Questions - Edexcel - A-Level Maths

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Edexcel AS Paper 1 2023 June Q11

6 marks Moderate -0.8

The height, $h$ metres, of a plant, $t$ years after it was first measured, is modelled by the equation

$$h = 2.3 - 1.7 \mathrm { e } ^ { - 0.2 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ Using the model,

find the height of the plant when it was first measured,
show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year. According to the model, there is a limit to the height to which this plant can grow.
Deduce the value of this limit.

Edexcel AS Paper 1 2023 June Q12

9 marks Moderate -0.3

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Show that the equation $$4 \tan x = 5 \cos x$$ can be written as $$5 \sin ^ { 2 } x + 4 \sin x - 5 = 0$$
Hence solve, for $0 < x \leqslant 360 ^ { \circ }$ $$4 \tan x = 5 \cos x$$ giving your answers to one decimal place.
Hence find the number of solutions of the equation $$4 \tan 3 x = 5 \cos 3 x$$ in the interval $0 < x \leqslant 1800 ^ { \circ }$, explaining briefly the reason for your answer.

Edexcel AS Paper 1 2023 June Q13

7 marks Moderate -0.8

Relative to a fixed origin $O$

point $A$ has position vector $10 \mathbf { i } - 3 \mathbf { j }$
point $B$ has position vector $- 8 \mathbf { i } + 9 \mathbf { j }$
point $C$ has position vector $- 2 \mathbf { i } + p \mathbf { j }$ where $p$ is a constant
1. Find $\overrightarrow { A B }$
2. Find $| \overrightarrow { A B } |$ giving your answer as a fully simplified surd.

Given that points $A , B$ and $C$ lie on a straight line,

find the value of $p$,
state the ratio of the area of triangle $A O C$ to the area of triangle $A O B$.

Edexcel AS Paper 1 2023 June Q14

5 marks Standard +0.3

Find, in simplest form, the coefficient of $x ^ { 5 }$ in the expansion of

$$\left( 5 + 8 x ^ { 2 } \right) \left( 3 - \frac { 1 } { 2 } x \right) ^ { 6 }$$

Edexcel AS Paper 1 2023 June Q15

7 marks Moderate -0.3

In this question you must show detailed reasoning.

\section*{Solutions relying on calculator technology are not acceptable.} The curve $C _ { 1 }$ has equation $y = 8 - 10 x + 6 x ^ { 2 } - x ^ { 3 }$ The curve $C _ { 2 }$ has equation $y = x ^ { 2 } - 12 x + 14$

Verify that when $x = 1$ the curves $C _ { 1 }$ and $C _ { 2 }$ intersect. The curves also intersect when $x = k$.
Given that $k < 0$
use algebra to find the exact value of $k$.

Edexcel AS Paper 1 2023 June Q16

6 marks Standard +0.3

A curve has equation $y = \mathrm { f } ( x ) , x \geqslant 0$

Given that

$\mathrm { f } ^ { \prime } ( x ) = 4 x + a \sqrt { x } + b$, where $a$ and $b$ are constants
the curve has a stationary point at $( 4,3 )$
the curve meets the $y$-axis at - 5
find $\mathrm { f } ( x )$, giving your answer in simplest form.

Edexcel AS Paper 1 2023 June Q17

5 marks Standard +0.3

In this question $p$ and $q$ are positive integers with $q > p$

Statement 1: $q ^ { 3 } - p ^ { 3 }$ is never a multiple of 5

Show, by means of a counter example, that Statement 1 is not true. Statement 2: When $p$ and $q$ are consecutive even integers $q ^ { 3 } - p ^ { 3 }$ is a multiple of 8
Prove, using algebra, that Statement 2 is true.

Edexcel AS Paper 1 2024 June Q1

4 marks Moderate -0.8

Find

$$\int \frac { 2 \sqrt { x } - 3 } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.

