Exponentials and logarithmsAQA A-Level Maths: Topic test

1. 1
   The curve $C$ has equation $y=5\mathrm{e}^{3x}$.

   (a)

   Which expression is $\dfrac{\mathrm{d}y}{\mathrm{d}x}$?

   [1 mark]

   • A$5\mathrm{e}^{3x}$
   • B$15x\mathrm{e}^{3x-1}$
   • C$15\mathrm{e}^{3x}$
   • D$\frac{5}{3}\mathrm{e}^{3x}$

   (b)

   What is the gradient of $C$ at the point where it crosses the $y$-axis?

   [1 mark]

   • A15
   • B5
   • C3
   • D45

   (c)

   Find the exact value of $x$ at which the gradient of $C$ is $30$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The variable $y$ is given by $y=\log_2\left(\dfrac{8}{x}\right)$ for $x>0$.

   (a)

   Which expression is equal to $y$?

   [1 mark]

   • A$8-\log_2x$
   • B$3-\log_2x$
   • C$\dfrac{3}{\log_2x}$
   • D$\log_2 8\times\log_2 x$

   (b)

   Which value of $x$ gives $y=5$?

   [1 mark]

   • A$\dfrac{1}{32}$
   • B$4$
   • C$-2$
   • D$\dfrac14$

   (c)

   Find the exact value of $x$ for which $y=\log_2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two equations in $x$ are to be solved. Equation (I) is $7^{x-2}=3^{x}$. Equation (II) is $\mathrm{e}^{2x}-7\mathrm{e}^{x}+12=0$.

   (a)

   Show that the solution of equation (I) can be written as $x=\dfrac{2\ln7}{\ln7-\ln3}$, and evaluate $x$ to 3 significant figures.

   [3 marks]

   (b)

   Solve equation (II), giving each solution as an exact value.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A biologist grows a mould colony and records its mass $M$ grams at time $t$ days. She models the mass by $M=kb^{t}$, where $k$ and $b$ are constants. A straight line is obtained when $\lg M$ is plotted against $t$ (where $\lg$ means $\log_{10}$). The line passes through the points $(1,\,0.50)$ and $(5,\,1.70)$.

   (a)

   Find the values of $k$ and $b$, each to 3 significant figures, and hence write down the model for $M$.

   [6 marks]

   (b)

   Use the model $\lg M=0.2+0.3t$ to find (i) the time for the mass to reach $500$ g, (ii) the doubling time of the colony, and (iii) comment on the validity of the model for large values of $t$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $f$ is defined by $f(x)=4\mathrm{e}^{-0.5x}+2$ for $x\in\mathbb{R}$.

   (a)

   What value does $f(x)$ approach as $x\to\infty$?

   [1 mark]

   • A0
   • B4
   • C6
   • D2

   (b)

   What is the value of $f(0)$?

   [1 mark]

   • A2
   • B6
   • C4
   • D8

   (c)

   Solve $f(x)=3$, giving your answer in the form $a\ln2$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The expression $E$ is defined by $E=\ln12-\ln3+2\ln5$.

   (a)

   Which expression is equal to $E$?

   [1 mark]

   • A$\ln100$
   • B$\ln40$
   • C$\ln20$
   • D$\ln19$

   (b)

   What is the value of $\mathrm{e}^{E}$?

   [1 mark]

   • A4
   • B25
   • C100
   • D10

   (c)

   Find the value of $x$ for which $\ln x+\ln4=E$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The value $V$ pounds of a car, $t$ years after it was bought new, is modelled by $V=18000\mathrm{e}^{-0.15t}$.

   (a)

   Find the time at which the model gives a value of $\pounds6000$. Give your answer in years to 2 decimal places.

   [3 marks]

   (b)

   Show that the rate of change of $V$ is proportional to $V$. Find the rate of change of $V$ when $t=2$, and state what your answer means in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The metabolic rate $R$ watts of a mammal of mass $M$ kg is modelled by $R=aM^{n}$, where $a$ and $n$ are constants. A graph of $\lg R$ against $\lg M$ is a straight line through the points $(0.6,\,1.0)$ and $(1.8,\,1.9)$.

   (a)

   Find $n$ and $a$, giving $a$ to 3 significant figures, and use the model to predict the metabolic rate of a mammal of mass $100$ kg.

   [6 marks]

   (b)

   A mammal has a metabolic rate of $50$ W. Find its mass. Show also that doubling the mass of a mammal multiplies its metabolic rate by about $1.68$, and interpret this.

   [6 marks]

   Total for question 8: 12 marks

End of questions