Exponential functionsAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=2^x$.

   (a)

   Which statement about the curve $C$ is correct?

   [1 mark]

   • AIt passes through $(0,1)$ and $y>0$ for all values of $x$.
   • BIt passes through $(0,2)$ and crosses the $x$-axis once.
   • CIt has the $y$-axis as an asymptote.
   • DIt is decreasing for $x<0$.

   (b)

   Which expression is equal to $\left(\frac12\right)^x$?

   [1 mark]

   • A$-2^x$
   • B$2^{-x}$
   • C$2^{\frac1x}$
   • D$\dfrac{1}{2^{-x}}$

   (c)

   Find the $x$-coordinate of each point where $C$ meets the lines $y=16$ and $y=\frac18$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number $N$ of bacteria in a culture, $t$ hours after it is first measured, is modelled by $N=500e^{0.4t}$.

   (a)

   Find $\dfrac{dN}{dt}$.

   [1 mark]

   • A$0.4e^{0.4t}$
   • B$500e^{0.4t}$
   • C$500\times0.4t\,e^{0.4t-1}$
   • D$200e^{0.4t}$

   (b)

   Which statement about the model is correct?

   [1 mark]

   • AThe rate of growth is constant.
   • BThe rate of growth is proportional to $t$.
   • CThe rate of growth is proportional to $N$.
   • DThe rate of growth decreases as $N$ increases.

   (c)

   Find the rate of increase of the number of bacteria when $t=5$, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=e^{2x}$ and passes through the point $P$ where $x=\frac12$.

   (a)

   Find the exact coordinates of $P$ and the exact gradient of $C$ at $P$.

   [3 marks]

   (b)

   Find the equation of the tangent to $C$ at $P$, and state the exact coordinates of the point where the tangent meets the $y$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cup of tea is left to cool in a room. Its temperature $T\,^\circ\mathrm{C}$, $t$ minutes after it is poured, is modelled by $T=20+70e^{-0.05t}$.

   (a)

   (i) State the temperature of the tea when it is poured, and the temperature the model predicts in the long term.
   (ii) Find the rate of change of $T$ when $t=10$, and interpret your answer in context.
   (iii) Show that $\dfrac{dT}{dt}=-0.05(T-20)$.

   [6 marks]

   (b)

   A second cup, made of metal, is modelled by $T=20+70e^{-0.1t}$.
   (i) Find the temperature of each cup when $t=10$.
   (ii) Find the initial rate of change of temperature for each cup.
   (iii) Use your answers to explain how the two cups compare, and what happens to both in the long term.

   [6 marks]

   Total for question 4: 12 marks

End of questions