$($a$)$ A stretched string, when plucked and then released, gives off a note of frequency f which depends on the length $1,$ tension $T$ and mass per unit length m of the string. Use dimension to derive the equation connecting the quantities.

$($b$)$ The position of a particle moving along $y$ axis is given by $y=4t^3-6t^2+3.$ Find the average velocity from period $3$ s to $5$ s

Given three vectors, $A=4i-5j-8k,B=2i-3j-k$ and $C=i-j+k$

Evaluate: $($a$)A(B+C)=(AB)+(AC)$

$($b$)A*(B+C)$

$($c$)A-3B+C$

$($d$)A+C-B$

$($a$)$ A projectile is fired horizontally from a gun that is $20$ m above flat ground, omerging from the gun with a speed of $25\frac{m}{s}.$ At what horizontal distance from the firing point does it strike the ground?

$($b$)$ A force of $50$ N is used to push a box through a distance $2\mathrm{.}4$ m $.$ A work of $45$ J was done on the box.

$($i$)$ What is the direction of the force?

$($ii$)$ Find the mass of the box if the velocity is $2\frac{m}{s}$