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Edexcel P1 2019 January Q6

7 marks Moderate -0.3

(Solutions based entirely on graphical or numerical methods are not acceptable.)

Given $$\mathrm { f } ( x ) = 2 x ^ { \frac { 5 } { 2 } } - 40 x + 8 \quad x > 0$$

solve the equation $\mathrm { f } ^ { \prime } ( x ) = 0$
solve the equation $\mathrm { f } ^ { \prime \prime } ( x ) = 5$

Edexcel P1 2019 January Q7

6 marks Moderate -0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-14_327_595_251_676} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Not to scale Figure 3 shows the design for a structure used to support a roof. The structure consists of four wooden beams, $A B , B D , B C$ and $A D$. Given $A B = 6.5 \mathrm {~m} , B C = B D = 4.7 \mathrm {~m}$ and angle $B A C = 35 ^ { \circ }$

find, to one decimal place, the size of angle $A C B$,
find, to the nearest metre, the total length of wood required to make this structure.

Edexcel P1 2019 January Q8

6 marks Moderate -0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-16_647_970_306_488} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The curve $C$ with equation $y = \mathrm { f } ( x )$ is shown in Figure 4. The curve $C$

has a single turning point, a maximum at ( 4,9 )
crosses the coordinate axes at only two places, $( - 3,0 )$ and $( 0,6 )$
has a single asymptote with equation $y = 4$ as shown in Figure 4.
1. State the equation of the asymptote to the curve with equation $y = \mathrm { f } ( - x )$.
2. State the coordinates of the turning point on the curve with equation $y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)$.

Given that the line with equation $y = k$, where $k$ is a constant, intersects $C$ at exactly one point,

state the possible values for $k$. The curve $C$ is transformed to a new curve that passes through the origin.

Given that the new curve has equation $y = \mathrm { f } ( x ) - a$, state the value of the constant $a$.
Write down an equation for another single transformation of $C$ that also passes through the origin.

Edexcel P1 2019 January Q9

7 marks Standard +0.3

The equation

$$\frac { 3 } { x } + 5 = - 2 x + c$$ where $c$ is a constant, has no real roots.
Find the range of possible values of $c$.

Edexcel P1 2019 January Q10

7 marks Standard +0.3

A sector $A O B$, of a circle centre $O$, has radius $r \mathrm {~cm}$ and angle $\theta$ radians.

Given that the area of the sector is $6 \mathrm {~cm} ^ { 2 }$ and that the perimeter of the sector is 10 cm ,

show that $$3 \theta ^ { 2 } - 13 \theta + 12 = 0$$
Hence find possible values of $r$ and $\theta$.
□ \includegraphics[max width=\textwidth, alt={}, center]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-21_131_19_2627_1882}

Edexcel P1 2019 January Q11

12 marks Moderate -0.3

11. (a) On Diagram 1 sketch the graphs of

$y = x ( 3 - x )$
$y = x ( x - 2 ) ( 5 - x )$ showing clearly the coordinates of the points where the curves cross the coordinate axes.
(b) Show that the $x$ coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation $x \left( x ^ { 2 } - 8 x + 13 \right) = 0$ The point $P$ lies on both curves. Given that $P$ lies in the first quadrant,
(c) find, using algebra and showing your working, the exact coordinates of $P$.

\includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
\section*{Diagram 1}

Edexcel P1 2019 January Q12

9 marks Moderate -0.3

12. The curve with equation $y = \mathrm { f } ( x ) , x > 0$, passes through the point $P ( 4 , - 2 )$. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x \sqrt { x } - 10 x ^ { - \frac { 1 } { 2 } }$$

find the equation of the tangent to the curve at $P$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are integers to be found.
Find $\mathrm { f } ( x )$.

Edexcel P1 2020 January Q1

4 marks Easy -1.2

Find, in simplest form,

$$\int \left( \frac { 8 x ^ { 3 } } { 3 } - \frac { 1 } { 2 \sqrt { x } } - 5 \right) \mathrm { d } x$$

Edexcel P1 2020 January Q2

5 marks Easy -1.2

2. Given $y = 3 ^ { x }$, express each of the following in terms of $y$. Write each expression in its simplest form.

$3 ^ { 3 x }$
$\frac { 1 } { 3 ^ { x - 2 } }$
$\frac { 81 } { 9 ^ { 2 - 3 x } }$

Edexcel P1 2020 January Q3

6 marks Easy -1.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{28839dd5-b9c1-4cbd-981e-8f79c43ba086-06_652_654_269_646} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of the curve with equation $y = x ^ { 2 } + 3 x - 2$ The point $P ( 3,16 )$ lies on the curve.

