Questions - Edexcel P3 - A-Level Maths

find the mass of this tree, giving your answer to 2 significant figures.
Show that the equation of the model can be written in the form $$M = p r ^ { q }$$ giving the values of the constants $p$ and $q$ to 3 significant figures.
With reference to the model, interpret the value of the constant $p$. Q

Edexcel P3 2021 October Q8

8. A curve $C$ has equation $y = \mathrm { f } ( x )$, where $$f ( x ) = \arcsin \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2 \quad - \frac { \pi } { 2 } \leqslant y \leqslant \frac { \pi } { 2 }$$

Sketch $C$.
Given $x = 2 \sin y$, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { A - x ^ { 2 } } }$$ where $A$ is a constant to be found. The point $P$ lies on $C$ and has $y$ coordinate $\frac { \pi } { 4 }$
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)

Edexcel P3 2021 October Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-26_698_744_255_593} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = x \left( x ^ { 2 } - 4 \right) e ^ { - \frac { 1 } { 2 } x }$$

Find $f ^ { \prime } ( x )$. The line $l$ is the normal to the curve at $O$ and meets the curve again at the point $P$. The point $P$ lies in the 3rd quadrant, as shown in Figure 3.
Show that the $x$ coordinate of $P$ is a solution of the equation $$x = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x } }$$
Using the iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x _ { n } } } \quad \text { with } x _ { 1 } = - 2$$ find, to 4 decimal places,
1. the value of $x _ { 2 }$
2. the $x$ coordinate of $P$.

Edexcel P3 2021 October Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-30_515_673_255_639} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = ( 1 + 2 \cos 2 x ) ^ { 2 }$$

Show that $$( 1 + 2 \cos 2 x ) ^ { 2 } \equiv p + q \cos 2 x + r \cos 4 x$$ where $p , q$ and $r$ are constants to be found. The curve touches the positive $x$-axis for the second time when $x = a$, as shown in Figure 4. The regions bounded by the curve, the $y$-axis and the $x$-axis up to $x = a$ are shown shaded in Figure 4.
Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-32_2255_51_313_1980}

Edexcel P3 2022 October Q1

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

$$f ( x ) = \frac { 2 x ^ { 3 } - 4 x - 15 } { x ^ { 2 } + 3 x + 4 }$$

Show that $$f ( x ) \equiv A x + B + \frac { C ( 2 x + 3 ) } { x ^ { 2 } + 3 x + 4 }$$ where $A , B$ and $C$ are integers to be found.
Hence, find $$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.

Edexcel P3 2022 October Q2

2. The functions f and g are defined by $$\begin{array} { l l } f ( x ) = 5 - \frac { 4 } { 3 x + 2 } & x \geqslant 0
g ( x ) = \left| 4 \sin \left( \frac { x } { 3 } + \frac { \pi } { 6 } \right) \right| & x \in \mathbb { R } \end{array}$$

Find the range of f
1. Find $\mathrm { f } ^ { - 1 } ( x )$
2. Write down the domain of $\mathrm { f } ^ { - 1 }$
Find $\mathrm { fg } ( - \pi )$

Edexcel P3 2022 October Q3

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

\includegraphics[alt={},max width=\textwidth]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-08_535_839_402_555} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } \mathrm { e } ^ { 3 x } \quad x \in \mathbb { R }$$ The curve has a maximum turning point at $A$ and a minimum turning point at $( 2,0 )$

Use calculus to find the exact coordinates of $A$. Given that the equation $\mathrm { f } ( x ) = k$, where $k$ is a constant, has at least two distinct roots,
state the range of possible values for $k$.

Edexcel P3 2022 October Q4

4. $$y = \log _ { 10 } ( 2 x + 1 )$$

Express $x$ in terms of $y$.
Hence, giving your answer in terms of $x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
4.
Express $x$ in terms of $y$.
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(2)
Hence, giving your answer in terms of $x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-10_2662_111_107_1950}

Edexcel P3 2022 October Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-12_479_551_214_699} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The profit made by a company, $\pounds P$ million, $t$ years after the company started trading, is modelled by the equation $$P = \frac { 4 t - 1 } { 10 } + \frac { 3 } { 4 } \ln \left[ \frac { t + 1 } { ( 2 t + 1 ) ^ { 2 } } \right]$$ The graph of $P$ against $t$ is shown in Figure 2. According to the model,

show that exactly one year after it started trading, the company had made a loss of approximately £ 830000 A manager of the company wants to know the value of $t$ for which $P = 0$
Show that this value of $t$ occurs in the interval [6,7]
Show that the equation $P = 0$ can be expressed in the form $$t = \frac { 1 } { 4 } + \frac { 15 } { 8 } \ln \left[ \frac { ( 2 t + 1 ) ^ { 2 } } { t + 1 } \right]$$
Using the iteration formula $$t _ { n + 1 } = \frac { 1 } { 4 } + \frac { 15 } { 8 } \ln \left[ \frac { \left( 2 t _ { n } + 1 \right) ^ { 2 } } { t _ { n } + 1 } \right] \text { with } t _ { 1 } = 6$$ find the value of $t _ { 2 }$ and the value of $t _ { 6 }$, giving your answers to 3 decimal places.
Hence find, according to the model, how many months it takes in total, from when the company started trading, for it to make a profit.
(2)

Edexcel P3 2022 October Q6

6. $$y = \frac { 2 + 3 \sin x } { \cos x + \sin x }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a \tan x + b \sec x + c } { \sec x + 2 \sin x }$$ where $a , b$ and $c$ are integers to be found.

