The law of motion of a body moving along a straight line is $x = \frac{1}{2} vt$, where $x$ is its distance from a fixed point on the line at time $t$ and $v$ is its velocity. Then:

$A$ stone is dropped into a quiet lake and waves move in circles at a speed of $8 \,cm/sec$. At the instant when the radius of the circular wave is $12 \,cm$, how fast is the enclosed area increasing?

If $y=5x^2+6x+6$,$x=2$,and $\Delta x=0.001$,then the value of $dy$ is:

In a Maha Kumbh, a drone camera is moving along the curve $3y = x^3 - 3$. The camera captures high-quality pictures when the $y$-coordinate changes $9$ times as fast as the $x$-coordinate. Which of the following is a precise position of the drone at that instant?

$A$ particle starts moving from rest from a fixed point in a fixed direction. The distance $s$ from the fixed point at a time $t$ is given by $s = t^{2} + at - b + 17$, where $a$ and $b$ are real numbers. If the particle comes to rest after $5 \ s$ at a distance of $s = 25$ units from the fixed point, then the values of $a$ and $b$ are respectively:

Moving along the $x$-axis are two points with $x = 10 + 6t$ and $x = 3 + t^2$. The speed with which they are receding from each other at the time of encounter is ........... $cm/sec$. ($x$ is in $cm$ and $t$ is in seconds)