Find the gradient of the tangent to the curve at $P$. The point $Q$ with $x$ coordinate $3 + h$ also lies on the curve.
Find, in terms of $h$, the gradient of the line $P Q$. Write your answer in simplest form.
Explain briefly the relationship between the answer to (b) and the answer to (a).

Edexcel P1 2020 January Q4

9 marks Moderate -0.5

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{28839dd5-b9c1-4cbd-981e-8f79c43ba086-08_622_894_258_683} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the plan view of a house $A B C D$ and a lawn $A P C D A$. $A B C D$ is a rectangle with $A B = 16 \mathrm {~m}$. $A P C O A$ is a sector of a circle centre $O$ with radius 12 m . The point $O$ lies on the line $D C$, as shown in Figure 2.

Show that the size of angle $A O D$ is 1.231 radians to 3 decimal places. The lawn $A P C D A$ is shown shaded in Figure 2.
Find the area of the lawn, in $\mathrm { m } ^ { 2 }$, to one decimal place.
Find the perimeter of the lawn, in metres, to one decimal place.

Edexcel P1 2020 January Q5

7 marks Standard +0.3

5. (a) Find, using algebra, all solutions of $$20 x ^ { 3 } - 50 x ^ { 2 } - 30 x = 0$$ (b) Hence find all real solutions of $$20 ( y + 3 ) ^ { \frac { 3 } { 2 } } - 50 ( y + 3 ) - 30 ( y + 3 ) ^ { \frac { 1 } { 2 } } = 0$$

Edexcel P1 2020 January Q6

8 marks Standard +0.3

6. The line $l _ { 1 }$ has equation $3 x - 4 y + 20 = 0$ The line $l _ { 2 }$ cuts the $x$-axis at $R ( 8,0 )$ and is parallel to $l _ { 1 }$

Find the equation of $l _ { 2 }$, writing your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found. The line $l _ { 1 }$ cuts the $x$-axis at $P$ and the $y$-axis at $Q$.
Given that $P Q R S$ is a parallelogram, find
the area of $P Q R S$,
the coordinates of $S$.

Edexcel P1 2020 January Q7

5 marks Easy -1.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{28839dd5-b9c1-4cbd-981e-8f79c43ba086-18_599_723_274_614} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows part of the curve $C _ { 1 }$ with equation $y = 3 \sin x$, where $x$ is measured in degrees. The point $P$ and the point $Q$ lie on $C _ { 1 }$ and are shown in Figure 3.

State
1. the coordinates of $P$,
2. the coordinates of $Q$. A different curve $C _ { 2 }$ has equation $y = 3 \sin x + k$, where $k$ is a constant.
  The curve $C _ { 2 }$ has a maximum $y$ value of 10
  The point $R$ is the minimum point on $C _ { 2 }$ with the smallest positive $x$ coordinate.
State the coordinates of $R$. Figure 3

Sketch $C _ { 1 }$ showing the coordinates of any point where the curve touches or crosses the coordinate axes.
Hence or otherwise
1. find the values of $x$ for which $\mathrm { f } \left( \frac { 1 } { 4 } x \right) = 0$
2. find the value of the constant $p$ such that the curve with equation $y = \mathrm { f } ( x ) + p$ passes through the origin. A second curve $C _ { 2 }$ has equation $y = \mathrm { g } ( x )$, where $\mathrm { g } ( x ) = \mathrm { f } ( x + 1 )$
1. Find, in simplest form, $\mathrm { g } ( x )$. You may leave your answer in a factorised form.
2. Hence, or otherwise, find the $y$ intercept of curve $C _ { 2 }$

Edexcel P1 2020 January Q11

11 marks Standard +0.3

11. A curve has equation $y = \mathrm { f } ( x )$, where $$f ^ { \prime \prime } ( x ) = \frac { 6 } { \sqrt { x ^ { 3 } } } + x \quad x > 0$$ The point $P ( 4 , - 50 )$ lies on the curve.
Given that $\mathrm { f } ^ { \prime } ( x ) = - 4$ at $P$,

find the equation of the normal at $P$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants,
(3)
find $\mathrm { f } ( x )$.
(8)
VIIIV SIHI NI IIIYM ION OC VIIV SIHI NI JIHMM ION OO VI4V SIHI NI JIIYM ION OO

