Questions - Edexcel P1 - A-Level Maths

Express $\mathrm { f } ( x )$ in the form
2. $\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }$
Express $\mathrm { f } ( x )$ in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 }
& \text { where } a , b \text { and } c \text { are integers to be found. }
& \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses }
& \text { te } y \text {-axis at the point } R \text {. }
& \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and }
& n \text { are integers to be found. } \end{aligned}$$ $\_\_\_\_$ "

Edexcel P1 2020 October Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-08_885_1388_260_287} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, $A O C$ and $B O C$, joined to a sector $A O B$ of a circle centre $O$.

Angle $A O B = \alpha$
$A O = O B = 3 \mathrm {~cm}$
$O C = 5 \mathrm {~cm}$

Given that the area of sector $A O B$ is $7.2 \mathrm {~cm} ^ { 2 }$

show that $\alpha = 1.6$ radians.
Hence find
1. the area of the badge, giving your answer in $\mathrm { cm } ^ { 2 }$ to 2 significant figures,
2. the perimeter of the badge, giving your answer in cm to one decimal place.
  VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P1 2020 October Q4

4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4
x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.

Edexcel P1 2020 October Q5

5. (i) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_572_1025_212_463} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$.
The curve passes through the points $( - 5,0 )$ and $( 0 , - 3 )$ and touches the $x$-axis at the point $( 2,0 )$. On separate diagrams sketch the curve with equation

$y = \mathrm { f } ( x + 2 )$
$y = \mathrm { f } ( - x )$ On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
(ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation $$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$ where $k$ is a constant.
The curve meets the $y$-axis at the point $( 0 , \sqrt { 3 } )$ and passes through the points $( p , 0 )$ and ( $q , 0$ ). Find
the value of $k$,
the exact value of $p$ and the exact value of $q$.

Edexcel P1 2020 October Q6

6. The point $A$ has coordinates $( - 4,11 )$ and the point $B$ has coordinates $( 8,2 )$.

Find the gradient of the line $A B$, giving your answer as a fully simplified fraction. The point $M$ is the midpoint of $A B$. The line $l$ passes through $M$ and is perpendicular to $A B$.
Find an equation for $l$, giving your answer in the form $p x + q y + r = 0$ where $p , q$ and $r$ are integers to be found. The point $C$ lies on $l$ such that the area of triangle $A B C$ is 37.5 square units.
Find the two possible pairs of coordinates of point $C$.

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P1 2020 October Q7

7. The curve $C$ has equation $$y = \frac { 1 } { 2 - x }$$

Sketch the graph of $C$. On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line $l$ has equation $y = 4 x + k$, where $k$ is a constant. Given that $l$ meets $C$ at two distinct points,
show that $$k ^ { 2 } + 16 k + 48 > 0$$
Hence find the range of possible values for $k$.

Edexcel P1 2020 October Q8

8. The curve $C$ has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line $l _ { 1 }$ is the tangent to $C$ at the point where $x = 6$
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 line $l _ { 2 }$ is the tangent to $C$ at the point where $x = \alpha$
Given that $l _ { 1 }$ and $l _ { 2 }$ are parallel and distinct,
find the value of $\alpha$

Edexcel P1 2020 October Q9

9. A curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 9,10 )$. Given that $$f ^ { \prime } ( x ) = 27 x ^ { 2 } - \frac { 21 x ^ { 3 } - 5 x } { 2 \sqrt { x } } \quad x > 0$$ find $\mathrm { f } ( x )$, fully simplifying each term.

Edexcel P1 2021 October Q1

Find

$$\int 12 x ^ { 3 } + \frac { 1 } { 6 \sqrt { x } } - \frac { 3 } { 2 x ^ { 4 } } \mathrm {~d} x$$ giving each term in simplest form.

Edexcel P1 2021 October Q2

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points $P$ and $Q$ lie on the curve.
The gradient of the curve at both point $P$ and point $Q$ is 2
Find the $x$ coordinates of $P$ and $Q$.

Edexcel P1 2021 October Q3

3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve $C$ and the straight line $l$.
The infinite region $R$, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by $C$ and $l$ only. Given that

$\quad l$ has a gradient of 3
$C$ has equation $y = 2 x ^ { 2 } - 50$
$\quad C$ and $l$ intersect on the negative $x$-axis
use inequalities to define the region $R$.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-06_643_652_575_648} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ and the straight line $l$.
The infinite region $R$, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by $C$ and $l$ only.
Given that

$l$ has a gradient of 3
$C$ has equation $y = 2 x ^ { 2 } - 50$
$C$ and $l$ intersect on the negative $x$-axis
use inequalities to define the region $R$.

Edexcel P1 2021 October Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-08_721_855_214_550} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = \cos 2 x ^ { \circ } \quad 0 \leqslant x \leqslant k$$ The point $Q$ and the point $R ( k , 0 )$ lie on the curve and are shown in Figure 2.

State
1. the coordinates of $Q$,
2. the value of $k$.
Given that there are exactly two solutions to the equation $$\cos 2 x ^ { \circ } = p \quad \text { in the region } 0 \leqslant x \leqslant k$$ find the range of possible values for $p$.

Edexcel P1 2021 October Q5

5. The line $l _ { 1 }$ has equation $3 y - 2 x = 30$ The line $l _ { 2 }$ passes through the point $A ( 24,0 )$ and is perpendicular to $l _ { 1 }$
Lines $l _ { 1 }$ and $l _ { 2 }$ meet at the point $P$

Find, using algebra and showing your working, the coordinates of $P$. Given that $l _ { 1 }$ meets the $x$-axis at the point $B$,
find the area of triangle $B P A$.

Edexcel P1 2021 October Q6

6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve $C$ has equation $y = \mathrm { f } ( x )$ where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

Sketch a graph of $C$. Show on your graph the coordinates of the points where $C$ cuts or meets the coordinate axes.
Write $\mathrm { f } ( x )$ in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d$, where $a , b , c$ and $d$ are constants to be found.
Hence, find the equation of the tangent to $C$ at the point where $x = \frac { 1 } { 3 }$

Edexcel P1 2021 October Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-18_428_894_210_525} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows the design for a sign at a bird sanctuary.
The design consists of a kite $O A B C$ joined to a sector $O C X A$ of a circle centre $O$.
In the design

$O A = O C = 0.6 \mathrm {~m}$
$A B = C B = 1.4 \mathrm {~m}$
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 $$f ( - x ) \geqslant 6 \text { and } x < 0$$

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