Questions - Edexcel P3 - A-Level Maths

Find $\frac { \mathrm { d } x } { \mathrm {~d} y }$ giving your answer as a fully simplified fraction. The tangents at points $P$ and $Q$ on the curve are parallel to the $y$-axis, as shown in Figure 4.
Use the answer to part (a) to find the equations of these two tangents.

Edexcel P3 2024 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-02_520_474_246_797} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the graph with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = 2 | x - 5 | + 10$$ The point $P$, shown in Figure 1, is the vertex of the graph.

State the coordinates of $P$
Use algebra to solve $$2 | x - 5 | + 10 > 6 x$$ (Solutions relying on calculator technology are not acceptable.)
Find the point to which $P$ is mapped, when the graph with equation $y = \mathrm { f } ( x )$ is transformed to the graph with equation $y = 3 \mathrm { f } ( x - 2 )$

Edexcel P3 2024 June Q2

2. $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x + 8 } { x - 2 }$$

Write $g ( x )$ in the form $$A x + B + \frac { C } { x - 2 }$$ where $A , B$ and $C$ are integers to be found.
Hence use algebraic integration to show that $$\int _ { 4 } ^ { 8 } \mathrm {~g} ( x ) \mathrm { d } x = \alpha + \beta \ln 3$$ where $\alpha$ and $\beta$ are integers to be found.

Edexcel P3 2024 June Q3

(i) The variables $x$ and $y$ are connected by the equation

$$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$ Sketch the graph of $\log _ { 10 } y$ against $\log _ { 10 } x$
Show on your sketch the coordinates of the points of intersection of the graph with the axes.
(ii) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-08_888_885_744_552} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the linear relationship between $\log _ { 3 } N$ and $t$.
Show that $N = a b ^ { t }$ where $a$ and $b$ are constants to be found.

Edexcel P3 2024 June Q4

4. $$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$

Write $\mathrm { f } ( x )$ in the form $$a \sin 2 x + b \cos 2 x + c$$ where $a$, $b$ and $c$ are integers to be found.
Use the answer to part (a) to write $\mathrm { f } ( x )$ in the form $$R \sin ( 2 x + \alpha ) + c$$ where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$
Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 significant figures.
Hence, or otherwise,
1. state the maximum value of $\mathrm { f } ( x )$
2. find the second smallest positive value of $x$ at which a maximum value of $\mathrm { f } ( x )$ occurs. Give your answer to 3 significant figures.

Edexcel P3 2024 June Q5

The functions $f$ and $g$ are defined by

$$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0
& \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$

Find $\mathrm { f } ^ { - 1 } ( 22 )$
Use differentiation to prove that g is an increasing function.
Find $\mathrm { g } ^ { - 1 }$
Find the range of fg

Edexcel P3 2024 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-18_856_990_246_539} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve with equation $$y = \sqrt { 4 x - 7 }$$ The line $l$, shown in Figure 3, is the normal to the curve at the point $P ( 8,5 )$

Use calculus to show that an equation of $l$ is $$5 x + 2 y - 50 = 0$$ The region $R$, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and $l$.
Use algebraic integration to find the exact area of $R$.

Edexcel P3 2024 June Q7

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

Solutions relying entirely on calculator technology are not acceptable.

Given that $$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$ show that $$\tan x = - 2 - \sqrt { 3 }$$
Hence or otherwise, solve, for $0 \leqslant \theta < 180 ^ { \circ }$ $$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$

Edexcel P3 2024 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_499_551_246_758} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 is a graph showing the path of a golf ball after the ball has been hit until it first hits the ground. The vertical height, $h$ metres, of the ball above the ground has been plotted against the horizontal distance travelled, $x$ metres, measured from where the ball was hit. The ball travels a horizontal distance of $d$ metres before it first hits the ground.
The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
The path of the ball is modelled by the equation $$h = 1.5 x - 0.5 x \mathrm { e } ^ { 0.02 x } \quad 0 \leqslant x \leqslant d$$ \section*{Use the model to answer parts (a), (b) and (c).}

Find the value of $d$, giving your answer to 2 decimal places.
(Solutions relying entirely on calculator technology are not acceptable.)
Show that the maximum value of $h$ occurs when $$x = 50 \ln \left( \frac { 150 } { x + 50 } \right)$$ Using the iteration formula $$x _ { n + 1 } = 50 \ln \left( \frac { 150 } { x _ { n } + 50 } \right) \quad \text { with } x _ { 1 } = 30$$
1. find the value of $x _ { 2 }$ to 2 decimal places,
2. find, by repeated iteration, the horizontal distance travelled by the golf ball before it reaches its maximum height. Give your answer to 2 decimal places.
  \includegraphics[max width=\textwidth, alt={}, center]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_2270_56_309_1981}

Edexcel P3 2024 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-30_714_1079_251_495} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} The curve shown in Figure 5 has equation $$x = 4 \sin ^ { 2 } y - 1 \quad 0 \leqslant y \leqslant \frac { \pi } { 2 }$$ The point $P \left( k , \frac { \pi } { 3 } \right)$ lies on the curve.

