1.06a - OCR Spec

Edexcel C3 Q2

8 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c8b85e00-4549-4219-a75d-85f67ccb8e16-2_638_675_644_445} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curves $y = 3 + 2 \mathrm { e } ^ { x }$ and $y = \mathrm { e } ^ { x + 2 }$ which cross the $y$-axis at the points $A$ and $B$ respectively.

Find the exact length $A B$. The two curves intersect at the point $C$.
Find an expression for the $x$-coordinate of $C$ and show that the $y$-coordinate of $C$ is $\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }$.

Edexcel C3 Q5

11 marks Moderate -0.3

5. $$\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , \quad x \in \mathbb { R } .$$

State the range of f .
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.
Solve the equation $\mathrm { f } ( x ) = 7$.
Find an equation for the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $y = 7$.

AQA C4 2011 January Q5

7 marks Moderate -0.3

5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after $t$ days is $m$ grams. Its initial mass is $m _ { 0 }$ grams.

Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
A mass of $m _ { 0 }$ grams of this form of iodine decays to $\frac { m _ { 0 } } { 16 }$ grams in $d$ days. Find the value of $d$.
After $n$ days, a mass of this form of iodine has decayed to less than $1 \%$ of its initial mass. Find the minimum integer value of $n$.

AQA C4 2012 January Q4

9 marks Standard +0.3

4 A scientist is testing models for the growth and decay of colonies of bacteria. For a particular colony, which is growing, the model is $P = A \mathrm { e } ^ { \frac { 1 } { 8 } t }$, where $P$ is the number of bacteria after a time $t$ minutes and $A$ is a constant.

This growing colony consists initially of 500 bacteria. Calculate the number of bacteria, according to the model, after one hour. Give your answer to the nearest thousand.
For a second colony, which is decaying, the model is $Q = 500000 \mathrm { e } ^ { - \frac { 1 } { 8 } t }$, where $Q$ is the number of bacteria after a time $t$ minutes. Initially, the growing colony has 500 bacteria and, at the same time, the decaying colony has 500000 bacteria.
1. Find the time at which the populations of the two colonies will be equal, giving your answer to the nearest 0.1 of a minute.
2. The population of the growing colony will exceed that of the decaying colony by 45000 bacteria at time $T$ minutes. Show that $$\left( \mathrm { e } ^ { \frac { 1 } { 8 } T } \right) ^ { 2 } - 90 \mathrm { e } ^ { \frac { 1 } { 8 } T } - 1000 = 0$$ and hence find the value of $T$, giving your answer to one decimal place.
  (4 marks)

AQA C4 2013 January Q7

13 marks Standard +0.3

7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model $$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$ for the number of rodents, $N$, in the population $t$ weeks after the start of the investigation. Use this model to answer the following questions.

1. Find the size of the population at the start of the investigation.
2. Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
3. Find the number of weeks that it will take the population to reach 400 . Give your answer in the form $t = r \ln s$, where $r$ and $s$ are integers.
1. Show that the rate of growth, $\frac { \mathrm { d } N } { \mathrm {~d} t }$, is given by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
2. The maximum rate of growth occurs after $T$ weeks. Find the value of $T$.

AQA C4 2011 June Q2

6 marks Moderate -0.8

2 The average weekly pay of a footballer at a certain club was $\pounds 80$ on 1 August 1960. By 1 August 1985, this had risen to $\pounds 2000$. The average weekly pay of a footballer at this club can be modelled by the equation $$P = A k ^ { t }$$ where $\pounds P$ is the average weekly pay $t$ years after 1 August 1960, and $A$ and $k$ are constants.

1. Write down the value of $A$.
2. Show that the value of $k$ is 1.137411 , correct to six decimal places.
Use this model to predict the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed $\pounds 100000$.

AQA C4 2012 June Q4

8 marks Easy -1.2

4 The value, $\pounds V$, of an initial investment, $\pounds P$, at the end of $n$ years is given by the formula $$V = P \left( 1 + \frac { r } { 100 } \right) ^ { n }$$ where $r \%$ per year is the fixed interest rate.
Mr Brown invests $\pounds 1000$ in Barcelona Bank at a fixed interest rate of $3 \%$ per year.

1. Find the value of Mr Brown's investment at the end of 5 years. Give your value to the nearest $\pounds 10$.
2. The value of Mr Brown's investment will first exceed $\pounds 2000$ after $N$ complete years. Find the value of $N$.
Mrs White invests $\pounds 1500$ in Bilbao Bank at a fixed interest rate of $1.5 \%$ per year. Mr Brown and Mrs White invest their money at the same time. The value of Mr Brown's investment will first exceed the value of Mrs White's investment after $T$ complete years. Find the value of $T$.

OCR MEI Further Numerical Methods 2021 November Q3

7 marks Standard +0.3

3 The method of False Position is used to find a sequence of approximations to the root of an equation. The spreadsheet output showing these approximations, together with some further analysis, is shown below.

