Questions P1 - A-Level Maths

$O A = O C = 0.6 \mathrm {~m}$
$A B = C B = 1.4 \mathrm {~m}$
Angle $O A B =$ Angle $O C B = 2$ radians
Angle $A O C = \theta$ radians, as shown in Figure 3

Making your method clear,

show that $\theta = 1.64$ radians to 3 significant figures,
find the perimeter of the sign, in metres to 2 significant figures,
find the area of the sign, in $\mathrm { m } ^ { 2 }$ to 2 significant figures.

Edexcel P1 2021 October Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-22_657_659_214_646} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the curve $C$ with equation $$y = 4 + 12 x - 3 x ^ { 2 }$$ The point $M$ is the maximum turning point on $C$.

1. Write $4 + 12 x - 3 x ^ { 2 }$ in the form $$a + b ( x + c ) ^ { 2 }$$ where $a , b$ and $c$ are constants to be found.
2. Hence, or otherwise, state the coordinates of $M$. The line $l _ { 1 }$ passes through $O$ and $M$, as shown in Figure 4.
  A line $l _ { 2 }$ touches $C$ and is parallel to $l _ { 1 }$
Find an equation for $l _ { 2 }$

Edexcel P1 2021 October Q9

9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-26_595_716_420_662} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \sqrt { x } \quad x > 0$$ The point $P ( 9,3 )$ lies on the curve and is shown in Figure 5.
On the next page there is a copy of Figure 5 called Diagram 1.

On Diagram 1, sketch and clearly label the graphs of $$y = \mathrm { f } ( 2 x ) \text { and } y = \mathrm { f } ( x ) + 3$$ Show on each graph the coordinates of the point to which $P$ is transformed. The graph of $y = \mathrm { f } ( 2 x )$ meets the graph of $y = \mathrm { f } ( x ) + 3$ at the point $Q$.
Show that the $x$ coordinate of $Q$ is the solution of $$\sqrt { x } = 3 ( \sqrt { 2 } + 1 )$$
Hence find, in simplest form, the coordinates of $Q$.

\includegraphics[max width=\textwidth, alt={}]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-27_599_720_274_660}
\section*{Diagram 1} Turn over for a copy of Diagram 1 if you need to redraw your graphs. Only use this copy if you need to redraw your graphs.
\includegraphics[max width=\textwidth, alt={}, center]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-29_600_718_1991_660} Copy of Diagram 1

Edexcel P1 2021 October Q10

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

$\mathrm { f } ^ { \prime } ( x ) = a x - 12 x ^ { \frac { 1 } { 3 } }$, where $a$ is a constant
$\mathrm { f } ^ { \prime \prime } ( x ) = 0$ when $x = 27$
the curve passes through the point $( 1 , - 8 )$
1. find the value of $a$.
2. Hence find $\mathrm { f } ( x )$.

Edexcel P1 2022 October Q1

The curve $C$ has equation

$$y = \frac { x ^ { 3 } } { 4 } - x ^ { 2 } + \frac { 17 } { x } \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer in simplest form. The point $R \left( 2 , \frac { 13 } { 2 } \right)$ lies on $C$.
Find the equation of the tangent to $C$ at the point $R$. Write your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.

Edexcel P1 2022 October Q2

Given that

$$( x - 5 ) ( 2 x + 1 ) ( x + 3 ) \equiv a x ^ { 3 } + b x ^ { 2 } - 32 x - 15$$ where $a$ and $b$ are constants,

find the value of $a$ and the value of $b$.
Hence find $$\int \frac { ( x - 5 ) ( 2 x + 1 ) ( x + 3 ) } { 5 \sqrt { x } } \mathrm {~d} x$$ writing each term in simplest form.

Edexcel P1 2022 October Q3

The share price of a company is monitored.

Exactly 3 years after monitoring began, the share price was $\pounds 1.05$
Exactly 5 years after monitoring began, the share price was $\pounds 1.65$
The share price, $\pounds V$, of the company is modelled by the equation $$V = p t + q$$ where $t$ is the number of years after monitoring began and $p$ and $q$ are constants.

Find the value of $p$ and the value of $q$. Exactly $T$ years after monitoring began, the share price was $\pounds 2.50$
Find the value of $T$, according to the model, giving your answer to one decimal place.

Edexcel P1 2022 October Q4

In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = x ^ { 2 } ( 2 x + 1 ) - 15 x$$

Solve $$\mathrm { f } ( x ) = 0$$
Hence solve $$y ^ { \frac { 4 } { 3 } } \left( 2 y ^ { \frac { 2 } { 3 } } + 1 \right) - 15 y ^ { \frac { 2 } { 3 } } = 0 \quad y > 0$$ giving your answer in simplified surd form.

Edexcel P1 2022 October Q5

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

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

$\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 12 } { \sqrt { \mathrm { x } } } + \frac { x } { 3 } - 4$
the point $P ( 9,8 )$ lies on $C$
1. find, in simplest form, $\mathrm { f } ( x )$

The line $l$ is the normal to $C$ at $P$

Find the coordinates of the point at which $l$ crosses the $y$-axis.