Verify that $k = 2$
1. Find $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$
2. Hence show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 \sqrt { x + 1 } \sqrt { 3 - x } }$ The normal to the curve at $P$ cuts the $x$-axis at the point $N$.
Find the exact area of triangle $O P N$, where $O$ is the origin. Give your answer in the form $a \pi + b \pi ^ { 2 }$ where $a$ and $b$ are constants.

Edexcel P3 2020 October Q1

Solve, for $0 \leqslant x < 360 ^ { \circ }$, the equation

$$2 \cos 2 x = 7 \cos x$$ giving your solutions to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel P3 2020 October Q2

A scientist monitored the growth of bacteria on a dish over a 30 -day period.

The area, $N \mathrm {~mm} ^ { 2 }$, of the dish covered by bacteria, $t$ days after monitoring began, is modelled by the equation $$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$

Show that this equation may be written in the form $$N = a b ^ { t }$$ where $a$ and $b$ are constants to be found. Give the value of $a$ to the nearest integer and give the value of $b$ to 3 significant figures.
Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in $\mathrm { mm } ^ { 2 }$, to 2 significant figures.

Edexcel P3 2020 October Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96948fd3-5438-4e95-b41b-2f649ca8dfac-06_828_828_210_557} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a curve with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { \sqrt { 4 x - 1 } } \quad x > \frac { 1 } { 4 }$$

Find, in simplest form, $\mathrm { f } ^ { \prime } ( x )$.
Hence find the range of f.
VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P3 2020 October Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96948fd3-5438-4e95-b41b-2f649ca8dfac-10_780_839_123_557} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the graph with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = 21 - 2 | 2 - x | \quad x \geqslant 0$$

Find ff(6)
Solve the equation $\mathrm { f } ( x ) = 5 x$ Given that the equation $\mathrm { f } ( x ) = k$, where $k$ is a constant, has exactly two roots,
state the set of possible values of $k$. The graph with equation $y = \mathrm { f } ( x )$ is transformed onto the graph with equation $y = a \mathrm { f } ( x - b )$ The vertex of the graph with equation $y = a \mathrm { f } ( x - b )$ is (6, 3). Given that $a$ and $b$ are constants,
find the value of $a$ and the value of $b$. \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-11_2255_50_314_34}

Edexcel P3 2020 October Q5

5. (a) Show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$ (b) Hence find, using algebraic integration, $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 3 } x d x$$

Edexcel P3 2020 October Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96948fd3-5438-4e95-b41b-2f649ca8dfac-16_565_844_217_552} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of curve $C _ { 1 }$ with equation $y = 5 \mathrm { e } ^ { x - 1 } + 3$
and curve $C _ { 2 }$ with equation $y = 10 - x ^ { 2 }$
The point $P$ lies on $C _ { 1 }$ and has $y$ coordinate 18

Find the $x$ coordinate of $P$, writing your answer in the form $\ln k$, where $k$ is a constant to be found. The curve $C _ { 1 }$ meets the curve $C _ { 2 }$ at $x = \alpha$ and at $x = \beta$, as shown in Figure 3.
Using a suitable interval and a suitable function that should be stated, show that to 3 decimal places $\alpha = 1.134$ The iterative equation $$x _ { n + 1 } = - \sqrt { 7 - 5 \mathrm { e } ^ { x _ { n } - 1 } }$$ is used to find an approximation to $\beta$. Using this iterative formula with $x _ { 1 } = - 3$
find the value of $x _ { 2 }$ and the value of $\beta$, giving each answer to 6 decimal places.

Edexcel P3 2020 October Q7

7. (a) Express $\cos x + 4 \sin x$ in the form $R \cos ( x - \alpha )$ where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$ and give the value of $\alpha$, in radians, to 3 decimal places. A scientist is studying the behaviour of seabirds in a colony. She models the height above sea level, $H$ metres, of one of the birds in the colony by the equation $$H = \frac { 24 } { 3 + \cos \left( \frac { 1 } { 2 } t \right) + 4 \sin \left( \frac { 1 } { 2 } t \right) } \quad 0 \leqslant t \leqslant 6.5$$ where $t$ seconds is the time after it leaves the nest. Find, according to the model,
(b) the minimum height of the seabird above sea level, giving your answer to the nearest cm,
(c) the value of $t$, to 2 decimal places, when $H = 10$ \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-21_2255_50_314_34}

Edexcel P3 2020 October Q8

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

$$g ( x ) = e ^ { 3 x } \sec 2 x \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$

Find $\mathrm { g } ^ { \prime } ( x )$
Hence find the $x$ coordinate of the stationary point of $C$.
(ii) A different curve has equation $$x = \ln ( \sin y ) \quad 0 < y < \frac { \pi } { 2 }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { x } } { \mathrm { f } ( x ) }$$ where $\mathrm { f } ( x )$ is a function of $\mathrm { e } ^ { x }$ that should be found.

