \: \bf with \: \: time$

Thus,

For option a (❌) ,
=-=-=-=-=-=-=-=-=-=

Since it's clearly seen that the velocity depends on the $\: \beta \:$ term.

For option b (❌),
=-=-=-=-=-=-=-=-=-=

Since being $\: \alpha = \beta \:$, the velocity can't be zero since a and b are other two constants too on which the velocity depends.

For option c(✔️),
=-=-=-=-=-=-=-=-=-=

The explanation given above may help u with it..

For option d(❌),
=-=-=-=-=-=-=-=-=-=

The assertion is opposite to the statement of option c and thus is false.

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Answer 2

C is the correct answer