$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 12, question 9)

Variables $\D N$ and $\D x$ are such that $\D N = 200 + 50e^{\frac{x}{100}}.$

(i) Find the value of $\D N$ when $\D x = 0.$ [1]

(ii) Find the value of $\D x$ when $\D N = 600.$ [3]

(iii) Find the value of $\D N$ when $\D \frac{dN}{dx}=45.$ [4]

2 (CIE 2014, w, paper 21, question 5)

The number of bacteria $\D B$ in a culture, $\D t$ days after the first observation, is given by

\[B= 500 +400e^{0.2t}.\]

(i) Find the initial number present. [1]

(ii) Find the number present after 10 days. [1]

(iii) Find the rate at which the bacteria are increasing after 10 days. [2]

(iv) Find the value of $\D t$ when $\D B = 10000.$ [3]

3 (CIE 2014, w, paper 23, question 4)

The profit \$ $\D P$ made by a company in its nth year is modelled by \[P=1000e^{an+b}.\]

In the first year the company made \$2000 profit.

(i) Show that $\D a + b = \ln 2.$ [1]

In the second year the company made \$3297 profit.

(ii) Find another linear equation connecting $\D a$ and $\D b.$ [2]

(iii) Solve the two equations from parts (i) and (ii) to find the value of $\D a$ and of $\D b.$ [2]

(iv) Using your values for $\D a$ and $\D b,$ find the profit in the 10th year. [2]

4 (CIE 2016, w, paper 21, question 4)

The number of bacteria, $\D N,$ present in a culture can be modelled by the equation $\D N= 7000+ 2000e^{-0.05t},$ where $\D t$ is measured in days. Find

(i) the number of bacteria when $\D t = 10,$ [1]

(ii) the value of $\D t$ when the number of bacteria reaches 7500, [3]

(iii) the rate at which the number of bacteria is decreasing after 8 days. [3]

5 (CIE 2017, march, paper 12, question 11)

It is given that $\D y = Ae^{bx} ,$ where $\D A$ and $\D b$ are constants. When $\D \ln y$ is plotted against $\D x$ a straight line graph is obtained which passes through the points (1.0, 0.7) and (2.5, 3.7).

(i) Find the value of $\D A$ and of $\D b.$ [6]

(ii) Find the value of $\D y$ when $\D x = 2.$ [2]

6 (CIE 2017, march, paper 22, question 2)

The value, $\D V$ dollars, of a car aged $\D t$ years is given by $\D V = 12000 e^{-0.2t}.$

(i) Write down the value of the car when it was new. [1]

(ii) Find the time it takes for the value to decrease to $\D \frac{2}{3}$ of the value when it was new. [2]

7 (CIE 2017, s, paper 12, question 7)

It is given that $\D y = A(10^{bx}),$ where $\D A$ and $\D b$ are constants. The straight line graph obtained when $\D \lg y$ is plotted against $\D x$ passes through the points (0.5, 2.2) and (1.0, 3.7).

(i) Find the value of $\D A$ and of $\D b.$ [5]

Using your values of $\D A$ and $\D b,$ find

(ii) the value of $\D y$ when $\D x = 0.6,$ [2]

(iii) the value of $\D x$ when $\D y = 600.$ [2]

8 (CIE 2018, s, paper 11, question 5)

The population, $\D P,$ of a certain bacterium $\D t$ days after the start of an experiment is modelled by $\D P = 800e^{kt},$ where $\D k$ is a constant.

(i) State what the figure 800 represents in this experiment. [1]

(ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of $\D k.$ [3]

(iii) Calculate the population three days after the start of the experiment. [2]

9 (CIE 2018, s, paper 12, question 7)

A population, $\D B,$ of a particular bacterium, $\D t$ hours after measurements began, is given by $\D B =1000e^{\frac{t}{4}}.$

(i) Find the value of $\D B$ when $\D t = 0.$ [1]

(ii) Find the time taken for $\D B$ to double in size. [3]

(iii) Find the value of $\D B$ when $\D t = 8.$ [1]

1. $\D 250;208;4700$

2. (i) $\D 900$

(ii) $\D 3456$

(iii) $\D 591$

(iv) $\D 15.8$

3. (ii) $\D 2a + b = \ln 3.297$

(iii) $\D a = 0.5; b = 0.193$

(iv) $\D n = 10; P = 180000$

4. (i) $\D 8213$

(ii) $\D 27.7$

(iii) $\D \pm 67$

5. (i) $\D b = 2;A = 0.273,$

(ii) $\D 14.9$

6. (i) $\D 12 000$

(ii) $\D 2$

7. (i) $\D b=3,A=5.01$ or $\D 10.7$

(ii) $\D y=315$ or $\D 102.5$

(iii) $\D x = 0.693$

8. The number of bacteria at the

start of the experiment

$\D 1.61,100 000$

9. (i) $\D 1000,$

(ii) $\D 2.77$

(iii) $\D 7389$

1 (CIE 2012, s, paper 12, question 9)