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

Edexcel P3 2020 October Q9

9. (a) Given that $$\frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \equiv x ^ { 2 } + P + \frac { Q } { x - 4 } \quad x > - 3$$ find the value of the constant $P$ and show that $Q = 5$ The curve $C$ has equation $y = \mathrm { g } ( x )$, where $$g ( x ) = \frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \quad - 3 < x < 3.5 \quad x \in \mathbb { R }$$ (b) Find the equation of the tangent to $C$ at the point where $x = 2$ Give your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96948fd3-5438-4e95-b41b-2f649ca8dfac-28_876_961_1055_495} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the curve $C$.
The region $R$, shown shaded in Figure 4, is bounded by $C$, the $y$-axis, the $x$-axis and the line with equation $x = 2$
(c) Find the exact area of $R$, writing your answer in the form $a + b \ln 2$, where $a$ and $b$ are constants to be found. \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-31_2255_50_314_34}

VIHV SIHII NI I IIIM I ON OC

VIAV SIHI NI JYHAM ION OO

VI4V SIHI NI JLIYM ION OO

\includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-32_106_113_2524_1832}

Edexcel P3 2021 October Q1

The function f is defined by

$$\mathrm { f } ( x ) = \frac { 5 x } { x ^ { 2 } + 7 x + 12 } + \frac { 5 x } { x + 4 } \quad x > 0$$

Show that $\mathrm { f } ( x ) = \frac { 5 x } { x + 3 }$
Find $\mathrm { f } ^ { - 1 }$
1. Find, in simplest form, $\mathrm { f } ^ { \prime } ( x )$.
2. Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.
  (3)

Edexcel P3 2021 October Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-06_570_604_255_673} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the graph with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = | 3 x - 13 | + 5 \quad x \in \mathbb { R }$$ The vertex of the graph is at point $P$, as shown in Figure 1.

State the coordinates of $P$.
1. State the range of f .
2. Find the value of ff(4)
Solve, using algebra and showing your working, $$16 - 2 x > | 3 x - 13 | + 5$$ The graph with equation $y = \mathrm { f } ( x )$ is transformed onto the graph with equation $y = a \mathrm { f } ( x + b )$ The vertex of the graph with equation $y = a \mathrm { f } ( x + b )$ is $( 4,20 )$ Given that $a$ and $b$ are constants,
find the value of $a$ and the value of $b$.

Edexcel P3 2021 October Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-10_541_618_248_671} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The total mass of gold, $G$ tonnes, extracted from a mine is modelled by the equation $$G = 40 - 30 \mathrm { e } ^ { 1 - 0.05 t } \quad t \geqslant k \quad G \geqslant 0$$ where $t$ is the number of years after 1st January 1800.
Figure 2 shows a sketch of $G$ against $t$. Use the equation of the model to answer parts (a), (b) and (c).

1. Find the value of $k$.
2. Hence find the year and month in which gold started being extracted from the mine.
Find the total mass of gold extracted from the mine up to 1st January 1870. There is a limit to the mass of gold that can be extracted from the mine.
State the value of this limit.
M

Edexcel P3 2021 October Q4

4. In this question you should show detailed reasoning. \section*{Solutions relying entirely on calculator technology are not acceptable.}

Show that the equation $$2 \sin \left( \theta - 30 ^ { \circ } \right) = 5 \cos \theta$$ can be written in the form $$\tan \theta = 2 \sqrt { 3 }$$
Hence, or otherwise, solve for $0 \leqslant x \leqslant 360 ^ { \circ }$ $$2 \sin \left( x - 10 ^ { \circ } \right) = 5 \cos \left( x + 20 ^ { \circ } \right)$$ giving your answers to one decimal place.

Edexcel P3 2021 October Q5

5. (i) Find, by algebraic integration, the exact value of $$\int _ { 2 } ^ { 4 } \frac { 8 } { ( 2 x - 3 ) ^ { 3 } } d x$$ (ii) Find, in simplest form, $$\int x \left( x ^ { 2 } + 3 \right) ^ { 7 } d x$$

Edexcel P3 2021 October Q6

6. (i) The curve $C _ { 1 }$ has equation $$y = 3 \ln \left( x ^ { 2 } - 5 \right) - 4 x ^ { 2 } + 15 \quad x > \sqrt { 5 }$$ Show that $C _ { 1 }$ has a stationary point at $x = \frac { \sqrt { p } } { 2 }$ where $p$ is a constant to be found.
(ii) A different curve $C _ { 2 }$ has equation $$y = 4 x - 12 \sin ^ { 2 } x$$

Show that, for this curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = A + B \sin 2 x$$ where $A$ and $B$ are constants to be found.
Hence, state the maximum gradient of this curve.

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