$($a$)$ Given vectors $=4i-2j+k$ and $\frac{?}{\text{b}\text{a}\text{r}}(b)=2i+5j+3k,$ calculate $\frac{*}{b}ar(b).$

$($b$)$ Given vectors $=4i-2j+k$ and $\frac{?}{\text{b}\text{a}\text{r}}(b)=2i+5j+3k,$ calculate $\frac{}{b}ar(b).$

$($c$)$ Given vectors $=i-tj$ and $\frac{?}{\text{b}\text{a}\text{r}}(b)=(t^2+1)i+(t+7)j,$ for what values of $t$ are $$ and $\frac{?}{\text{b}\text{a}\text{r}}(b)$ perpendicular?

$($d$)$ For a triangle ABC with sides of length $|AB|=c,|AC|=b$ and $|BC|=a,$ and angle $\frac{?}{{?}_{C}AB}=,$ use the scalar product to prove the cosine rule

$a^2=b^2+c^2-2$bccos$$

$($a$)$ For a body moving in a straight line with uniform acceleration $a,$ derive the following expressions $($where $v$ is velocity, $u$ is initial velocity, $t$ is time and $s$ is displacement$)$:

$($i$)v=u+at$

$($ii$)s=ut+\frac{1}{2}a{t}^2$

$($iii$)v^2=u^2+2as$

$($b$)$ A rocket is launched vertically upwards from ground level with its thruster providing an acceleration of $3gm{s}^{-2}$ for a period of twenty seconds. The thruster then shuts off, leaving the rocket solely under the effect of acceleration due to gravity $gm{s}^{-2}.$

$($i$)$ What is the maximum height reached by the rocket?

$($ii$)$ What is the elapsed time between launching the rocket and the rocket hitting the ground?

$($a$)$ A straight river flows at $4k\frac{m}{h}$ and its width is $300$ m $.$ A rower can row a boat at $5k\frac{m}{h}$ in still water.

$($i$)$ Find the direction in which the rower must row in order to go straight across.

$($ii$)$ Find the time takes them to cross.

$($b$)$ A child kicks a ball with velocity $\sqrt[\phantom{2}]{5gh}$ at an angle $$ above the horizontal, aiming to hit a wall of height $h$ at a distance $2h.$ The ball is affected by acceleration due to gravity $(g),$ and is not affected by air resistance.

$($i$)$ Calculate the trajectory of the particle $($i$.$e$.$ the relationship between the height $z$ of the projectile and the distance travelled $x).$

$($ii$)$ For what values of $$ will the ball hit the top of the wall?

$($iii$)$ For what values of $$ will the ball hit the base of the wall directly, without first bouncing off the ground?

$($iv$)$ For what values of $$ will the ball hit the wall? Explain your answer.