	C	D	E	F	G	H	I	J
4	a	f(a)	b	$\mathrm { f } ( b )$	$x _ { \text {new } }$	$\mathrm { f } \left( x _ { \text {new } } \right)$	difference	ratio
5	1	-1.8248	2	17.2899	1.09547	-1.80507
6	1.09547	-1.80507	2	17.2899	1.18097	-1.75418	0.08551
7	1.18097	-1.75418	2	17.2899	1.25641	-1.66246	0.07544	0.88229
8	1.25641	-1.66246	2	17.2899	1.32164	-1.52781	0.06523	0.86458
9	1.32164	-1.52781	2	17.2899	1.37672	-1.35706	0.05508	0.84439
10	1.37672	-1.35706	2	17.2899	1.42208	-1.1642	0.04536	0.8236
11	1.42208	-1.1642	2	17.2899	1.45853	-0.96616	0.03646	0.80376
12	1.45853	-0.96616	2	17.2899	1.48719	-0.77825	0.02866	0.78598
13	1.48719	-0.77825	2	17.2899
14

The formula in cell D5 is $\quad = \mathrm { SINH } \left( \mathrm { C5 } ^ { \wedge } 2 \right) - \mathrm { C5 } ^ { \wedge } 3 - 2$.

Write down the equation which is being solved. The formula in cell C 6 is $\quad = \mathrm { IF } ( \mathrm { H } 5 < 0 , \mathrm { G } 5 , \mathrm { C } 5 )$.
Write down a similar formula for cell E6.
Calculate the values which would be displayed in cells G13 and G14 to find further approximations to the root.
Explain what the values in column J tell you about

Edexcel FM2 AS 2022 June Q4

10 marks Standard +0.3

A particle $P$ moves on the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ in the direction of $x$ increasing, where

$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of $P$ at time $t$ seconds is $a \mathrm {~ms} ^ { - 2 }$

Show that $a = 2 v + 1$
Find the acceleration of $P$ when $t = 0$
Find the exact distance travelled by $P$ in accelerating from a speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to a speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Edexcel FM2 AS 2023 June Q2

8 marks Standard +0.3

A particle $P$ is moving along the $x$-axis.

At time $t$ seconds, $t \geqslant 0 , P$ has acceleration $a \mathrm {~ms} ^ { - 2 }$ and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing, where $$v = \mathrm { e } ^ { 2 t } + 6 \mathrm { e } ^ { t } - k t$$ and $k$ is a positive constant.
When $t = \ln 2$, $a = 0$

Find the value of $k$. When $t = 0$, the particle passes through the fixed point $A$.
When $t = \ln 2$, the particle is $d$ metres from $A$.
Showing all stages of your working, find the value of $d$ correct to 2 significant figures.
[0pt] [Solutions relying entirely on calculator technology are not acceptable.]

OCR MEI C2 Q6

5 marks Moderate -0.8

On the copy of Fig. 5, draw by eye a tangent to the curve at the point where $x = 2$. Hence find an estimate of the gradient of $y = 2 ^ { x }$ when $x = 2$.
Calculate the $y$-values on the curve when $x = 1.8$ and $x = 2.2$. Hence calculate another approximation to the gradient of $y = 2 ^ { x }$ when $x = 2$.

AQA C2 2007 June Q6

10 marks Moderate -0.8

6 The diagram shows a sketch of the curve with equation $y = 3 \left( 2 ^ { x } + 1 \right)$. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-5_465_851_390_607} The curve $y = 3 \left( 2 ^ { x } + 1 \right)$ intersects the $y$-axis at the point $A$.

Find the $y$-coordinate of the point $A$.
Use the trapezium rule with four ordinates (three strips) to find an approximate value for $\int _ { 0 } ^ { 6 } 3 \left( 2 ^ { x } + 1 \right) d x$.
The line $y = 21$ intersects the curve $y = 3 \left( 2 ^ { x } + 1 \right)$ at the point $P$.
1. Show that the $x$-coordinate of $P$ satisfies the equation $$2 ^ { x } = 6$$
2. Use logarithms to find the $x$-coordinate of $P$, giving your answer to three significant figures.

AQA C3 Q5

Standard +0.3

5 The diagram shows part of the graph of $y = \mathrm { e } ^ { 2 x } - 9$. The graph cuts the coordinate axes at ( $0 , a$ ) and ( $b , 0$ ). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}

State the value of $a$, and show that $b = \ln 3$.
Show that $y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81$.
The shaded region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the volume of the solid formed, giving your answer in the form $\pi ( p \ln 3 + q )$, where $p$ and $q$ are integers.
Sketch the curve with equation $y = \left| \mathrm { e } ^ { 2 x } - 9 \right|$ for $x \geqslant 0$.

AQA C3 2006 January Q5

12 marks Standard +0.3

5 The diagram shows part of the graph of $y = \mathrm { e } ^ { 2 x } - 9$. The graph cuts the coordinate axes at $( 0 , a )$ and $( b , 0 )$. \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-03_826_924_477_541}

State the value of $a$, and show that $b = \ln 3$.
Show that $y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81$.
The shaded region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the volume of the solid formed, giving your answer in the form $\pi ( p \ln 3 + q )$, where $p$ and $q$ are integers.
Sketch the curve with equation $y = \left| \mathrm { e } ^ { 2 x } - 9 \right|$ for $x \geqslant 0$.

