Questions P1 - A-Level Maths

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Edexcel P1 2022 June Q4

6 marks Moderate -0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-08_604_1207_251_370} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a curve with equation $y = \mathrm { f } ( x )$ The curve has a minimum at $P ( - 1,0 )$ and a maximum at $Q \left( \frac { 3 } { 2 } , 2 \right)$ The line with equation $y = 1$ is the only asymptote to the curve. On separate diagrams sketch the curves with equation

$y = \mathrm { f } ( x ) - 2$
$y = \mathrm { f } ( - x )$ On each sketch you must clearly state
- the coordinates of the maximum and minimum points
- the equation of the asymptote

Edexcel P1 2022 June Q5

9 marks Moderate -0.8

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

Given that

$\mathrm { f } ( x )$ is a quadratic expression
the maximum turning point on $C$ has coordinates $( - 2,12 )$
$C$ cuts the negative $x$-axis at - 5
1. find $\mathrm { f } ( x )$

The line $l _ { 1 }$ has equation $y = \frac { 4 } { 5 } x$ Given that the line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through $( - 5,0 )$

find an equation for $l _ { 2 }$, writing your answer in the form $y = m x + c$ where $m$ and $c$ are constants to be found.
(3) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ and the lines $l _ { 1 }$ and $l _ { 2 }$

Define the region $R$, shown shaded in Figure 2, using inequalities.

Edexcel P1 2022 June Q6

7 marks Standard +0.3

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

Given that $$2 x y - 3 x ^ { 2 } = 50$$ and $$y - x ^ { 3 } + 6 x = 0$$ show that $$2 x ^ { 4 } - 15 x ^ { 2 } - 50 = 0$$
Hence solve the simultaneous equations $$\begin{aligned} 2 x y - 3 x ^ { 2 } & = 50 \\ y - x ^ { 3 } + 6 x & = 0 \end{aligned}$$ Give your answers in fully simplified surd form. \includegraphics[max width=\textwidth, alt={}, center]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-14_2257_52_312_1982}

Edexcel P1 2022 June Q7

9 marks Moderate -0.3

7. The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$ Given that

$\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \sqrt { x } } + \frac { A } { x ^ { 2 } } + 3$, where $A$ is a constant
$\mathrm { f } ^ { \prime \prime } ( x ) = 0$ when $x = 4$
1. find the value of $A$.

Given also that

Edexcel P1 2022 June Q8

10 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-22_922_876_246_539} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the outline of the face of a ceiling fan viewed from below.
The fan consists of three identical sections congruent to $O A B C D O$, shown in Figure 3, where

$O A B O$ is a sector of a circle with centre $O$ and radius 9 cm
$O B C D O$ is a sector of a circle with centre $O$ and radius 84 cm
angle $A O D = \frac { 2 \pi } { 3 }$ radians

Given that the length of the arc $A B$ is 15 cm ,

show that the length of the arc $C D$ is 35.9 cm to one decimal place. The face of the fan is modelled to be a flat surface.
Find, according to the model,
the perimeter of the face of the fan, giving your answer to the nearest cm,
the surface area of the face of the fan. Give your answer to 3 significant figures and make your units clear.

Edexcel P1 2022 June Q9

8 marks Moderate -0.3

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-26_428_1354_251_287} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows part of the graph of the curve with equation $y = \sin x$ Given that $\sin \alpha = p$, where $0 < \alpha < 90 ^ { \circ }$

state, in terms of $p$, the value of
1. $2 \sin \left( 180 ^ { \circ } - \alpha \right)$
2. $\sin \left( \alpha - 180 ^ { \circ } \right)$
3. $3 + \sin \left( 180 ^ { \circ } + \alpha \right)$ A copy of Figure 4, labelled Diagram 1, is shown on page 27. On Diagram 1,
sketch the graph of $y = \sin 2 x$
Hence find, in terms of $\alpha$, the $x$ coordinates of any points in the interval $0 < x < 180 ^ { \circ }$ where $$\sin 2 x = p$$
\includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-27_433_1331_296_310}
\section*{Diagram 1}

Edexcel P1 2022 June Q10

12 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-28_655_869_255_541} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the curve $C$ with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where $k$ is a constant.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ The line $l$, shown in Figure 5, is the normal to $C$ at the point $A$ with $x$ coordinate $- \frac { 7 } { 2 }$ Given that $l$ is also a tangent to $C$ at the point $B$,
show that the $x$ coordinate of the point $B$ is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
Hence find the $x$ coordinate of $B$, justifying your answer. Given that the $y$ intercept of $l$ is - 1
find the value of $k$.
\includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}