Edexcel P1 2021 January Q1

8 marks Moderate -0.8

A curve has equation

$$y = 2 x ^ { 3 } - 5 x ^ { 2 } - \frac { 3 } { 2 x } + 7 \quad x > 0$$

Find, in simplest form, $\frac { \mathrm { d } y } { \mathrm {~d} x }$ The point $P$ lies on the curve and has $x$ coordinate $\frac { 1 } { 2 }$
Find an equation of the normal to the curve at $P$, writing your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.
VILU SIHI NI JIIIM ION OC VIUV SIHI NI III M M I ON OO VIAV SIHI NI JIIIM I ION OC

Edexcel P1 2021 January Q2

6 marks Moderate -0.8

A tree was planted.

Exactly 3 years after it was planted, the height of the tree was 2 m . Exactly 5 years after it was planted, the height of the tree was 2.4 m . Given that the height, $H$ metres, of the tree, $t$ years after it was planted, can be modelled by the equation $$H ^ { 3 } = p t ^ { 2 } + q$$ where $p$ and $q$ are constants,

find, to 3 significant figures where necessary, the value of $p$ and the value of $q$. Exactly $T$ years after the tree was planted, its height was 5 m .
Find the value of $T$ according to the model, giving your answer to one decimal place.

Edexcel P1 2021 January Q3

5 marks Easy -1.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-08_625_835_264_557} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve $C _ { 1 }$ with equation $y = 4 \cos x ^ { \circ }$ The point $P$ and the point $Q$ lie on $C _ { 1 }$ and are shown in Figure 1.

State
1. the coordinates of $P$,
2. the coordinates of $Q$. The curve $C _ { 2 }$ has equation $y = 4 \cos x ^ { \circ } + k$, where $k$ is a constant.
  Curve $C _ { 2 }$ has a minimum $y$ value of - 1
  The point $R$ is the maximum point on $C _ { 2 }$ with the smallest positive $x$ coordinate.
State the coordinates of $R$.

Edexcel P1 2021 January Q4

8 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-10_583_866_260_539} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The points $P$ and $Q$, as shown in Figure 2, have coordinates ( $- 2,13$ ) and ( $4 , - 5$ ) respectively. The straight line $l$ passes through $P$ and $Q$.

Find an equation for $l$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are integers to be found. The quadratic curve $C$ passes through $P$ and has a minimum point at $Q$.
Find an equation for $C$. The region $R$, shown shaded in Figure 2, lies in the second quadrant and is bounded by $C$ and $l$ only.
Use inequalities to define region $R$. \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-11_2255_50_314_34}

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

Edexcel P1 2021 January Q5

9 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-14_470_940_246_500} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector $A B C O A$ of a circle, centre $O$, joined to a triangle $A O D$. Given that

$O A = O C = 6 \mathrm {~m}$
$A D = 14 \mathrm {~m}$
angle $A D C = 0.43$ radians
angle $A O D$ is an obtuse angle
$O C D$ is a straight line
find
1. the size of angle $A O D$, in radians, to 3 decimal places,
2. the length of arc $A B C$, in metres, to one decimal place,
3. the total area of the viewing platform, in $\mathrm { m } ^ { 2 }$, to one decimal place.

Edexcel P1 2021 January Q6

10 marks Standard +0.3

6. (a) Sketch the curve with equation $$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$ (b) On a separate diagram, sketch the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ stating the coordinates of the point of intersection with the $x$-axis and, in terms of $k$, the equation of the horizontal asymptote.
(c) Find the range of possible values of $k$ for which the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ does not touch or intersect the line with equation $y = 3 x + 4$ \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}

Edexcel P1 2021 January Q7

9 marks Moderate -0.3

7. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = 2 x - 3 \sqrt { x } - 5 \quad x > 0$$

Solve the equation $$f ( x ) = 9$$
Solve the equation $$\mathrm { f } ^ { \prime \prime } ( x ) = 6$$

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Edexcel P1 2019 January Q6

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