Questions - Edexcel P1 - A-Level Maths

the curve with equation $y = \tan x$
the straight line l with equation $y = \pi x$
in the interval $- \pi < x < \pi$
1. State the period of $\tan x$
2. Write down the number of roots of the equation
  1. $\tan x = ( \pi + 2 ) x$ in the interval $- \pi < x < \pi$
  2. $\tan x = \pi x$ in the interval $- 2 \pi < x < 2 \pi$
  3. $\tan x = \pi x$ in the interval $- 100 \pi < x < 100 \pi$

Edexcel P1 2023 January Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bb21001f-fe68-4776-992d-ede1aae233d7-26_902_896_248_587} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C$ with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = ( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$$

Use the given information to state the values of $x$ for which $$f ( x ) > 0$$
Expand $( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$, writing your answer as a polynomial in simplest form. The straight line $l$ is the tangent to $C$ at the point where $C$ cuts the $y$-axis.
Given that $l$ cuts $C$ at the point $P$, as shown in Figure 4,
find, using algebra, the $x$ coordinate of $P$
(Solutions based on calculator technology are not acceptable.)

Edexcel P1 2023 January Q11

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

Given that

$\mathrm { f } ^ { \prime \prime } ( x ) = 4 x + \frac { 1 } { \sqrt { x } }$
the point $P$ has $x$ coordinate 4 and lies on $C$
the tangent to $C$ at $P$ has equation $y = 3 x + 4$
1. find an equation of the normal to $C$ at $P$
2. find $\mathrm { f } ( x )$, writing your answer in simplest form.

Edexcel P1 2024 January Q1

Find

$$\int ( 2 x - 5 ) ( 3 x + 2 ) ( 2 x + 5 ) \mathrm { d } x$$ writing your answer in simplest form.

Edexcel P1 2024 January Q2

The triangle $A B C$ is such that

$A B = 15 \mathrm {~cm}$
$A C = 25 \mathrm {~cm}$
angle $B A C = \theta ^ { \circ }$
area triangle $A B C = 100 \mathrm {~cm} ^ { 2 }$
1. Find the value of $\sin \theta ^ { \circ }$

Given that $\theta > 90$

find the length of $B C$, in cm , to 3 significant figures.

Edexcel P1 2024 January Q3

The curve $C$ has equation

$$y = \frac { 5 x ^ { 3 } - 8 } { 2 x ^ { 2 } } \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ writing your answer in simplest form. The point $P ( 2,4 )$ lies on $C$.
Find an equation for the tangent to $C$ at $P$ writing your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel P1 2024 January Q4

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

Solutions relying on calculator technology are not acceptable.

By substituting $p = 2 ^ { x }$, show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$

Edexcel P1 2024 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2043b938-ed3f-4b69-9ea9-b4ab62e2a8ce-10_891_850_295_609} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The straight line $l _ { 1 }$, shown in Figure 1, passes through the points $P ( - 2,9 )$ and $Q ( 10,6 )$.

Find the equation of $l _ { 1 }$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found. The straight line $l _ { 2 }$ passes through the origin $O$ and is perpendicular to $l _ { 1 }$
The lines $l _ { 1 }$ and $l _ { 2 }$ meet at the point $R$ as shown in Figure 1.
Find the coordinates of $R$
Find the exact area of triangle $O P Q$.

Edexcel P1 2024 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2043b938-ed3f-4b69-9ea9-b4ab62e2a8ce-14_919_954_299_559} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a plot of part of the curve $C _ { 1 }$ with equation $$y = 5 \cos x$$ with $x$ being measured in degrees.
The point $P$, shown in Figure 2, is a minimum point on $C _ { 1 }$

State the coordinates of $P$ The point $Q$ lies on a different curve $C _ { 2 }$
Given that point $Q$
- is a maximum point on the curve
- is the maximum point with the smallest $x$ coordinate, $x > 0$
- find the coordinates of $Q$ when
  1. $C _ { 2 }$ has equation $y = 5 \cos x - 2$
  2. $C _ { 2 }$ has equation $y = - 5 \cos x$

Edexcel P1 2024 January Q7

(a) Sketch the graph of the curve $C$ with equation

$$y = \frac { 4 } { x - k }$$ where $k$ is a positive constant.
Show on your sketch

the coordinates of any points where $C$ cuts the coordinate axes
the equation of the vertical asymptote to $C$

Given that the straight line with equation $y = 9 - x$ does not cross or touch $C$
(b) find the range of values of $k$.