Variables $\D N$ and $\D x$ are such that $\D N = 200 + 50e^{\frac{x}{100}}.$

(i) Find the value of $\D N$ when $\D x = 0.$ [1]

(ii) Find the value of $\D x$ when $\D N = 600.$ [3]

(iii) Find the value of $\D N$ when $\D \frac{dN}{dx}=45.$ [4]

2 (CIE 2014, w, paper 21, question 5)

The number of bacteria $\D B$ in a culture, $\D t$ days after the first observation, is given by

\[B= 500 +400e^{0.2t}.\]

(i) Find the initial number present. [1]

(ii) Find the number present after 10 days. [1]

(iii) Find the rate at which the bacteria are increasing after 10 days. [2]

(iv) Find the value of $\D t$ when $\D B = 10000.$ [3]

3 (CIE 2014, w, paper 23, question 4)

The profit \$ $\D P$ made by a company in its nth year is modelled by \[P=1000e^{an+b}.\]

In the first year the company made \$2000 profit.

(i) Show that $\D a + b = \ln 2.$ [1]

In the second year the company made \$3297 profit.

(ii) Find another linear equation connecting $\D a$ and $\D b.$ [2]

(iii) Solve the two equations from parts (i) and (ii) to find the value of $\D a$ and of $\D b.$ [2]

(iv) Using your values for $\D a$ and $\D b,$ find the profit in the 10th year. [2]

4 (CIE 2016, w, paper 21, question 4)

The number of bacteria, $\D N,$ present in a culture can be modelled by the equation $\D N= 7000+ 2000e^{-0.05t},$ where $\D t$ is measured in days. Find

(i) the number of bacteria when $\D t = 10,$ [1]

(ii) the value of $\D t$ when the number of bacteria reaches 7500, [3]

(iii) the rate at which the number of bacteria is decreasing after 8 days. [3]

5 (CIE 2017, march, paper 12, question 11)

It is given that $\D y = Ae^{bx} ,$ where $\D A$ and $\D b$ are constants. When $\D \ln y$ is plotted against $\D x$ a straight line graph is obtained which passes through the points (1.0, 0.7) and (2.5, 3.7).

(i) Find the value of $\D A$ and of $\D b.$ [6]

(ii) Find the value of $\D y$ when $\D x = 2.$ [2]

6 (CIE 2017, march, paper 22, question 2)

The value, $\D V$ dollars, of a car aged $\D t$ years is given by $\D V = 12000 e^{-0.2t}.$

(i) Write down the value of the car when it was new. [1]

(ii) Find the time it takes for the value to decrease to $\D \frac{2}{3}$ of the value when it was new. [2]

7 (CIE 2017, s, paper 12, question 7)

It is given that $\D y = A(10^{bx}),$ where $\D A$ and $\D b$ are constants. The straight line graph obtained when $\D \lg y$ is plotted against $\D x$ passes through the points (0.5, 2.2) and (1.0, 3.7).

(i) Find the value of $\D A$ and of $\D b.$ [5]

Using your values of $\D A$ and $\D b,$ find

(ii) the value of $\D y$ when $\D x = 0.6,$ [2]

(iii) the value of $\D x$ when $\D y = 600.$ [2]

8 (CIE 2018, s, paper 11, question 5)

The population, $\D P,$ of a certain bacterium $\D t$ days after the start of an experiment is modelled by $\D P = 800e^{kt},$ where $\D k$ is a constant.

(i) State what the figure 800 represents in this experiment. [1]

(ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of $\D k.$ [3]

(iii) Calculate the population three days after the start of the experiment. [2]

9 (CIE 2018, s, paper 12, question 7)

A population, $\D B,$ of a particular bacterium, $\D t$ hours after measurements began, is given by $\D B =1000e^{\frac{t}{4}}.$

(i) Find the value of $\D B$ when $\D t = 0.$ [1]

(ii) Find the time taken for $\D B$ to double in size. [3]

(iii) Find the value of $\D B$ when $\D t = 8.$ [1]

1. $\D 250;208;4700$

2. (i) $\D 900$

(ii) $\D 3456$

(iii) $\D 591$

(iv) $\D 15.8$

3. (ii) $\D 2a + b = \ln 3.297$

(iii) $\D a = 0.5; b = 0.193$

(iv) $\D n = 10; P = 180000$

4. (i) $\D 8213$

(ii) $\D 27.7$

(iii) $\D \pm 67$

5. (i) $\D b = 2;A = 0.273,$

(ii) $\D 14.9$

6. (i) $\D 12 000$

(ii) $\D 2$

7. (i) $\D b=3,A=5.01$ or $\D 10.7$

(ii) $\D y=315$ or $\D 102.5$

(iii) $\D x = 0.693$

8. The number of bacteria at the

start of the experiment

$\D 1.61,100 000$

9. (i) $\D 1000,$

(ii) $\D 2.77$

(iii) $\D 7389$

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