Questions P1 - A-Level Maths

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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. $3 \left( 9 ^ { 3 z } \right) + 26 \left( 9 ^ { 2 z } \right) - 9 \left( 9 ^ { z } \right) = 0$

Edexcel P1 2021 June Q3

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]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-08_351_999_383_598} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle $A B C$ with

$A B = p$ metres
$A C = q$ metres
$B C = 2 \sqrt { 2 }$ metres
angle $B A C = 60 ^ { \circ }$
1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side $A C$ is 2 metres longer than side $A B$, use algebra to find

the exact value of $p$,
the exact value of $q$. Using the answers to part (b),

calculate the exact area of the flower bed.

Edexcel P1 2021 June Q4

4. Find $$\int \frac { ( 3 \sqrt { x } + 2 ) ( x - 5 ) } { 4 \sqrt { x } } d x$$ writing each term in simplest form.

Edexcel P1 2021 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-14_563_671_255_657} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The share value of two companies, company $A$ and company $B$, has been monitored over a 15-year period. The share value $P _ { A }$ of company $\boldsymbol { A }$, in millions of pounds, is modelled by the equation $$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$ where $t$ is the number of years after monitoring began. The share value $P _ { B }$ of company $B$, in millions of pounds, is modelled by the equation $$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$ where $t$ is the number of years after monitoring began. Figure 2 shows a graph of both models. Use the equations of one or both models to answer parts (a) to (d).

Find the difference between the share value of company $\boldsymbol { A }$ and the share value of company $\boldsymbol { B }$ at the point monitoring began.
State the maximum share value of company $\boldsymbol { A }$ during the 15-year period.
Find, using algebra and showing your working, the times during this 15-year period when the share value of company $\boldsymbol { A }$ was greater than the share value of company $\boldsymbol { B }$.
Explain why the model for company $\boldsymbol { A }$ should not be used to predict its share value when $t = 20$
\includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}

Edexcel P1 2021 June Q6

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

$C$ passes through the point $P ( 8,2 )$
$\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 3 x ^ { 2 } } + 3 - 2 ( \sqrt [ 3 ] { x } )$
1. find the equation of the tangent to $C$ at $P$. Write your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found.
  (3)
2. Find, in simplest form, $\mathrm { f } ( x )$.
  \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-21_2647_1840_118_111}

Edexcel P1 2021 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-22_775_837_251_557} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The line $l _ { 1 }$ has equation $4 y + 3 x = 48$
The line $l _ { 1 }$ cuts the $y$-axis at the point $C$, as shown in Figure 3.

State the $y$ coordinate of $C$. The point $D ( 8,6 )$ lies on $l _ { 1 }$
The line $l _ { 2 }$ passes through $D$ and is perpendicular to $l _ { 1 }$ The line $l _ { 2 }$ cuts the $y$-axis at the point $E$ as shown in Figure 3.
Show that the $y$ coordinate of $E$ is $- \frac { 14 } { 3 }$ A sector $B C E$ of a circle with centre $C$ is also shown in Figure 3. Given that angle $B C E$ is 1.8 radians,
find the length of arc $B E$. The region $C B E D$, shown shaded in Figure 3, consists of the sector $B C E$ joined to the triangle $C D E$.
Calculate the exact area of the region $C B E D$.

Edexcel P1 2021 June Q8

8. The curve $C _ { 1 }$ has equation $$y = 3 x ^ { 2 } + 6 x + 9$$

Write $3 x ^ { 2 } + 6 x + 9$ in the form $$a ( x + b ) ^ { 2 } + c$$ where $a$, $b$ and $c$ are constants to be found. The point $P$ is the minimum point of $C _ { 1 }$
Deduce the coordinates of $P$. A different curve $C _ { 2 }$ has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where $A$, $B$, $C$ and $D$ are constants. Given that $C _ { 2 }$
- passes through $P$
- intersects the $x$-axis at $- 4 , - 2$ and 3
- find, making your method clear, the values of $A , B , C$ and $D$.
  \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-27_2644_1840_118_111}
\includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-29_2646_1838_121_116}

Edexcel P1 2021 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-30_707_1034_251_456} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the curve with equation $$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ The line $l$, shown in Figure 4, is an asymptote to $y = \tan x$

State an equation for $l$. A copy of Figure 4, labelled Diagram 1, is shown on the next page.
1. On Diagram 1, sketch the curve with equation $$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ stating the equation of the horizontal asymptote of this curve.
2. Hence, giving a reason, state the number of solutions of the equation
State the number of solutions of the equation $\tan x = \frac { 1 } { x } + 1$ in the region
1. $0 \leqslant x \leqslant 40 \pi$
2. $- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi$ $$\begin{aligned} & \qquad \tan x = \frac { 1 } { x } + 1
  & \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi \end{aligned}$$" \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447} \captionsetup{labelformat=empty} \caption{Diagram 1}
  \end{figure}
  \includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}

Edexcel P1 2022 June Q1

Find

$$\int \left( 10 x ^ { 5 } + 6 x ^ { 3 } - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in simplest form.

Edexcel P1 2022 June Q2

2. In the triangle $A B C$,

$A B = 21 \mathrm {~cm}$
$B C = 13 \mathrm {~cm}$
angle $B A C = 25 ^ { \circ }$
angle $A C B = x ^ { \circ }$
1. Use the sine rule to find the value of $\sin x ^ { \circ }$, giving your answer to 4 decimal places.

Given also that $A B$ is the longest side of the triangle,

find the value of $x$, giving your answer to 2 decimal places.

Edexcel P1 2022 June Q3

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

Show that $\frac { \sqrt { 180 } - \sqrt { 80 } } { \sqrt { 5 } }$ is an integer and find its value.
Simplify $$\frac { 4 \sqrt { 5 } - 5 } { 7 - 3 \sqrt { 5 } }$$ giving your answer in the form $a + b \sqrt { 5 }$ where $a$ and $b$ are rational numbers.

Edexcel P1 2022 June Q4

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

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

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

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

$\mathrm { f } ( x ) = 8 \sqrt { 3 }$, when $x = 12$
find $\mathrm { f } ( x )$, giving each term in simplest form.

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