Edexcel P3 2022 October Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-20_624_798_219_575} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the graph of $C _ { 1 }$ with equation $$y = 5 - | 3 x - 22 |$$

Write down the coordinates of
1. the vertex of $C _ { 1 }$
2. the intersection of $C _ { 1 }$ with the $y$-axis.
Find the $x$ coordinates of the intersections of $C _ { 1 }$ with the $x$-axis. Diagram 1, shown on page 21, is a copy of Figure 3.
On Diagram 1, sketch the curve $C _ { 2 }$ with equation $$y = \frac { 1 } { 9 } x ^ { 2 } - 9$$ Identify clearly the coordinates of any points of intersection of $C _ { 2 }$ with the coordinate axes.
Find the coordinates of the points of intersection of $C _ { 1 }$ and $C _ { 2 }$ (Solutions relying entirely on calculator technology are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-21_629_803_1137_573} \section*{Diagram 1} Solutions relying entirely on calculator technology are not acceptable.

Edexcel P3 2022 October Q9

9. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Given that $\cos 2 \theta - \sin 3 \theta \neq 0$

prove that $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } \equiv \frac { 1 + \sin \theta } { 1 - 2 \sin \theta - 4 \sin ^ { 2 } \theta }$$
Hence solve, for $0 < \theta \leqslant 360 ^ { \circ }$ $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } = 2 \operatorname { cosec } \theta$$ Give your answers to one decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-28_2257_52_309_1983}

Edexcel P3 2023 October Q1

A curve has equation $y = \mathrm { f } ( x )$ where

$$\mathrm { f } ( x ) = x ^ { 2 } - 5 x + \mathrm { e } ^ { x } \quad x \in \mathbb { R }$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root, $\alpha$, in the interval [1,2] The iterative formula $$x _ { n + 1 } = \sqrt { 5 x _ { n } - \mathrm { e } ^ { x _ { n } } }$$ with $x _ { 1 } = 1$ is used to find an approximate value for the root $\alpha$.
1. Find the value of $x _ { 2 }$ to 4 decimal places.
2. Find, by repeated iteration, the value of $\alpha$, giving your answer to 4 decimal places.

Edexcel P3 2023 October Q2

The function f is defined by

$$\mathrm { f } ( x ) = \frac { x + 3 } { x - 4 } \quad x \in \mathbb { R } , x \neq 4$$

Find ff(6)
Find $f ^ { - 1 }$ The function $g$ is defined by $$g ( x ) = x ^ { 2 } + 5 \quad x \in \mathbb { R } , x > 0$$
Find the exact value of $a$ for which $$\operatorname { gf } ( a ) = 7$$

Edexcel P3 2023 October Q3

(a) Using the identity for $\cos ( A + B )$, prove that

$$\cos 2 A \equiv 2 \cos ^ { 2 } A - 1$$ (b) Hence, using algebraic integration, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 8 } } \left( 5 - 4 \cos ^ { 2 } 3 x \right) d x$$

Edexcel P3 2023 October Q4

A new mobile phone is released for sale.

The total sales $N$ of this phone, in thousands, is modelled by the equation $$N = 125 - A \mathrm { e } ^ { - 0.109 t } \quad t \geqslant 0$$ where $A$ is a constant and $t$ is the time in months after the phone was released for sale.
Given that when $t = 0 , N = 32$

state the value of $A$. Given that when $t = T$ the total sales of the phone was 100000
find, according to the model, the value of $T$. Give your answer to 2 decimal places.
Find, according to the model, the rate of increase in total sales when $t = 7$, giving your answer to 3 significant figures.
(Solutions relying entirely on calculator technology are not acceptable.) The total sales of the mobile phone is expected to reach 150000
Using this information,
give a reason why the given equation is not suitable for modelling the total sales of the phone.

Edexcel P3 2023 October Q5

The curve $C$ has equation

$$y = \frac { \ln \left( x ^ { 2 } + k \right) } { x ^ { 2 } + k } \quad x \in \mathbb { R }$$ where $k$ is a positive constant.