Edexcel P1 2022 October Q6

(a) Given that $k$ is a positive constant such that $0 < k < 4$ sketch, on separate axes, the graphs of
1. $y = ( 2 x - k ) ( x + 4 ) ^ { 2 }$
2. $y = \frac { k } { x ^ { 2 } }$
  showing the coordinates of any points where the graphs cross or meet the coordinate axes, leaving coordinates in terms of $k$, where appropriate.
  (b) State, with a reason, the number of roots of the equation
$$( 2 x - k ) ( x + 4 ) ^ { 2 } = \frac { k } { x ^ { 2 } }$$

Edexcel P1 2022 October Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-16_732_1071_248_497} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = \mathrm { f } ( x )$.
The points $P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )$ and $S ( 3,6 )$ lie on the curve.

Using Figure 1, find the range of values of $x$ for which $$\mathrm { f } ( x ) < 6$$
State the largest solution of the equation $$f ( 2 x ) = 6$$
1. Sketch the curve with equation $y = \mathrm { f } ( - x )$. On your sketch, state the coordinates of the points to which $P , Q , R$ and $S$ are transformed.
2. Hence find the set of values of $x$ for which $$f ( - x ) \geqslant 6 \text { and } x < 0$$

Edexcel P1 2022 October Q8

8. \section*{Diagram NOT to scale} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-20_461_1036_296_534} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector $A O B X$ of a circle centre $O$ joined to a quadrilateral $A O B C$.

$B C = 8 \mathrm {~m}$
$O A = O B = 3 \mathrm {~m}$
angle $A O B$ is $\frac { 2 \pi } { 3 }$ radians
angle $B C A$ is $\frac { \pi } { 6 }$ radians
1. Calculate (i) the exact area of the sector $A O B X$,
  (ii) the exact perimeter of the sector $A O B X$.
2. Calculate the exact area of the triangle $A O B$.
3. Show that the length $A B$ is $3 \sqrt { 3 } \mathrm {~m}$.
4. Find the total surface area of the pond. Give your answer in $\mathrm { m } ^ { 2 }$ correct to 2 significant figures.

Edexcel P1 2022 October Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-24_889_666_258_703} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with equation $$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$

Write $\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$ in the form $$a ( x + b ) ^ { 2 } + c$$ where $a , b$ and $c$ are constants to be found. The point $M$ is the minimum turning point of $C$, as shown in Figure 3.
Deduce the coordinates of $M$ The line $l$ is the normal to $C$ at the point $P$, as shown in Figure 3.
Given that $l$ has equation $y = k - \frac { 1 } { 8 } x$, where $k$ is a constant,
1. find the coordinates of $P$
2. find the value of $k$ Question 9 continues on the next page \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605} \captionsetup{labelformat=empty} \caption{Figure 4}
  \end{figure} Figure 4 is a copy of Figure 3. The finite region $R$, shown shaded in Figure 4, is bounded by $l , C$ and the line through $M$ parallel to the $y$-axis.
Identify the inequalities that define $R$.

Edexcel P1 2023 October Q1

Given that

$$y = 5 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } - 7 x \quad x > 0$$ find, in simplest form,

$\frac { \mathrm { d } y } { \mathrm {~d} x }$
$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$

Edexcel P1 2023 October Q2

Given that

$$a = \frac { 1 } { 64 } x ^ { 2 } \quad b = \frac { 16 } { \sqrt { x } }$$ express each of the following in the form $k x ^ { n }$ where $k$ and $n$ are simplified constants.

$a ^ { \frac { 1 } { 2 } }$
$\frac { 16 } { b ^ { 3 } }$
$\left( \frac { a b } { 2 } \right) ^ { - \frac { 4 } { 3 } }$

Edexcel P1 2023 October Q3

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

Solutions relying on calculator technology are not acceptable.

Write $\frac { 8 - \sqrt { 15 } } { 2 \sqrt { 3 } + \sqrt { 5 } }$ in the form $a \sqrt { 3 } + b \sqrt { 5 }$ where $a$ and $b$ are integers to be found.
Hence, or otherwise, solve $$( x + 5 \sqrt { 3 } ) \sqrt { 5 } = 40 - 2 x \sqrt { 3 }$$ giving your answer in simplest form.

Edexcel P1 2023 October Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-08_687_775_248_646} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $y = \frac { 1 } { x + 2 }$

State the equation of the asymptote of $C$ that is parallel to the $y$-axis.
Factorise fully $x ^ { 3 } + 4 x ^ { 2 } + 4 x$ A copy of Figure 1, labelled Diagram 1, is shown on the next page.
On Diagram 1, add a sketch of the curve with equation $$y = x ^ { 3 } + 4 x ^ { 2 } + 4 x$$ On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.
Hence state the number of real solutions of the equation $$( x + 2 ) \left( x ^ { 3 } + 4 x ^ { 2 } + 4 x \right) = 1$$ giving a reason for your answer.
\includegraphics[max width=\textwidth, alt={}]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-09_800_1700_1053_185}
Only use the copy of Diagram 1 if you need to redraw your answer to part (c).

