Exponential functions and e^xEdexcel A-Level Maths: Subtopic test

1. 1
   A function is given by $f(x)=a^x$, where $a>0$, and its graph passes through the point $(2,\,25)$.

   (a)

   Find the value of $a$.

   [1 mark]

   • A$12.5$
   • B$625$
   • C$-5$
   • D$5$

   (b)

   Find the value of $f(-1)$.

   [1 mark]

   • A$\frac15$
   • B$-5$
   • C$5$
   • D$-\frac15$

   (c)

   Solve $f(x)=\dfrac{1}{125}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the curve $y=e^{2x-1}+3$.

   (a)

   Which line is the horizontal asymptote of the curve?

   [1 mark]

   • A$y=0$
   • B$y=3$
   • C$y=-1$
   • D$x=\frac12$

   (b)

   Find the $y$-coordinate where the curve crosses the $y$-axis.

   [1 mark]

   • A$e+3$
   • B$4$
   • C$\frac1e+3$
   • D$e^{-1}$

   (c)

   Explain why the curve never crosses the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The mass $M$ grams of a radioactive sample $t$ days after it is first measured is modelled by $M=80e^{-0.05t}$.

   (a)

   Find $\dfrac{dM}{dt}$ and hence show that the rate of change of $M$ is proportional to $M$.

   [3 marks]

   (b)

   Find the rate at which the mass is changing when $t=10$, to 3 significant figures, and interpret your answer. Explain why this shows an exponential model suits radioactive decay.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population $P$ of a colony of bacteria $t$ hours after the start of an experiment is modelled by $P=500e^{0.3t}$.

   (a)

   (i) State the initial population.
   (ii) Find the population after $5$ hours, to the nearest whole number.
   (iii) Find the rate at which the population is increasing when $t=5$, to 3 significant figures.

   [6 marks]

   (b)

   A second colony is modelled by $Q=3000e^{-0.1t}$.
   (i) Find $\dfrac{dQ}{dt}$ when $t=0$ and interpret your answer.
   (ii) Describe what happens to $P$ and to $Q$ as $t$ becomes very large, and comment on how realistic each model is in the long term.

   [6 marks]

   Total for question 4: 12 marks

End of questions