Edexcel AS Paper 1 2024 June Q2

8 marks Moderate -0.8

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$ where $a$ and $b$ are constants.
Given that ( $x - 4$ ) is a factor of $\mathrm { f } ( x )$,

use the factor theorem to show that $$10 a = 32 + b$$ Given also that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$,
express $\mathrm { f } ( x )$ in the form $$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$ where $k$ is a constant to be found.
Hence,
1. state the number of real roots of the equation $\mathrm { f } ( x ) = 0$
2. write down the largest root of the equation $\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0$

Edexcel AS Paper 1 2024 June Q3

8 marks Moderate -0.8

Relative to a fixed origin $O$,

point $P$ has position vector $9 \mathbf { i } - 8 \mathbf { j }$
point $Q$ has position vector $3 \mathbf { i } - 5 \mathbf { j }$
1. Find $\overrightarrow { P Q }$

Given that $R$ is the point such that $\overrightarrow { Q R } = 9 \mathbf { i } + 18 \mathbf { j }$

show that angle $P Q R = 90 ^ { \circ }$ Given also that $S$ is the point such that $\overrightarrow { P S } = 3 \overrightarrow { Q R }$

find the exact area of $P Q R S$

Edexcel AS Paper 1 2024 June Q4

5 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-10_547_1475_306_294} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of triangle $A B D$ and triangle $B C D$ Given that

$A D C$ is a straight line
$B D = ( x + 3 ) \mathrm { cm }$
$B C = x \mathrm {~cm}$
angle $B D C = 30 ^ { \circ }$
angle $B C D = 140 ^ { \circ }$
1. show that $x = 10.5$ correct to 3 significant figures.

Given also that $A D = ( x - 2 ) \mathrm { cm }$

find the length of $A B$, giving your answer to 3 significant figures.

Edexcel AS Paper 1 2024 June Q5

9 marks Moderate -0.3

The curve $C _ { 1 }$ has equation

$$y = \frac { 6 } { x } + 3$$

1. Sketch $C _ { 1 }$ stating the coordinates of any points where the curve cuts the coordinate axes.
2. State the equations of any asymptotes to the curve $C _ { 1 }$ The curve $C _ { 2 }$ has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
Show that $C _ { 1 }$ and $C _ { 2 }$ intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the $x$ coordinate of one of the points of intersection is $- \frac { 2 } { 3 }$
use algebra to find the $x$ coordinates of the other points of intersection between $C _ { 1 }$ and $C _ { 2 }$ (Solutions relying on calculator technology are not acceptable.)

Edexcel AS Paper 1 2024 June Q6

6 marks Standard +0.3

The binomial expansion of

$$( 1 + a x ) ^ { 12 }$$ up to and including the term in $x ^ { 2 }$ is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where $a$ and $k$ are constants.

Show that $a = - \frac { 5 } { 8 }$
Hence find the value of $k$ Using the expansion and making your method clear,
find an estimate for the value of $\left( \frac { 17 } { 16 } \right) ^ { 12 }$, giving your answer to 4 decimal places.

Edexcel AS Paper 1 2024 June Q7

5 marks Moderate -0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-18_614_878_296_555} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A chimney emits smoke particles.
On a particular day, the concentration of smoke particles in the air emitted by this chimney, $P$ parts per million, is measured at various distances, $x \mathrm {~km}$, from the chimney. Figure 2 shows a sketch of the linear relationship between $\log _ { 10 } P$ and $x$ that is used to model this situation. The line passes through the point ( $0,3.3$ ) and the point ( $6,2.1$ )

Find a complete equation for the model in the form $$P = a b ^ { x }$$ where $a$ and $b$ are constants. Give the value of $a$ and the value of $b$ each to 4 significant figures.
With reference to the model, interpret the value of $a b$

