P2: Exponentials and logarithmsEdexcel International A Level Maths: Topic test

1. 1
   The curve $y=k\times2^x$, where $k$ is a constant, passes through the point $(3,40)$.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$5$
   • B$\frac{20}{3}$
   • C$320$
   • D$20$

   (b)

   What is the value of $y$ when $x=-2$?

   [1 mark]

   • A$-20$
   • B$\frac54$
   • C$20$
   • D$\frac1{20}$

   (c)

   Find the value of $x$ at which the curve meets the line $y=80$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $P=2\log_3 6-\log_3 4+\frac12\log_3 81$.

   (a)

   Find the exact value of $2\log_3 6-\log_3 4$.

   [1 mark]

   • A$1$
   • B$\log_3 32$
   • C$2$
   • D$\dfrac{\log_3 36}{\log_3 4}$

   (b)

   Find the exact value of $\frac12\log_3 81$.

   [1 mark]

   • A$40.5$
   • B$4$
   • C$\frac12$
   • D$2$

   (c)

   Given that $\log_3 x=P+\log_3 2$, find the value of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The concentration of a drug in a patient's blood is modelled by $C=40\times0.82^t$ mg l$^{-1}$, where $t$ is the time in hours after the drug is given.

   (a)

   Find the time taken for the concentration to fall to $5$ mg l$^{-1}$, giving your answer to $3$ significant figures.

   [3 marks]

   (b)

   A second dose is given when the concentration falls to $12$ mg l$^{-1}$. Find, to the nearest minute, how many hours and minutes after the first dose this happens.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\mathrm{f}(x)=2^{2x}-9\times2^{x}+8$.

   (a)

   Solve $\mathrm{f}(x)=0$.

   [6 marks]

   (b)

   Find the minimum value of $\mathrm{f}(x)$, and show that it occurs when $x=2\log_2 3-1$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The value, in pounds, of a machine $t$ years after it was bought is modelled by $V=12000\times0.85^t$.

   (a)

   What is the value of the machine after $4$ years, to the nearest pound?

   [1 mark]

   • A£4800
   • B£6264
   • C£40800
   • D£10200

   (b)

   Which statement about this model is correct?

   [1 mark]

   • AThe value is never zero but gets as close to zero as we like.
   • BThe value falls by £1800 every year.
   • CThe value becomes negative for large $t$.
   • DThe value halves every $4$ years.

   (c)

   Find the time taken for the value of the machine to fall to half of its original value, giving your answer to $3$ significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the equation $\log_2(x+6)-\log_2(x-2)=3$.

   (a)

   Which expression is equal to the left-hand side of the equation?

   [1 mark]

   • A$\log_2 8$
   • B$\dfrac{\log_2(x+6)}{\log_2(x-2)}$
   • C$\log_2\left[(x+6)(x-2)\right]$
   • D$\log_2\dfrac{x+6}{x-2}$

   (b)

   What is the solution of the equation?

   [1 mark]

   • A$x=3$
   • B$x=\frac87$
   • C$x=\frac{22}{7}$
   • D$x=6$

   (c)

   Explain why the equation can have no solution with $x<2$, and verify that the solution found in part (b) satisfies the original equation.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The number of subscribers to a streaming service is modelled by $S=50000\times a^t$, where $t$ is the number of years after launch and $a$ is a positive constant. After $2$ years there are $72000$ subscribers.

   (a)

   Find the value of $a$.

   [3 marks]

   (b)

   Find the number of subscribers after $5$ years, to the nearest hundred, and the time for the number of subscribers to reach $200000$, giving your answer in years to $2$ decimal places.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two cultures of bacteria are grown. After $t$ hours the populations are $A=200\times2^{t}$ and $B=1600\times1.5^{t}$.

   (a)

   Find the time at which the two populations are equal, to $3$ significant figures, and the population at that time to the nearest hundred.

   [6 marks]

   (b)

   Use logarithms to base $10$ to show that $\log_{10}A=t\log_{10}2+\log_{10}200$, and hence find the time at which $A$ first exceeds $10^6$, giving your answer to $3$ significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions