Questions P3 - A-Level Maths

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.

Edexcel P3 2021 October Q7

7 The mass, $M \mathrm {~kg}$, of a species of tree can be modelled by the equation $$\log _ { 10 } M = 1.93 \log _ { 10 } r + 0.684$$ where $r \mathrm {~cm}$ is the base radius of the tree.
The base radius of a particular tree of this species is 45 cm .
According to the model,

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 $y$.
$\begin{array} { c } \text { Leave }
\text { blank } \end{array}$
(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.

	\(\begin{array} { c } \text { Leave }
\text { blank } \end{array}\)

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