1. 1
   The cost of fuel is directly proportional to the distance travelled.

   (a)

   The cost of fuel, $C$ dirhams, is directly proportional to the distance travelled, $d$ km, so $C = kd$. If $C = 60$ when $d = 40$, find the value of $k$.

   [1 mark]

   • A$k = 1.5$
   • B$k = 0.67$
   • C$k = 2400$
   • D$k = 20$

   (b)

   Using $C = 1.5d$, find the cost of travelling 90 km.

   [1 mark]

   • A91.5 dirhams
   • B135 dirhams
   • C60 dirhams
   • D150 dirhams

   (c)

   Using $C = 1.5d$, find the distance travelled if the fuel cost was 225 dirhams.

   [1 mark]

   • A75 km
   • B337.5 km
   • C150 km
   • D223.5 km

   Total for question 1: 3 marks

2. 2
   The time taken to fill a swimming pool is inversely proportional to the number of pumps used.

   (a)

   The time, $t$ hours, taken to fill a swimming pool is inversely proportional to the number of pumps used, $p$, so $t = \frac{k}{p}$. If it takes 6 hours with 4 pumps, find the value of $k$ and write the formula connecting $t$ and $p$.

   [2 marks]

   (b)

   Using $t = \frac{24}{p}$, find how many pumps are needed to fill the pool in 2 hours.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A car's braking distance is proportional to the square of its speed, and reaction distance is proportional to speed itself.

   (a)

   The braking distance, $D$ metres, of a car is proportional to the square of its speed, $v$ km/h, so $D = kv^2$. If $D = 20$ when $v = 40$, find $k$ and hence find the braking distance when $v = 60$.

   [3 marks]

   (b)

   The car's reaction distance, $R$ metres, before braking begins, is proportional to the speed, so $R = mv$. If $R = 12$ when $v = 40$, find $m$, and hence find the total stopping distance (reaction plus braking) when $v = 60$, using $D = 0.0125v^2$ from part (a).

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A scientist studies how gas pressure and temperature relate to volume, using inverse and square-root proportion.

   (a)

   The pressure, $P$ pascals, exerted by a gas is inversely proportional to its volume, $V$ litres, so $P = \frac{k}{V}$. A scientist finds that when $V = 8$, $P = 15$. Find the value of $k$, then find $P$ when $V = 5$.

   [4 marks]

   (b)

   The scientist also finds that the time, $T$ seconds, for the gas to reach equilibrium is proportional to the square root of the volume, so $T = c\sqrt{V}$. If $T = 12$ when $V = 16$, find $c$, and hence find $T$ when $V = 25$.

   [4 marks]

   (c)

   The scientist combines both relationships to define efficiency $E = \frac{P}{T}$ using the formulas $P = \frac{120}{V}$ and $T = 3\sqrt{V}$ found previously. Show that $E$ is proportional to $V^{-\frac{3}{2}}$, stating the constant of proportionality, and find $E$ when $V = 4$.

   [5 marks]

   Total for question 4: 13 marks

End of questions