Edexcel P1 2023 June Q1

4 marks Moderate -0.8

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

Solve the inequality $$4 x ^ { 2 } - 3 x + 7 \geq 4 x + 9$$

Edexcel P1 2023 June Q2

6 marks Easy -1.3

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

A rectangular sports pitch has length $x$ metres and width $y$ metres, where $x > y$ Given that the perimeter of the pitch is 350 m ,

write down an equation linking $x$ and $y$ Given also that the area of the pitch is $7350 \mathrm {~m} ^ { 2 }$
write down a second equation linking $x$ and $y$
hence find the value of $x$ and the value of $y$

Edexcel P1 2023 June Q3

6 marks Moderate -0.8

(a) Express $3 x ^ { 2 } + 12 x + 13$ in the form

$$a ( x + b ) ^ { 2 } + c$$ where $a$, $b$ and $c$ are integers to be found.
(b) Hence sketch the curve with equation $y = 3 x ^ { 2 } + 12 x + 13$ On your sketch show clearly

the coordinates of the $y$ intercept
the coordinates of the turning point of the curve

Edexcel P1 2023 June Q4

7 marks Moderate -0.8

In this question you must show all stages of your working.
1. Write
$$y = \frac { 5 x ^ { 2 } + \sqrt { x ^ { 3 } } } { \sqrt [ 3 ] { 8 x } }$$ in the form $$y = A x ^ { p } + B x ^ { q }$$ where $A , B , p$ and $q$ are constants to be found.
Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ giving each coefficient in simplest form.

Edexcel P1 2023 June Q5

10 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-10_488_784_310_667} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the plan for a garden.
In the plan

$O A$ and $C D$ are perpendicular to $O D$
$A B$ is an arc of the circle with centre $O$ and radius 4 metres
$\quad B C$ is parallel to $O D$
$O D$ is 6 metres, $O A$ is 4 metres and $C D$ is 1.5 metres
1. Show that angle $A O B$ is 1.186 radians to 4 significant figures.
2. Find the perimeter of the garden, giving your answer in metres to 3 significant figures.
3. Find the area of the garden, giving your answer in square metres to 3 significant figures.

Edexcel P1 2023 June Q6

7 marks Moderate -0.8

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
1. Expand and simplify
$$\left( r - \frac { 1 } { r } \right) ^ { 2 }$$
Express $\frac { 1 } { 3 + 2 \sqrt { 2 } }$ in the form $p + q \sqrt { 2 }$ where $p$ and $q$ are integers.
Use the results of parts (a) and (b), or otherwise, to show that $$\sqrt { 3 + 2 \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 + 2 \sqrt { 2 } } } = 2$$

Edexcel P1 2023 June Q7

6 marks Moderate -0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-18_737_951_301_587} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The region $R _ { 1 }$, shown shaded in Figure 2, is defined by the inequalities $$0 \leqslant y \leqslant 2 \quad y \leqslant 10 - 2 x \quad y \leqslant k x$$ where $k$ is a constant.
The line $x = a$, where $a$ is a constant, passes through the intersection of the lines $y = 2$ and $y = k x$ Given that the area of $R _ { 1 }$ is $\frac { 27 } { 4 }$ square units,

find
1. the value of $a$
2. the value of $k$
Define the region $R _ { 2 }$, also shown shaded in Figure 2, using inequalities.

Edexcel P1 2023 June Q8

10 marks Moderate -0.3

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

Solutions relying entirely on calculator technology are not acceptable.

Find the equation of the tangent to the curve with equation $$y = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$ at the point $P ( 4,12 )$ Give your answer in the form $a x + b y + c = 0$ where $a$, $b$ and $c$ are integers. The curve with equation $y = \mathrm { f } ( x )$ also passes through the point $P ( 4,12 )$ Given that $$f ^ { \prime } ( x ) = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$
find $\mathrm { f } ( x )$ giving the coefficients in simplest form.

Edexcel P1 2023 June Q9

9 marks Moderate -0.3

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_536_933_255_568} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows part of the graph of the trigonometric function with equation $y = \mathrm { f } ( x )$

Write down an expression for $\mathrm { f } ( x )$ On a separate diagram,
sketch, for $- 2 \pi < x < 2 \pi$, the graph of the curve with equation $y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)$ Show clearly the coordinates of all the points where the curve intersects the coordinate axes.
(ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows part of the graph of the trigonometric function with equation $y = \mathrm { g } ( x )$
Write down an expression for $\mathrm { g } ( x )$ On a separate diagram,
sketch, for $- 2 \pi < x < 2 \pi$, the graph of the curve with equation $y = \mathrm { g } ( x ) - 2$ Show clearly the coordinates of the $y$ intercept.

