P3: Exponentials and logarithmsEdexcel International A Level Maths: Topic test

1. 1
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=4-3\mathrm{e}^{-2x}$, $x\in\mathbb{R}$.

   (a)

   Which of the following is the range of $\mathrm{f}$?

   [1 mark]

   • A$\mathrm{f}(x)>4$
   • B$\mathrm{f}(x)<1$
   • C$\mathrm{f}(x)<4$
   • D$\mathrm{f}(x)>-3$

   (b)

   The graph of $y=\mathrm{f}(x)$ crosses the $y$-axis at the point $(0,k)$. Find $k$.

   [1 mark]

   • A$1$
   • B$4$
   • C$-3$
   • D$7$

   (c)

   Solve $\mathrm{f}(x)=0$, giving your answer in an exact form.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=\ln(7-2x)$, where $x$ takes the largest possible domain.

   (a)

   Which of the following is the domain of $\mathrm{g}$?

   [1 mark]

   • A$x>\dfrac72$
   • B$x<\dfrac72$
   • C$x<7$
   • D$x\leqslant\dfrac72$

   (b)

   Solve $\mathrm{g}(x)=2$.

   [1 mark]

   • A$x=\dfrac{7+\mathrm{e}^2}{2}$
   • B$x=\dfrac52$
   • C$x=\dfrac{7-\mathrm{e}}{2}$
   • D$x=\dfrac{7-\mathrm{e}^2}{2}$

   (c)

   Find $\mathrm{g}^{-1}(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\mathrm{e}^{2x}-7\mathrm{e}^{x}+10$.

   (a)

   Find the exact $x$-coordinates of the points where $C$ crosses the $x$-axis.

   [3 marks]

   (b)

   Find the $y$-coordinate of the point where $C$ crosses the $y$-axis. Find also the exact coordinates of the minimum point of $C$. (Hint: the equation is a quadratic in $\mathrm{e}^{x}$.)

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A scientist studies two drugs. For drug A, the concentration $C$ mg l$^{-1}$ in the blood $t$ hours after an injection is modelled by $C=18\mathrm{e}^{-0.35t}$. For drug B, $C=kb^{t}$, where $k$ and $b$ are constants, and a graph of $\lg C$ against $t$ is a straight line through the points $(0,1.6)$ and $(5,0.6)$.

   (a)

   For drug A, write down the initial concentration and find, to $3$ significant figures, the time at which the concentration is $5$ mg l$^{-1}$. Find also the time taken for the concentration to halve, to $3$ significant figures.

   [6 marks]

   (b)

   For drug B, find the values of $k$ and $b$ to $3$ significant figures, and find the time at which $C=10$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The power output $P$ kW of a wind turbine at wind speed $v$ m s$^{-1}$ is modelled by $P=av^{n}$, where $a$ and $n$ are constants. A graph of $\lg P$ against $\lg v$ is a straight line with gradient $3$ that passes through the point $(1,\,2.6)$.

   (a)

   Find the value of $a$ to $3$ significant figures.

   [1 mark]

   • A$0.398$
   • B$398$
   • C$2.6$
   • D$-0.4$

   (b)

   Find the power output when $v=10$.

   [1 mark]

   • A$1000$ kW
   • B$39.8$ kW
   • C$398$ kW
   • D$3980$ kW

   (c)

   Find the wind speed, to $3$ significant figures, at which the power output is $50$ kW.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The population $N$ of a colony of seals, $t$ years after it was first surveyed, is modelled by $N=250\mathrm{e}^{0.08t}$.

   (a)

   Find the population predicted by the model $10$ years after the first survey, to the nearest whole number.

   [1 mark]

   • A$450$
   • B$2000$
   • C$270$
   • D$556$

   (b)

   Find the time, in years to $3$ significant figures, for the population to double.

   [1 mark]

   • A$12.5$
   • B$8.66$
   • C$5.55$
   • D$17.3$

   (c)

   Show that, according to the model, the population increases by approximately $8.3\%$ each year.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2\ln(x-1)+3$, $x>1$.

   (a)

   Find the exact $x$-coordinate of the point where the graph of $y=\mathrm{f}(x)$ crosses the $x$-axis.

   [3 marks]

   (b)

   Find $\mathrm{f}^{-1}(x)$ and state its domain.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A hot metal bar is left to cool in a workshop at $20^{\circ}$C. Its temperature $T^{\circ}$C, $t$ minutes after it is removed from a furnace, is modelled by $T=20+A\mathrm{e}^{-kt}$, where $A$ and $k$ are positive constants. The initial temperature is $180^{\circ}$C and after $15$ minutes the temperature is $100^{\circ}$C.

   (a)

   Find the value of $A$ and the exact value of $k$. Hence find the time at which the temperature of the bar is $30^{\circ}$C.

   [6 marks]

   (b)

   A student plots $\ln(T-20)$ against $t$. (i) Show that this gives a straight line, and state its gradient and its intercept on the vertical axis. (ii) Use the straight-line equation to find the time at which $T=60$.

   [6 marks]

   Total for question 8: 12 marks

End of questions