Edexcel AS Paper 1 2024 June Q8

10 marks Moderate -0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-20_915_924_303_580} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of the curve $C$ with equation $$y = x ^ { 3 } - 14 x + 23$$ The line $l$ is the tangent to $C$ at the point $A$, also shown in Figure 3.
Given that $l$ has equation $y = - 2 x + 7$

show, using calculus, that the $x$ coordinate of $A$ is 2 The line $l$ cuts $C$ again at the point $B$.
Verify that the $x$ coordinate of $B$ is - 4 The finite region, $R$, shown shaded in Figure 3, is bounded by $C$ and $l$.
Using algebraic integration,
show that the area of $R$ is 108

Edexcel AS Paper 1 2024 June Q9

5 marks Moderate -0.3

9. $$\begin{aligned} p & = \log _ { a } 16 \\ q & = \log _ { a } 25 \end{aligned}$$ where $a$ is a constant.
Find in terms of $p$ and/or $q$,

$\log _ { a } 256$
$\log _ { a } 100$
$\log _ { a } 80 \times \log _ { a } 3.2$

Edexcel AS Paper 1 2024 June Q10

8 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-26_748_764_296_646} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the circle $C$

the point $P ( - 1 , k + 8 )$ is the centre of $C$
the point $Q \left( 3 , k ^ { 2 } - 2 k \right)$ lies on $C$
$k$ is a positive constant
the line $l$ is the tangent to $C$ at $Q$

Given that the gradient of $l$ is - 2

show that $$k ^ { 2 } - 3 k - 10 = 0$$
Hence find an equation for $C$

Edexcel AS Paper 1 2024 June Q11

6 marks Standard +0.3

The prices of two precious metals are being monitored.

The price per gram of metal $A , \pounds V _ { A }$, is modelled by the equation $$V _ { A } = 100 + 20 \mathrm { e } ^ { 0.04 t }$$ where $t$ is the number of months after monitoring began.
The price per gram of metal $B , \pounds V _ { B }$, is modelled by the equation $$V _ { B } = p \mathrm { e } ^ { - 0.02 t }$$ where $p$ is a positive constant and $t$ is the number of months after monitoring began.
Given that $V _ { B } = 2 V _ { A }$ when $t = 0$

find the value of $p$ When $t = T$, the rate of increase in the price per gram of metal $A$ was equal to the rate of decrease in the price per gram of metal $B$
Find the value of $T$, giving your answer to one decimal place.
(Solutions based entirely on calculator technology are not acceptable.)

Edexcel AS Paper 1 2024 June Q12

13 marks Standard +0.3

12. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-34_494_499_306_778} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows the plan view of the design for a swimming pool.
The pool is modelled as a quarter of a circle joined to two equal sized rectangles as shown. Given that

the quarter circle has radius $x$ metres
the rectangles each have length $x$ metres and width $y$ metres
the total surface area of the swimming pool is $100 \mathrm {~m} ^ { 2 }$
1. show that, according to the model, the perimeter $P$ metres of the swimming pool is given by

$$P = 2 x + \frac { 200 } { x }$$

Use calculus to find the value of $x$ for which $P$ has a stationary value.

Prove, by further calculus, that this value of $x$ gives a minimum value for $P$ Access to the pool is by side $A B$ shown in Figure 5.
Given that $A B$ must be at least one metre,

determine, according to the model, whether the swimming pool with the minimum perimeter would be suitable.

Edexcel AS Paper 1 2024 June Q13

9 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
1. Show that the equation
$$\sin \theta ( 7 \sin \theta - 4 \cos \theta ) = 4$$ can be written as $$3 \tan ^ { 2 } \theta - 4 \tan \theta - 4 = 0$$
Hence solve, for $0 < x < 360 ^ { \circ }$ $$\sin x ( 7 \sin x - 4 \cos x ) = 4$$ giving your answers to one decimal place.
Hence find the smallest solution of the equation $$\sin 4 \alpha ( 7 \sin 4 \alpha - 4 \cos 4 \alpha ) = 4$$ in the range $720 ^ { \circ } < \alpha < 1080 ^ { \circ }$, giving your answer to one decimal place.