Questions - Edexcel P2 - A-Level Maths

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Hence find the coordinates of the stationary points of the curve.

Edexcel P2 2022 June Q9

9. A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k \lambda ^ { t }$$ where

$N$ grams is the amount of carbon-14 currently present in the item
$k$ grams was the initial amount of carbon-14 present in the item
$t$ is the number of years since the item was made
$\lambda$ is a constant, with $0 < \lambda < 1$
1. Sketch the graph of $N$ against $t$ for $k = 1$

Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,

show that the value of the constant $\lambda$ is 0.999878 to 6 decimal places. Given that Item $A$

is known to have had 15 grams of carbon-14 present initially
is thought to be 3250 years old
calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item $A$.

Item $B$ is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.

Use algebra to calculate the age of Item $B$ to the nearest 100 years.

Edexcel P2 2022 June Q10

10. The circle $C$ has centre $X ( 3,5 )$ and radius $r$ The line $l$ has equation $y = 2 x + k$, where $k$ is a constant.

Show that $l$ and $C$ intersect when $$5 x ^ { 2 } + ( 4 k - 26 ) x + k ^ { 2 } - 10 k + 34 - r ^ { 2 } = 0$$ Given that $l$ is a tangent to $C$,
show that $5 r ^ { 2 } = ( k + p ) ^ { 2 }$, where $p$ is a constant to be found. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db4ec300-8081-4d29-acd5-0aae789d8f95-28_636_572_902_687} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line $l$
- cuts the $y$-axis at the point $A$
- touches the circle $C$ at the point $B$
  as shown in Figure 2.
  Given that $A B = 2 r$
- find the value of $k$

Edexcel P2 2023 June Q1

The continuous curve $C$ has equation $y = \mathrm { f } ( x )$.

A table of values of $x$ and $y$ for $y = \mathrm { f } ( x )$ is shown below.

$x$	4.0	4.2	4.4	4.6	4.8	5.0
$y$	9.2	8.4556	3.8512	5.0342	7.8297	8.6

Use the trapezium rule with all the values of $y$ in the table to find an approximation for $$\int _ { 4 } ^ { 5 } f ( x ) d x$$ giving your answer to 3 decimal places.

Edexcel P2 2023 June Q2

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } + 5 x + a$$ where $a$ is a constant.
Given that ( $2 x - 3$ ) is a factor of $\mathrm { f } ( x )$,

use the factor theorem to show that $a = - 3$
Hence show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.

Edexcel P2 2023 June Q3

A circle $C$ has centre $( 2,5 )$

Given that the point $P ( 8 , - 3 )$ lies on $C$

1. find the radius of $C$
2. find an equation for $C$
Find the equation of the tangent to $C$ at $P$ giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers to be found.

Edexcel P2 2023 June Q4

The binomial expansion, in ascending powers of $x$, of

$$( 3 + p x ) ^ { 5 }$$ where $p$ is a constant, can be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$ where $A$, $B$, $C$ and $D$ are constants.

Find the value of $A$ Given that
- $B = 18 D$
- $p < 0$
- find
  1. the value of $p$
  2. the value of $C$

Edexcel P2 2023 June Q5

Use the laws of logarithms to solve

$$\log _ { 2 } ( 16 x ) + \log _ { 2 } ( x + 1 ) = 3 + \log _ { 2 } ( x + 6 )$$

Edexcel P2 2023 June Q6

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A software developer released an app to download.
The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were $$2 k - 15 \quad k \quad k + 4$$ where $k$ is a constant.
Given that the numbers of downloads each month are modelled as a geometric series,

show that $k ^ { 2 } - 7 k - 60 = 0$
predict the number of downloads in the 4th month. The total number of all downloads of the app is predicted to exceed 3 million for the first time in the $N$ th month.
Calculate the value of $N$ according to the model.

Edexcel P2 2023 June Q7

The height of a river above a fixed point on the riverbed was monitored over a 7-day period.

The height of the river, $H$ metres, $t$ days after monitoring began, was given by $$H = \frac { \sqrt { t } } { 20 } \left( 20 + 6 t - t ^ { 2 } \right) + 17 \quad 0 \leqslant t \leqslant 7$$ Given that $H$ has a stationary value at $t = \alpha$

use calculus to show that $\alpha$ satisfies the equation $$5 \alpha ^ { 2 } - 18 \alpha - 20 = 0$$
Hence find the value of $\alpha$, giving your answer to 3 decimal places.
Use further calculus to prove that $H$ is a maximum at this value of $\alpha$.

Edexcel P2 2023 June Q8

(i) A student writes the following statement:
"When $a$ and $b$ are consecutive prime numbers, $a ^ { 2 } + b ^ { 2 }$ is never a multiple of 10 "
Prove by counter example that this statement is not true.
(ii) Given that $x$ and $y$ are even integers greater than 0 and less than 6 , prove by exhaustion, that

$$1 < x ^ { 2 } - \frac { x y } { 4 } < 15$$

Edexcel P2 2023 June Q9

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
Hence solve, for $0 < x < 360 ^ { \circ }$ $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.

Edexcel P2 2023 June Q10

The curve $C$ has equation

$$y = \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$ where $k$ is a positive constant.

Show that $$\int _ { 1 } ^ { 16 } \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \mathrm {~d} x = a k ^ { 2 } + b k + \frac { 2046 } { 5 }$$ where $a$ and $b$ are integers to be found. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e3b364c-151b-471d-acb6-01afb018fb75-26_645_670_904_699} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve $C$ and the line $l$.
Given that $l$ intersects $C$ at the point $A ( 1,9 )$ and at the point $B ( 16 , q )$ where $q$ is a constant,
show that $k = 4$ The region $R$, shown shaded in Figure 1, is bounded by $C$ and $l$
Using the answers to parts (a) and (b),
find the area of region $R$

Edexcel P2 2023 June Q11

A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n }
u _ { 1 } & = 3 \end{aligned}$$ where $a$ and $b$ are constants.

