The exponential function e^xEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=e^{2x-1}+3$ for all real $x$.

   (a)

   Which line is the horizontal asymptote of the graph of $y=f(x)$?

   [1 mark]

   • A$y=0$
   • B$y=3$
   • C$y=-1$
   • D$y=e^{-1}$

   (b)

   Where does the graph of $y=f(x)$ meet the $y$-axis?

   [1 mark]

   • A$(0,\,e^{-1})$
   • B$(0,\,3)$
   • C$(0,\,3+e^{-1})$
   • D$(0,\,4)$

   (c)

   Write down the range of $f$ and give a reason why $f(x)$ can never equal $3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass $M$ grams of a radioactive sample is modelled by $M=80e^{-0.05t}$, where $t$ is the time in hours after the sample was first measured.

   (a)

   What is the mass of the sample when it is first measured?

   [1 mark]

   • A$1$ g
   • B$0$ g
   • C$80e^{-0.05}$ g
   • D$80$ g

   (b)

   What is the mass of the sample after $10$ hours?

   [1 mark]

   • A$48.5$ g
   • B$47.9$ g
   • C$0.539$ g
   • D$79.5$ g

   (c)

   Find the time at which the mass of the sample is $20$ g. Give your answer in hours to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=e^{3x}-5$ and the line $l$ has equation $y=20$.

   (a)

   Find the exact coordinates of the points where $C$ crosses the coordinate axes.

   [3 marks]

   (b)

   (i) Find the exact $x$-coordinate of the point where $C$ meets $l$.
   (ii) Explain whether $C$ meets the line $y=-6$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The temperature $\theta\,^\circ$C of a cup of coffee, $t$ minutes after it is poured, is modelled by $\theta=20+75e^{-0.08t}$.

   (a)

   (i) State the temperature of the coffee when it is poured.
   (ii) Explain what the model predicts for the temperature of the coffee in the long term.
   (iii) Find the time taken for the coffee to cool to $50\,^\circ$C. Give your answer in minutes to 3 significant figures.

   [6 marks]

   (b)

   A second cup of coffee in a different room is modelled by $\theta=18+72e^{-kt}$, where $k$ is a positive constant. Its temperature is $60\,^\circ$C after $5$ minutes. Find $k$ to 3 significant figures, and hence find the time for this coffee to reach $30\,^\circ$C.

   [6 marks]

   Total for question 4: 12 marks

End of questions