24. The maximum acceleration attained on the interval $0 \leq t \leq 3$ by the particle whose velocity is given by $v ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 12 t + 4$ is
(A) 9
(B) 12
(C) 14
(D) 21
(E) 40

A particle moves along the $x$-axis with its position at time $t$ given by $x ( t ) = ( t - a ) ( t - b )$, where $a$ and $b$ are constants and $a \neq b$. For which of the following values of $t$ is the particle at rest?
(A) $t = a b$
(B) $t = \frac { a + b } { 2 }$
(C) $t = a + b$
(D) $t = 2 ( a + b )$
(E) $t = a$ and $t = b$

For $t \geq 0$, the position of a particle moving along the $x$-axis is given by $x ( t ) = \sin t - \cos t$. What is the acceleration of the particle at the point where the velocity is first equal to 0 ?
(A) $- \sqrt { 2 }$
(B) $-1$
(C) 0
(D) 1
(E) $\sqrt { 2 }$

A particle moves along a line so that its acceleration for $t \geq 0$ is given by $a ( t ) = \frac { t + 3 } { \sqrt { t ^ { 3 } + 1 } }$. If the particle's velocity at $t = 0$ is 5, what is the velocity of the particle at $t = 3$ ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
(E) 11.710

A point P moving on the coordinate plane has position $( x , y )$ at time $t \left( 0 < t < \frac { \pi } { 2 } \right)$ given by
$$x = t + \sin t \cos t , \quad y = \tan t$$
What is the minimum speed of point P for $0 < t < \frac { \pi } { 2 }$? [3 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 2
(4) $2 \sqrt { 2 }$
(5) $2 \sqrt { 3 }$

A point P starts at time $t = 0$ and moves on a number line. At time $t$ ($t \geq 0$), its position $x$ is given by $$x = t^{3} - \frac{3}{2}t^{2} - 6t$$ What is the acceleration of point P at the time when its direction of motion changes after starting? [4 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18

121. In triangle ABC, $BC = 20$, $AH = 12$, and $AB = 12$ units. Line $\Delta$ parallel to BC moves at a constant speed of $0.2$ units per second. The rate of increase of the area of the trapezoid at the moment when the distance between the two parallel lines is 9 units is which of the following?
[Figure: Triangle ABC with vertex A at top, base BC at bottom, height AH drawn, points D and E on sides AB and AC respectively forming a trapezoid BCED, with H the foot of the altitude on BC]
(1) $0.8$ [4pt] (2) $0.9$ [4pt] (3) $1$ [4pt] (4) $1.2$
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The machine as shown has 2 rods of length 1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :
(1) Speed which is $\frac { 3 } { 4 }$ th of that of the roller when the weight is 0.4 m above the ground
(2) Constant speed
(3) Decreasing speed
(4) Increasing speed

A constant power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time $t$ is proportional to:-
(1) $t ^ { \frac { 2 } { 3 } }$
(2) $t ^ { \frac { 3 } { 2 } }$
(3) $t$
(4) $t ^ { \frac { 1 } { 2 } }$

A particle is moving in one dimension (along $x$ axis) under the action of a variable force. It's initial position was 16 m right of origin. The variation of its position $x$ with time $t$ is given as $x = - 3 t ^ { 3 } + 18 t ^ { 2 } + 16t$, where $x$ is in m and $t$ is in s. The velocity of the particle when its acceleration becomes zero is $\_\_\_\_$ $\mathrm { m } \mathrm { s } ^ { - 1 }$.

The co-ordinates of a particle moving in $x - y$ plane are given by : $x = 2 + 4 \mathrm { t } , y = 3 \mathrm { t } + 8 \mathrm { t } ^ { 2 }$. The motion of the particle is :
(1) uniformly accelerated having motion along a parabolic path.
(2) uniform motion along a straight line.
(3) uniformly accelerated having motion along a straight line.
(4) non-uniformly accelerated.

Q5. A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $t$ is proportional to :
(1) $t$
(2) $t ^ { 3 / 2 }$
(3) $t ^ { 2 }$
(4) $t ^ { 2 / 3 }$