A particle is moving in one-dimensional motion on the x-axis

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It slows down from $x = - 3 m$ to $x = - 2 m$ and speeds up from $x = - 2 m$ to $x = - 1 m$ . Sign of acceleration changes from negative to positive

If it approaches the origin from negative x-axis, then its velocity is negative

If its acceleration is negative, it must be slowing down

If it slows down while moving from $x = 4 m$ to $x = 3 m$ , then its velocity is negative and acceleration is positive

answer is A, D.

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Detailed Solution

$v = \frac{d x}{d t}$ . If $x$ increases with time, then velocity is positive. It means for motion towards positive x-axis (independent of whether the particle is to the left of origin or to the right), velocity is positive. For motion towards negative x-axis (independent of whether the particle is to the left of origin or to the right), velocity is negative. $x = 4 m$
Slowing down means velocity and acceleration point in opposite directions.
Speeding up means velocity and acceleration point in same direction.
a) When it slows down while moving from $x = 4 m$ to $x = 3 m$ , then velocity is negative and acceleration is positive. Option (a) is correct.
b) Negative acceleration accompanied with negative velocity means speeding up. Negative acceleration with positive velocity means slowing down.
Option (b) is wrong.
c) From above discussion, if the particle approaches origin from negative x-axis, then its velocity is positive. Option (c) is wrong.
d) slows down from $x = - 3 m$ to $x = - 2 m$ : v is positive, a is negative
speeds up from $x = - 2 m$ to $x = - 1 m$ : v is positive, a is positive
Option (d) is correct.