Edexcel P1 2024 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2043b938-ed3f-4b69-9ea9-b4ab62e2a8ce-18_680_933_294_589} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the plan view of a platform.
The plan view of the platform consists of a sector $D O C$ of a circle centre $O$ joined to a sector $A O B E A$ of a different circle, also with centre $O$. Given that

angle $A O B = 0.8$ radians
arc length $C D = 9 \mathrm {~m}$
$D A : A O = 3 : 5$
1. show that $A O = 7.03 \mathrm {~m}$ to 3 significant figures.
2. Find the perimeter of the platform, in m , to 3 significant figures.
3. Find the total area of the platform, giving your answer in $\mathrm { m } ^ { 2 }$ to the nearest whole number.

Edexcel P1 2024 January Q9

The curve $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$.

Given that

$\mathrm { f } ( x )$ is a quadratic expression
$C _ { 1 }$ has a maximum turning point at $( 2,20 )$
$C _ { 1 }$ passes through the origin
1. sketch a graph of $C _ { 1 }$ showing the coordinates of any points where $C _ { 1 }$ cuts the coordinate axes,
2. find an expression for $\mathrm { f } ( x )$.

The curve $C _ { 2 }$ has equation $y = x \left( x ^ { 2 } - 4 \right)$
Curve $C _ { 1 }$ and $C _ { 2 }$ meet at the origin, and at the points $P$ and $Q$
Given that the $x$ coordinate of the point $P$ is negative,

using algebra and showing all stages of your working, find the coordinates of $P$

Edexcel P1 2024 January Q10

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

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

the point $P ( 2,8 \sqrt { 2 } )$ lies on $C$
$\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }$ where $k$ is a constant
$\mathrm { f } ^ { \prime \prime } ( x ) = 0$ at $P$
1. find the exact value of $k$,
2. find $\mathrm { f } ( x )$, giving your answer in simplest form.

Edexcel P1 2019 June Q1

The curve $C$ has equation $y = \frac { 1 } { 8 } x ^ { 3 } - \frac { 24 } { \sqrt { x } } + 1$
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving the answer in its simplest form.
  (3)
The point $P ( 4 , - 3 )$ lies on $C$.
Find the equation of the tangent to $C$ at the point $P$. Write your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found.

Edexcel P1 2019 June Q2

Answer this question showing each stage of your working.
1. Simplify $\frac { 1 } { 4 - 2 \sqrt { 2 } }$
  giving your answer in the form $a + b \sqrt { 2 }$ where $a$ and $b$ are rational numbers.
2. Hence, or otherwise, solve the equation
$$4 x = 2 \sqrt { 2 } x + 20 \sqrt { 2 }$$ giving your answer in the form $p + q \sqrt { 2 }$ where $p$ and $q$ are rational numbers.

Edexcel P1 2019 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-06_881_974_255_495} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 29 m ,

show that $x > 1.5 \mathrm {~m}$ Given also that the area of the garden is less than $72 \mathrm {~m} ^ { 2 }$,
form and solve a quadratic inequality in $x$.
Hence state the range of possible values of $x$.
\href{http://www.dynamicpapers.com}{www.dynamicpapers.com}

Edexcel P1 2019 June Q4

Find

$$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } d x$$ giving the answer in its simplest form. $$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } \mathrm {~d} x$$ giving the answer in its simplest form.

Edexcel P1 2019 June Q5

(a) Find, using algebra, all real solutions of

$$2 x ^ { 3 } + 3 x ^ { 2 } - 35 x = 0$$ (b) Hence find all real solutions of $$2 ( y - 5 ) ^ { 6 } + 3 ( y - 5 ) ^ { 4 } - 35 ( y - 5 ) ^ { 2 } = 0$$

Edexcel P1 2019 June Q6

The line with equation $y = 4 x + c$, where $c$ is a constant, meets the curve with equation $y = x ( x - 3 )$ at only one point.
1. Find the value of $c$.
2. Hence find the coordinates of the point of intersection.

Edexcel P1 2019 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-16_661_999_246_603} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The shape $A B C D A$ consists of a sector $A B C O A$ of a circle, centre $O$, joined to a triangle $A O D$, as shown in Figure 2. The point $D$ lies on $O C$.
The radius of the circle is 6 cm , length $A D$ is 5 cm and angle $A O D$ is 0.7 radians.