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x \left( B - \ln \left( x ^ { 2 } + k \right) \right) } { \left( x ^ { 2 } + k \right) ^ { 2 } }$$ where $A$ and $B$ are constants to be found. Given that $C$ has exactly three turning points,
find the $x$ coordinate of each of these points. Give your answer in terms of $k$ where appropriate.
find the upper limit to the value for $k$.

Edexcel P3 2023 October Q6

An area of sea floor is being monitored.

The area of the sea floor, $S \mathrm {~km} ^ { 2 }$, covered by coral reefs is modelled by the equation $$S = p q ^ { t }$$ where $p$ and $q$ are constants and $t$ is the number of years after monitoring began.
Given that $$\log _ { 10 } S = 4.5 - 0.006 t$$

find, according to the model, the area of sea floor covered by coral reefs when $t = 2$
find a complete equation for the model in the form $$S = p q ^ { t }$$ giving the value of $p$ and the value of $q$ each to 3 significant figures.
With reference to the model, interpret the value of the constant $q$

Edexcel P3 2023 October Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08291ac1-bdd4-4241-8959-7c89318fa5eb-18_554_1129_248_468} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = e ^ { - x ^ { 2 } } \left( 2 x ^ { 2 } - 3 \right) ^ { 2 }$$

Find the range of f
Show that $$\mathrm { f } ^ { \prime } ( x ) = 2 x \left( 2 x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x ^ { 2 } } \left( A - B x ^ { 2 } \right)$$ where $A$ and $B$ are constants to be found. Given that the line $y = k$, where $k$ is a constant, $k > 0$, intersects the curve at exactly two distinct points,
find the exact range of values of $k$

Edexcel P3 2023 October Q8

(a) Prove that

$$2 \operatorname { cosec } ^ { 2 } 2 \theta ( 1 - \cos 2 \theta ) \equiv 1 + \tan ^ { 2 } \theta$$ (b) Hence solve for $0 < x < 360 ^ { \circ }$, where $x \neq ( 90 n ) ^ { \circ } , n \in \mathbb { N }$, the equation $$2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = 4 + 3 \sec x$$ giving your answers to one decimal place.
(Solutions relying entirely on calculator technology are not acceptable.)

Edexcel P3 2023 October Q9

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]{08291ac1-bdd4-4241-8959-7c89318fa5eb-26_613_729_386_667} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $$y = | 2 - 4 \ln ( x + 1 ) | \quad x > k$$ where $k$ is a constant.
Given that the curve

has an asymptote at $x = k$
cuts the $y$-axis at point $A$
meets the $x$-axis at point $B$
as shown in Figure 2,
1. state the value of $k$
2. 1. find the $y$ coordinate of $A$
  2. find the exact $x$ coordinate of $B$
3. Using algebra and showing your working, find the set of values of $x$ such that

$$| 2 - 4 \ln ( x + 1 ) | > 3$$

Edexcel P3 2023 October Q10

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

A curve $C$ has equation $$x = \sin ^ { 2 } 4 y \quad 0 \leqslant y \leqslant \frac { \pi } { 8 } \quad 0 \leqslant x \leqslant 1$$ The point $P$ with $x$ coordinate $\frac { 1 } { 4 }$ lies on $C$

Find the exact $y$ coordinate of $P$
Find $\frac { \mathrm { d } x } { \mathrm {~d} y }$
Hence show that $\frac { \mathrm { d } y } { \mathrm {~d} x }$ can be written in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { q + r ( x + s ) ^ { 2 } } }$$ where $q , r$ and $s$ are constants to be found. Using the answer to part (c),
1. state the $x$ coordinate of the point where the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is a minimum,
2. state the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at this point.

Edexcel P3 2018 Specimen Q1

Express

$$\frac { 6 x + 4 } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.

Edexcel P3 2018 Specimen Q2

2. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$

Show that the equation $\mathrm { f } ( x ) = 0$ can be written as $$x = \sqrt { \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) } \quad x \neq - 3$$ The equation $x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0$ has a single root which is between 1 and 2
Use the iteration formula $$x _ { n + 1 } = \sqrt { \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) } \quad n \geqslant 0$$ with $x _ { 0 } = 1$ to find, to 2 decimal places, the value of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ The root of $\mathrm { f } ( x ) = 0$ is $\alpha$.
By choosing a suitable interval, prove that $\alpha = 1.272$ to 3 decimal places.

VIIIV SIHI NI JIIIM ION OC VIIV SIHI NI JAHAM ION OO VJ4V SIHIL NI JIIIM IONOO

Edexcel P3 2018 Specimen Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-08_542_540_269_696} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the graph $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 \quad x \geqslant 0$$

Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation $\mathrm { f } ( x ) = k$, where $k$ is a constant, has two distinct roots,
state the set of possible values for $k$.

VIIIV SIHI NI JIIIM ION OC VIIV SIHI NI JAHAM ION OO VI4V SIHIL NI JIIIM ION OC

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Questions — Edexcel P3 (133 questions)

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Edexcel P3 2021 October Q7

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