Pure: Exponentials and logarithmsEdexcel A-Level Maths: Topic test

1. 1
   A curve has equation $y=2\mathrm{e}^{3x}-5$.

   (a)

   At which value of $y$ does the curve cross the $y$-axis?

   [1 mark]

   • A$-5$
   • B$-3$
   • C$2$
   • D$-7$

   (b)

   What is the gradient of the curve at the point where $x=0$?

   [1 mark]

   • A$2$
   • B$3$
   • C$6$
   • D$18$

   (c)

   Find the exact value of $x$ at the point where the curve crosses the $x$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The positive numbers $a$ and $b$ satisfy $\log_2a=3$ and $\log_2b=5$.

   (a)

   What is the value of $\log_2(ab)$?

   [1 mark]

   • A$15$
   • B$-2$
   • C$256$
   • D$8$

   (b)

   What is the value of $\log_2\left(\dfrac{a^2}{b}\right)$?

   [1 mark]

   • A$1$
   • B$-4$
   • C$11$
   • D$\frac95$

   (c)

   Find the value of $\log_2\left(a^3\sqrt{b}\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In this question give any answer that is not exact to 3 significant figures.

   (a)

   Solve $7^{x-2}=50$.

   [3 marks]

   (b)

   Solve $2\log_2x=3+\log_2(x-2)$, where $x>2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Variables $x$ and $y$ are modelled by $y=ab^x$, where $a$ and $b$ are constants. Here $y$ is the number of subscribers, in thousands, to a streaming service $x$ years after 1 January 2015. The values of $\log_{10}y$ are linear in $x$: $\log_{10}y=0.9$ when $x=1$ and $\log_{10}y=1.5$ when $x=4$.

   (a)

   Show that $\log_{10}y=\log_{10}a+x\log_{10}b$. Hence find $a$ and $b$ to 3 significant figures.

   [6 marks]

   (b)

   Use the model $y=ab^x$, with the unrounded values of $a$ and $b$, to answer the following. (i) Find the value of $x$ for which $y=100$. (ii) Interpret the value of $b$ in context. (iii) Predict the number of subscribers when $x=10$. (iv) State one limitation of the model for large $x$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $f$ is defined by $f(x)=\ln(3x-2)$ for $x>\frac23$.

   (a)

   For which value of $x$ is $f(x)=0$?

   [1 mark]

   • A$x=1$
   • B$x=\frac23$
   • C$x=\frac{\mathrm{e}+2}{3}$
   • D$x=0$

   (b)

   What is the solution of $f(x)=2$?

   [1 mark]

   • A$x=\frac{\mathrm{e}^2-2}{3}$
   • B$x=\frac{\mathrm{e}^2}{3}+2$
   • C$x=2$
   • D$x=\frac{\mathrm{e}^2+2}{3}$

   (c)

   Find an expression for $f^{-1}(x)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A cup of tea cools in a room. Its temperature $T$ degrees Celsius, $t$ minutes after it is poured, is modelled by $T=20+70\mathrm{e}^{-0.08t}$.

   (a)

   What is the initial temperature of the tea?

   [1 mark]

   • A$20^\circ$C
   • B$70^\circ$C
   • C$90^\circ$C
   • D$0^\circ$C

   (b)

   According to the model, what value does $T$ approach for large $t$?

   [1 mark]

   • A$0^\circ$C
   • B$20^\circ$C
   • C$70^\circ$C
   • D$90^\circ$C

   (c)

   Find the time taken for the tea to cool to $50^\circ$C, giving your answer in minutes to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A car is bought new for $\pounds18\,000$. Its value, $\pounds V$, after $t$ years is modelled by $V=18\,000\times0.85^t$.

   (a)

   Find the value of the car after 3 years, to the nearest pound. State what happens to $V$ as $t$ becomes very large, and explain why the base $0.85$ means that the model shows decay rather than growth.

   [3 marks]

   (b)

   Find how long it takes for the value of the car to fall to $\pounds6000$, giving your answer in years to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A yeast culture contains $N$ cells $t$ hours after the start of an experiment. It is modelled by $N=N_0\mathrm{e}^{kt}$, where $N_0$ and $k$ are constants. Initially there are $2000$ cells and after $3$ hours there are $5400$ cells.

   (a)

   Find $N_0$ and $k$, giving $k$ to 3 significant figures. Hence find, in hours to 3 significant figures, when the culture first reaches $20\,000$ cells.

   [6 marks]

   (b)

   A second culture has $M=8000\mathrm{e}^{0.1t}$ cells at time $t$ hours. (i) Show that the two cultures have equal numbers of cells when $t=\dfrac{\ln4}{k-0.1}$, and find this time to 3 significant figures. (ii) Find the rate at which the first culture is increasing at $t=0$. (iii) State one limitation of using the model $N=N_0\mathrm{e}^{kt}$ for large $t$.

   [6 marks]

   Total for question 8: 12 marks

End of questions