Edexcel P1 2023 October Q5

5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle $A B D$ joined to triangle $B C D$.
Given that

$B D = x \mathrm {~m}$
$C D = ( 1 + x ) \mathrm { m }$
$B C = 5 \mathrm {~m}$
angle $B C D = \theta ^ { \circ }$
1. show that $\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }$

Given also that

$x = 2 \sqrt { 3 }$
angle $B A C = 30 ^ { \circ }$
$A D C$ is a straight line
find the area of triangle $A B C$, giving your answer, in $\mathrm { m } ^ { 2 }$, to one decimal place.

Edexcel P1 2023 October Q6

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

\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where $p$ is a constant, has two distinct real roots.

Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
Hence, using algebra, find the range of possible values of $p$

Edexcel P1 2023 October Q7

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

Given that

$f ^ { \prime } ( x ) = \frac { 4 x ^ { 2 } + 10 - 7 x ^ { \frac { 1 } { 2 } } } { 4 x ^ { \frac { 1 } { 2 } } }$
the point $P ( 4 , - 1 )$ lies on $C$
1. (i) find the value of the gradient of $C$ at $P$
  (ii) Hence find the equation of the normal 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.
2. Find $\mathrm { f } ( x )$.

Edexcel P1 2023 October Q8

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

\section*{Solutions relying on calculator technology are not acceptable.} The curve $C _ { 1 }$ has equation $$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$ The curve $C _ { 2 }$ has equation $$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$

Show that $C _ { 1 }$ and $C _ { 2 }$ meet when $$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$ Given that $C _ { 1 }$ and $C _ { 2 }$ meet at points $P$ and $Q$
find, using algebra, the exact distance $P Q$

Edexcel P1 2023 October Q9

9. Diagram NOT accurately drawn \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-24_581_1491_340_296} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows the plan view of the area being used for a ball-throwing competition.
Competitors must stand within the circle $C$ and throw a ball as far as possible into the target area, $P Q R S$, shown shaded in Figure 3. Given that

circle $C$ has centre $O$
$P$ and $S$ are points on $C$
$O P Q R S O$ is a sector of a circle with centre $O$
the length of arc $P S$ is 0.72 m
the size of angle $P O S$ is 0.6 radians
1. show that $O P = 1.2 \mathrm {~m}$

Given also that

the target area, $P Q R S$, is $90 \mathrm {~m} ^ { 2 }$
length $P Q = x$ metres
show that

$$5 x ^ { 2 } + 12 x - 1500 = 0$$

Hence calculate the total perimeter of the target area, $P Q R S$, giving your answer to the nearest metre.

Edexcel P1 2023 October Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-28_538_652_255_708} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C _ { 1 }$ with equation $$y = 3 \cos \left( \frac { x } { n } \right) ^ { \circ } \quad x \geqslant 0$$ where $n$ is a constant.
The curve $C _ { 1 }$ cuts the positive $x$-axis for the first time at point $P ( 270,0 )$, as shown in Figure 4.

1. State the value of $n$
2. State the period of $C _ { 1 }$ The point $Q$, shown in Figure 4, is a minimum point of $C _ { 1 }$
State the coordinates of $Q$. The curve $C _ { 2 }$ has equation $y = 2 \sin x ^ { \circ } + k$, where $k$ is a constant.
The point $R \left( a , \frac { 12 } { 5 } \right)$ and the point $S \left( - a , - \frac { 3 } { 5 } \right)$, both lie on $C _ { 2 }$
Given that $a$ is a constant less than 90
find the value of $k$.

Edexcel P1 2023 October Q11

11. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-30_595_869_255_568} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows part of the curve $C$ with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$

Write $2 x ^ { 2 } - 12 x + 14$ in the form $$a ( x + b ) ^ { 2 } + c$$ where $a$, $b$ and $c$ are constants to be found. Given that $C$ has a minimum at the point $P$
state the coordinates of $P$ The line $l$ intersects $C$ at $( - 1,28 )$ and at $P$ as shown in Figure 5.
Find the equation of $l$ giving your answer in the form $y = m x + c$ where $m$ and $c$ are constants to be found. The finite region $R$, shown shaded in Figure 5, is bounded by the $x$-axis, $l$, the $y$-axis, and $C$.
Use inequalities to define the region $R$.

Edexcel P1 2018 Specimen Q1

Given that $y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0$, find in their simplest form
1. $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
2. $\int y \mathrm {~d} x$
  a) $y = 4 x ^ { 3 } - 5 x ^ { - 2 }$
  $\frac { d y } { d x } = 12 x ^ { 2 } + 10 x ^ { - 3 }$
  b) $\int 4 x ^ { 3 } - 5 x ^ { - 2 } d x$
  $= \frac { 4 x ^ { 4 } } { 4 } - \frac { 5 x ^ { - 1 } } { - 1 } + c = x ^ { 4 } + 5 x ^ { - 1 } + c$

Questions P1 (1374 questions)

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Edexcel P1 2021 October Q7

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Edexcel P1 2018 Specimen Q1