Find the area of the sector $A B C O A$, giving your answer to one decimal place. Given angle $A D O$ is obtuse,
find the size of angle $A D O$, giving your answer to 3 decimal places.
Hence find the perimeter of shape $A B C D A$, giving your answer to one decimal place.
\href{http://www.dynamicpapers.com}{www.dynamicpapers.com}

Edexcel P1 2019 June Q8

The curve $C$ with equation $y = \mathrm { f } ( x ) , \quad x > 0$, passes through the point $P ( 4,1 )$.

Given that $\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x } - 2 - \frac { 8 } { 3 x ^ { 2 } }$

find the equation of the normal to $C$ at $P$. Write your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.
(4)
Find $\mathrm { f } ( x )$.
(5)
\href{http://www.dynamicpapers.com}{www.dynamicpapers.com}

Edexcel P1 2019 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-24_741_806_255_577} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a plot of the curve with equation $y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$

State the coordinates of the minimum point on the curve with equation $$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$ A copy of Figure 3, called Diagram 1, is shown on the next page.
On Diagram 1, sketch and label the curves
1. $y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$
2. $y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$
Hence find the number of solutions of the equation
1. $\tan \theta = 1 + \sin \theta$ that lie in the region $0 \leqslant \theta \leqslant 2160 ^ { \circ }$
2. $\tan \theta = 1 + \sin \theta$ that lie in the region $0 \leqslant \theta \leqslant 1980 ^ { \circ }$
  
  \includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
  \section*{Diagram 1}

Edexcel P1 2019 June Q10

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

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the $x$-axis at the point $P$ and crosses the $x$-axis at the point $Q$.

State the coordinates of the point $P$.
Find $f ^ { \prime } ( x )$.
Hence show that the equation of the tangent to the curve at the point where $x = \frac { 5 } { 2 }$ can be expressed in the form $y = k$, where $k$ is a constant to be found. The curve with equation $y = \mathrm { f } ( x + a )$, where $a$ is a constant, passes through the origin $O$.
State the possible values of $a$.
\href{http://www.dynamicpapers.com}{www.dynamicpapers.com}
$\_\_\_\_$ "

Edexcel P1 2021 June Q1

The curve $C$ has equation

$$y = \frac { x ^ { 2 } } { 3 } + \frac { 4 } { \sqrt { x } } + \frac { 8 } { 3 x } - 5 \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer in simplest form. The point $P ( 4,3 )$ lies on $C$.
Find the equation of the normal to $C$ at the point $P$. Write your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.

Edexcel P1 2021 June Q2

2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = a x ^ { 3 } + ( 6 a + 8 ) x ^ { 2 } - a ^ { 2 } x$$ where $a$ is a positive constant. Given $\mathrm { f } ( - 1 ) = 32$

1. show that the only possible value for $a$ is 3
2. Using $a = 3$ solve the equation $$\mathrm { f } ( x ) = 0$$
Hence find all real solutions of
1. $3 y + 26 y ^ { \frac { 2 } { 3 } } - 9 y ^ { \frac { 1 } { 3 } } = 0$
2. $3 \left( 9 ^ { 3 z } \right) + 26 \left( 9 ^ { 2 z } \right) - 9 \left( 9 ^ { z } \right) = 0$

Questions — Edexcel P1 (172 questions)

Browse by module

Browse by board

Edexcel P1 2023 January Q9

Edexcel P1 2023 January Q10

Edexcel P1 2023 January Q11

Edexcel P1 2024 January Q1

Edexcel P1 2024 January Q2

Edexcel P1 2024 January Q3

Edexcel P1 2024 January Q4

Edexcel P1 2024 January Q5

Edexcel P1 2024 January Q6

Edexcel P1 2024 January Q7

Edexcel P1 2024 January Q8

Edexcel P1 2024 January Q9

Edexcel P1 2024 January Q10

Edexcel P1 2019 June Q1

Edexcel P1 2019 June Q2

Edexcel P1 2019 June Q3

Edexcel P1 2019 June Q4

Edexcel P1 2019 June Q5

Edexcel P1 2019 June Q6

Edexcel P1 2019 June Q7

Edexcel P1 2019 June Q8

Edexcel P1 2019 June Q9

Edexcel P1 2019 June Q10

Edexcel P1 2021 June Q1

Edexcel P1 2021 June Q2