$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	Total
Marks	10	11	10	31
Score

Question 1 Code: 9709/41/O/N/11/7, Topic: -

A particle $P$ starts from a point $O$ and moves along a straight line. $P$ 's velocity $t$ s after leaving $O$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where

$$ v=0.16 t^{\frac{3}{2}}-0.016 t^{2} $$

$P$ comes to rest instantaneously at the point $A$.

$\text{(i)}$ Verify that the value of $t$ when $P$ is at $A$ is 100. $[1]$

$\text{(ii)}$ Find the maximum speed of $P$ in the interval $0< t <100$. $[4]$

$\text{(iii)}$ Find the distance $O A$. $[3]$

$\text{(iv)}$ Find the value of $t$ when $P$ passes through $O$ on returning from $A$. $[2]$

Question 2 Code: 9709/42/O/N/11/7, Topic: -

A tractor travels in a straight line from a point $A$ to a point $B$. The velocity of the tractor is $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t \mathrm{~s}$ after leaving $A$.

$\text{(i)}$

The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for

$\text{(a)}$ the distance $A B$, $[2]$

$\text{(b)}$ the acceleration of the tractor for $0< t <400$ and for $400< t <800$. $[2]$

$\text{(ii)}$ The actual velocity of the tractor is given by $v=0.04 t-0.00005 t^{2}$ for $0 \leqslant t \leqslant 800$.

$\text{(a)}$ Find the values of $t$ for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part $\text{(i)}$. $[3]$

For the interval $0 \leqslant t \leqslant 400$, the approximate velocity of the tractor in part $\text{(i)}$ is denoted by $v_{1} \mathrm{~m} \mathrm{~s}^{-1}$

$\text{(b)}$ Express $v_{1}$ in terms of $t$ and hence show that $v_{1}-v=0.00005(t-200)^{2}-1$. $[2]$

$\text{(c)}$ Deduce that $-1 \leqslant v_{1}-v \leqslant 1$. $[2]$

Question 3 Code: 9709/43/O/N/11/7, Topic: -

A car of mass $600 \mathrm{~kg}$ travels along a straight horizontal road starting from a point $A$. The resistance to motion of the car is $750 \mathrm{~N}$.

$\text{(i)}$ The car travels from $A$ to $B$ at constant speed in $100 \mathrm{~s}$. The power supplied by the car's engine is constant and equal to $30 \mathrm{~kW}$. Find the distance $A B$. $[3]$

$\text{(ii)}$ The car's engine is switched off at $B$ and the car's speed decreases until the car reaches $C$ with a speed of $20 \mathrm{~m} \mathrm{~s}^{-1}$. Find the distance $B C$. $[3]$

$\text{(iii)}$ The car's engine is switched on at $C$ and the power it supplies is constant and equal to $30 \mathrm{~kW}$. The car takes $14 \mathrm{~s}$ to travel from $C$ to $D$ and reaches $D$ with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$. Find the distance $C D.$ $[4]$