Edexcel P1 2023 June Q10

10 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-28_903_1010_219_539} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the quadratic curve $C$ with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line $l _ { 1 }$ also shown in Figure 5,

has gradient $\frac { 1 } { 2 }$
intersects $C$ on the negative $x$-axis and at the point $P$
find, in terms of $b$, an equation for $l _ { 2 }$ Given also that $l _ { 2 }$ intersects $C$ at the point $P$
show that another equation for $l _ { 2 }$ is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
Hence, or otherwise, find the value of $b$

Edexcel P1 2024 June Q1

3 marks Easy -1.2

Find

$$\int \left( 10 x ^ { 4 } - \frac { 3 } { 2 x ^ { 3 } } - 7 \right) \mathrm { d } x$$ giving each term in simplest form.

Edexcel P1 2024 June Q2

6 marks Easy -1.2

1. Given that $m = 2 ^ { n }$, express each of the following in simplest form in terms of $m$.
  1. $2 ^ { n + 3 }$
2. $16 ^ { 3 n }$ (ii) In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. Solve the equation $$x \sqrt { 3 } - 3 = x + \sqrt { 3 }$$ giving your answer in the form $p + q \sqrt { 3 }$ where $p$ and $q$ are integers.

Edexcel P1 2024 June Q3

6 marks Moderate -0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-06_688_771_251_648} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$.
The curve passes through the points $( - 1,0 )$ and $( 0,2 )$ and touches the $x$-axis at the point $( 3,0 )$. On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ( \mathrm { x } + 3 )$
$y = \mathrm { f } ( - 3 x )$ On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.

Edexcel P1 2024 June Q4

6 marks Standard +0.3

The curve $C _ { 1 }$ has equation

$$y = x ^ { 2 } + k x - 9$$ and the curve $C _ { 2 }$ has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where $k$ is a constant.
Given that $C _ { 1 }$ and $C _ { 2 }$ meet at a single point $P$

show that $$k ^ { 2 } + 26 k + 169 = 0$$
Hence find the coordinates of $P$

Edexcel P1 2024 June Q5

7 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-10_529_1403_255_267} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden $A B C D E A$ consists of a triangle $A B E$ and a right-angled triangle $B C D$ joined to a sector $B D E$ of a circle with radius 6 m and centre $B$. The points $A , B$ and $C$ lie on a straight line with $A B = 10.8 \mathrm {~m}$ Angle $B C D = \frac { \pi } { 2 }$ radians, angle $E B D = 1.3$ radians and $A E = 12.2 \mathrm {~m}$

Find the area of the sector $B D E$, giving your answer in $\mathrm { m } ^ { 2 }$
Find the size of angle $A B E$, giving your answer in radians to 2 decimal places.
Find the area of the garden, giving your answer in $\mathrm { m } ^ { 2 }$ to 3 significant figures.

Edexcel P1 2024 June Q6

7 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-14_899_901_251_584} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows

the line $l$ with equation $y - 5 x = 75$
the curve $C$ with equation $y = 2 x ^ { 2 } + x - 21$

The line $l$ intersects the curve $C$ at the points $P$ and $Q$, as shown in Figure 3 .

Find, using algebra, the coordinates of $P$ and the coordinates of $Q$. The region $R$, shown shaded in Figure 3, is bounded by $C , l$ and the $x$-axis.
Use inequalities to define the region $R$.

Edexcel P1 2024 June Q7

8 marks Moderate -0.3

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

$$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and $k$ is a constant.

Find, in simplest form,
1. $\mathrm { f } ^ { \prime } ( x )$
2. $\mathrm { f } ^ { \prime \prime } ( x )$ The curve with equation $y = \mathrm { f } ^ { \prime } ( x )$ intersects the curve with equation $y = \mathrm { f } ^ { \prime \prime } ( x )$ at the points $A$ and $B$. Given that the $x$ coordinate of $A$ is 5
find the value of $k$.
Hence find the coordinates of $B$.

Edexcel P1 2024 June Q8

7 marks Standard +0.3

The curve $C _ { 1 }$ has equation

$$y = x \left( 4 - x ^ { 2 } \right)$$

Sketch the graph of $C _ { 1 }$ showing the coordinates of any points of intersection with the coordinate axes. The curve $C _ { 2 }$ has equation $y = \frac { A } { x }$ where $A$ is a constant.
Show that the $x$ coordinates of the points of intersection of $C _ { 1 }$ and $C _ { 2 }$ satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
Hence find the range of possible values of $A$ for which $C _ { 1 }$ meets $C _ { 2 }$ at 4 distinct points.

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