AQA C3 2010 January Q6

11 marks Moderate -0.3

6 The functions $f$ and $g$ are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - 3 , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { 3 x + 4 } , & \text { for real values of } x , x \neq - \frac { 4 } { 3 } \end{array}$$

Find the range of $f$.
The inverse of f is $\mathrm { f } ^ { - 1 }$.
1. Find $\mathrm { f } ^ { - 1 } ( x )$.
2. Solve the equation $\mathrm { f } ^ { - 1 } ( x ) = 0$.
1. Find an expression for $\operatorname { gf } ( x )$.
2. Solve the equation $\mathrm { gf } ( x ) = 1$, giving your answer in an exact form.

AQA C4 2005 June Q8

14 marks Moderate -0.3

8

A cup of coffee is cooling down in a room. At time $t$ minutes after the coffee is made, its temperature is $x ^ { \circ } \mathrm { C }$, where $$x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }$$
1. Find the temperature of the coffee when it is made.
2. Find the temperature of the coffee 30 minutes after it is made.
3. Find how long it will take for the coffee to cool down to $60 ^ { \circ } \mathrm { C }$.
1. Use integration to solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 40 } ( x - 15 ) , \quad x > 15$$ given that $x = 85$ when $t = 0$, expressing $t$ in terms of $x$.
2. Hence show that $x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }$.

AQA C4 2007 June Q4

11 marks Moderate -0.8

4 A biologist is researching the growth of a certain species of hamster. She proposes that the length, $x \mathrm {~cm}$, of a hamster $t$ days after its birth is given by $$x = 15 - 12 \mathrm { e } ^ { - \frac { t } { 14 } }$$

Use this model to find:
1. the length of a hamster when it is born;
2. the length of a hamster after 14 days, giving your answer to three significant figures.
1. Show that the time for a hamster to grow to 10 cm in length is given by $t = 14 \ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers.
2. Find this time to the nearest day.
1. Show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 14 } ( 15 - x )$$
2. Find the rate of growth of the hamster, in cm per day, when its length is 8 cm .
  (1 mark)

AQA C4 2008 June Q8

9 marks Moderate -0.5

8

The number of fish in a lake is decreasing. After $t$ years, there are $x$ fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
1. Formulate a differential equation, in the variables $x$ and $t$ and a constant of proportionality $k$, where $k > 0$, to model the rate at which the number of fish in the lake is decreasing.
2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of $k$.
The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, $P$, in another lake, where $t$ is the time in years and $A$ is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
1. Taking 1 January 2008 as $t = 0$, find the value of $A$.
2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.

AQA M2 Q3

Moderate -0.8

3 A particle moves in a straight line and at time $t$ has velocity $v$, where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. Find an expression for the acceleration of the particle at time $t$.
2. State the range of values of the acceleration of the particle.
When $t = 0$, the particle is at the origin. Find an expression for the displacement of the particle from the origin at time $t$.
(4 marks)

AQA Paper 1 2020 June Q14

9 marks Standard +0.3

14 The function f is defined by $$f ( x ) = 3 ^ { x } \sqrt { x } - 1 \quad \text { where } x \geq 0$$ 14

$\quad \mathrm { f } ( x ) = 0$ has a single solution at the point $x = \alpha$ By considering a suitable change of sign, show that $\alpha$ lies between 0 and 1
14

(i) Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 ^ { x } ( 1 + x \ln 9 ) } { 2 \sqrt { x } }$$

14 (b) (ii) Use the Newton-Raphson method with $x _ { 1 } = 1$ to find $x _ { 3 }$, an approximation for $\alpha$.

Give your answer to five decimal places.

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

14 (b) (iii) Explain why the Newton-Raphson method fails to find $\alpha$ with $x _ { 1 } = 0$

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

Edexcel PURE 2024 October Q8

Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-24_472_595_246_735} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The heart rate of a horse is being monitored.
The heart rate $H$, measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where $t$ minutes is the time after monitoring began.
Figure 4 is a sketch of $H$ against $t$. \section*{Use the equation of the model to answer parts (a) to (e).}

State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches $L \mathrm { bpm }$.
State the value of $L$. The heart rate of the horse reaches its maximum value after $T$ minutes.
Find the value of $T$, giving your answer to 3 decimal places.
(Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after $M$ minutes.
Show that $M$ is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
1. find, to 4 decimal places, the value of $t _ { 2 }$
2. find, to 4 decimal places, the value of $M$

Pre-U Pre-U 9794/2 2016 Specimen Q6

8 marks Moderate -0.3

6 Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time $t$ hours after the injection, the concentration of Antiflu in Diane's bloodstream is $3 \mathrm { e } ^ { - 0.02 t }$ units and the concentration of Coldcure is $5 \mathrm { e } ^ { - 0.07 t }$ units. Each drug becomes ineffective when its concentration falls below 1 unit.

Show that Coldcure becomes ineffective before Antiflu.
Sketch, on the same diagram, the graphs of concentration against time for each drug.
20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later.

Pre-U Pre-U 9794/2 2019 Specimen Q6