Find, in terms of $a$ and $b$,
1. $u _ { 2 }$
2. $u _ { 3 }$ Given
  - $\sum _ { n = 1 } ^ { 3 } u _ { n } = 153$
$b = a + 9$
show that
Hence find the larger possible value of $u _ { 2 }$

Edexcel P2 2024 June Q1

(a) Find the first four terms, in ascending powers of $x$, of the binomial expansion of

$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of $x ^ { 3 }$ in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.

Edexcel P2 2024 June Q2

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

In an arithmetic series,

the sixth term is 2
the sum of the first ten terms is - 80

For this series,

find the value of the first term and the value of the common difference.
Hence find the smallest value of $n$ for which $$S _ { n } > 8000$$

Edexcel P2 2024 June Q3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
1. Using the laws of logarithms, solve
$$2 \log _ { 2 } ( 2 - x ) = 4 + \log _ { 2 } ( x + 10 )$$
Find the value of $$\log _ { \sqrt { a } } a ^ { 6 }$$ where $a$ is a positive constant greater than 1

Edexcel P2 2024 June Q4

4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where $k$ is a constant.

State the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ) Given that ( $2 x - 1$ ) is a factor of $\mathrm { f } ( x )$
show that $k = 11$
Hence
1. fully factorise $\mathrm { f } ( x )$,
2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.

Edexcel P2 2024 June Q5

In this question you must show detailed reasoning.
1. Given that $x$ and $y$ are positive numbers such that
$$( x - y ) ^ { 3 } > x ^ { 3 } - y ^ { 3 }$$ prove that $$y > x$$
Using a counter example, show that the result in part (a) is not true for all real numbers.

Edexcel P2 2024 June Q6

(a) Sketch the curve with equation

$$y = a ^ { x } + 4$$ where $a$ is a positive constant greater than 1
On your sketch, show

the coordinates of the point of intersection of the curve with the $y$-axis
the equation of the asymptote of the curve

$x$	2	2.3	2.6	2.9	3.2	3.5
$y$	0	0.3246	0.8629	1.6643	2.7896	4.3137

The table shows corresponding values of $x$ and $y$ for $$y = 2 ^ { x } - 2 x$$ with the values of $y$ given to 4 decimal places as appropriate.
Using the trapezium rule with all the values of $y$ in the given table,
(b) obtain an estimate for $\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } - 2 x \right) \mathrm { d } x$, giving your answer to 2 decimal places.
(c) Using your answer to part (b) and making your method clear, estimate

$\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } + 2 x \right) \mathrm { d } x$
$\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x + 1 } - 4 x \right) \mathrm { d } x$

Edexcel P2 2024 June Q7

The circle $C _ { 1 }$ has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$

1. Find the coordinates of the centre of $C _ { 1 }$
2. Find the exact value of the radius of $C _ { 1 }$ In part (b) you must show detailed reasoning.
  The circle $C _ { 2 }$ has equation $$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
Prove that the circles $C _ { 1 }$ and $C _ { 2 }$ neither touch nor intersect.

Edexcel P2 2024 June Q8

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
1. Solve, for $0 < x \leqslant \pi$, the equation
$$5 \sin x \tan x + 13 = \cos x$$ giving your answer in radians to 3 significant figures.
The temperature inside a greenhouse is monitored on one particular day. The temperature, $H ^ { \circ } \mathrm { C }$, inside the greenhouse, $t$ hours after midnight, is modelled by the equation $$H = 10 + 12 \sin ( k t + 18 ) ^ { \circ } \quad 0 \leqslant t < 24$$ where $k$ is a constant.
Use the equation of the model to answer parts (a) to (c).
Given that
- the temperature inside the greenhouse was $20 ^ { \circ } \mathrm { C }$ at 6 am
- $0 < k < 20$
  (a) find all possible values for $k$, giving each answer to 2 decimal places.
Given further that $0 < k < 10$
(b) find the maximum temperature inside the greenhouse,
(c) find the time of day at which this maximum temperature occurs. Give your answer to the nearest minute.

Edexcel P2 2024 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b822842d-ee62-40ce-a8de-967e556a80a8-26_915_912_255_580} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 is a sketch of the curve $C$ with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point $P$ is the stationary point of $C$.

Find, using calculus, the $x$ coordinate of $P$. The region $R _ { 1 }$, shown shaded in Figure 1, is bounded by $C$ and the $x$-axis.
The region $R _ { 2 }$, also shown shaded in Figure 1, is bounded by $C$, the $x$-axis and the line with equation $x = k$, where $k$ is a constant. Given that the area of $R _ { 1 }$ is equal to the area of $R _ { 2 }$
find, using calculus, the exact value of $k$.

Edexcel P2 2024 June Q10

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by $3 \%$ each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
Given that
- 4 years after monitoring began there were 3690 voles on the island
- 7 years after monitoring began there were 3470 voles on the island
- the number of voles on the island at the end of each year is modelled as a geometric sequence
- find the equation of this model in the form
$$N = a b ^ { t }$$ where $N$ is the number of voles, $t$ years after monitoring began and $a$ and $b$ are constants. Give the value of $a$ and the value of $b$ to 2 significant figures. When $t = T$, the number of dormice on the island is equal to the number of voles on the island.
Find, according to the models, the value of $T$, giving your answer to one decimal place.

Edexcel P2 2019 October Q1

A curve $C$ has equation $y = 2 x ^ { 2 } ( x - 5 )$
1. Find, using calculus, the $x$ coordinates of the stationary points of $C$.
2. Hence find the values of $x$ for which $y$ is increasing.

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