1. 1
   An engineer is investigating how different quantities in a mechanical system vary directly or inversely with one another.

   (a)

   If $y \propto x$ and $y = 20$ when $x = 5$, find $y$ when $x = 8$.

   [1 mark]

   • A$13$
   • B$25$
   • C$40$
   • D$32$

   (b)

   If $y \propto \frac{1}{x}$ and $y = 6$ when $x = 4$, find $y$ when $x = 3$.

   [1 mark]

   • A$2$
   • B$4.5$
   • C$8$
   • D$18$

   (c)

   If $y \propto x^2$ and $y = 18$ when $x = 3$, find $y$ when $x = 6$.

   [1 mark]

   • A$36$
   • B$72$
   • C$54$
   • D$9$

   Total for question 1: 3 marks

2. 2
   A physics student is investigating the relationship between the extension of a spring and the force applied to it.

   (a)

   The extension $E$ of a spring is directly proportional to the force $F$ applied to it. When $F = 12$ newtons, $E = 4.8$ cm. Find the formula connecting $E$ and $F$.

   [2 marks]

   (b)

   Using the formula from part (a), calculate the force needed to extend the spring by $9$ cm.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A pool maintenance company is investigating how the time to fill a swimming pool depends inversely on the number of pumps used, and wants to plan pump hire for different jobs.

   (a)

   The time $T$ taken to fill a swimming pool is inversely proportional to the number of identical pumps $n$ used. When $n = 3$, $T = 8$ hours. Find the formula connecting $T$ and $n$.

   [3 marks]

   (b)

   Using the formula from part (a), calculate how many pumps would be needed to fill the pool in 4.8 hours.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A road safety researcher is investigating how a car's braking distance depends on its speed, using a square proportionality relationship, and comparing braking distances at different speeds.

   (a)

   The braking distance $d$ of a car is directly proportional to the square of its speed $v$, i.e. $d \propto v^2$. When $v = 20$ mph, $d = 12$ metres. Find the formula connecting $d$ and $v$.

   [4 marks]

   (b)

   Using the formula from part (a), calculate the braking distance when the car is travelling at 50 mph.

   [4 marks]

   (c)

   A second car has a braking distance that is directly proportional to the cube of its speed, i.e. $d \propto v^3$, with the same constant of proportion behaviour such that at $v = 20$ mph its braking distance is also 12 metres. Find this car's formula, then determine at what speed this car would have the same braking distance (75 metres) as the first car at 50 mph (from part (b)), giving your answer to 1 decimal place.

   [5 marks]

   Total for question 4: 13 marks

End of questions