Questions - Edexcel P4 - A-Level Maths

Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the doorknob is given by $$V = K \int _ { 0 } ^ { 4 } \left( x ^ { 2 } - 8 x + 16 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ where $K$ is a constant to be found.
Hence, find the exact value of the volume of the doorknob. Give your answer in the form $p \pi \left( \mathrm { e } ^ { q } + r \right) \mathrm { cm } ^ { 3 }$ where $p , q$ and $r$ are simplified rational numbers to be found.

Edexcel P4 2022 January Q8

8. With respect to a fixed origin $O$ the points $A$ and $B$ have position vectors $$\left( \begin{array} { l } 6
6
2 \end{array} \right) \text { and } \left( \begin{array} { l } 6
0
7 \end{array} \right)$$ respectively. The line $l _ { 1 }$ passes through the points $A$ and $B$.

Write down an equation for $l _ { 1 }$ Give your answer in the form $\mathbf { r } = \mathbf { p } + \lambda \mathbf { q }$, where $\lambda$ is a scalar parameter. The line $l _ { 2 }$ has equation $$\mathbf { r } = \left( \begin{array} { l } 3
1
4 \end{array} \right) + \mu \left( \begin{array} { l } 1
5

Edexcel P4 2022 January Q9

9 \end{array} \right)$$ where $\mu$ is a scalar parameter.
(b) Show that $l _ { 1 }$ and $l _ { 2 }$ do not meet. The point $C$ is on $l _ { 2 }$ where $\mu = - 1$
(c) Find the acute angle between $A C$ and $l _ { 2 }$ Give your answer in degrees to one decimal place.

(a) Find the derivative with respect to $y$ of

$$\frac { 1 } { ( 1 + 2 \ln y ) ^ { 2 } }$$ (b) Hence find a general solution to the differential equation $$3 \operatorname { cosec } ( 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + 2 \ln y ) ^ { 3 } \quad y > 0 \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$ (c) Show that the particular solution of this differential equation for which $y = 1$ at $x = \frac { \pi } { 6 }$ is given by $$y = \mathrm { e } ^ { A \sec x - \frac { 1 } { 2 } }$$ where $A$ is an irrational number to be found.
\includegraphics[max width=\textwidth, alt={}, center]{fe07afad-9cfc-48c0-84f1-5717f81977d4-32_2649_1894_109_173}

Edexcel P4 2023 January Q1

1. $$f ( x ) = \frac { 5 x + 10 } { ( 1 - x ) ( 2 + 3 x ) }$$

Write $\mathrm { f } ( x )$ in partial fraction form.
1. Hence find, in ascending powers of $x$ up to and including the terms in $x ^ { 2 }$, the binomial series expansion of $\mathrm { f } ( x )$. Give each coefficient as a simplified fraction.
2. Find the range of values of $x$ for which this expansion is valid.

Edexcel P4 2023 January Q2

A set of points $P ( x , y )$ is defined by the parametric equations

$$x = \frac { t - 1 } { 2 t + 1 } \quad y = \frac { 6 } { 2 t + 1 } \quad t \neq - \frac { 1 } { 2 }$$

Show that all points $P ( x , y )$ lie on a straight line.
Hence or otherwise, find the $x$ coordinate of the point of intersection of this line and the line with equation $y = x + 12$

Edexcel P4 2023 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-08_419_665_255_708} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} Figure 1 shows a sketch of the curve with equation $$y = \sqrt { \frac { 3 x } { 3 x ^ { 2 } + 5 } } \quad x \geqslant 0$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the lines with equations $x = \sqrt { 5 }$ and $x = 5$ The region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
Use integration to find the exact volume of the solid generated. Give your answer in the form $a \ln b$, where $a$ is an irrational number and $b$ is a prime number.

Edexcel P4 2023 January Q4

(a) Using the substitution $u = \sqrt { 2 x + 1 }$, show that

$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } \mathrm { e } ^ { \sqrt { 2 x + 1 } } \mathrm {~d} x$$ may be expressed in the form $$\int _ { a } ^ { b } k u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u$$ where $a$, $b$ and $k$ are constants to be found.
(b) Hence find, by algebraic integration, the exact value of $$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } e ^ { \sqrt { 2 x + 1 } } d x$$ giving your answer in simplest form.

Edexcel P4 2023 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-14_940_881_251_593} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $$y ^ { 2 } = 2 x ^ { 2 } + 15 x + 10 y$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The curve is not defined for values of $x$ in the interval ( $p , q$ ), as shown in Figure 2.
Using your answer to part (a) or otherwise, find the value of $p$ and the value of $q$.
(Solutions relying entirely on calculator technology are not acceptable.)

Edexcel P4 2023 January Q6

Relative to a fixed origin $O$.

the point $A$ has position vector $2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }$
the point $B$ has position vector $8 \mathbf { i } + 3 \mathbf { j } - 7 \mathbf { k }$

The line $l$ passes through $A$ and $B$.

1. Find $\overrightarrow { A B }$
2. Find a vector equation for the line $l$ The point $C$ has position vector $3 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k }$
  The point $P$ lies on $l$
  Given that $\overrightarrow { C P }$ is perpendicular to $l$
find the position vector of the point $P$

Edexcel P4 2023 January Q7

The volume $V \mathrm {~cm} ^ { 3 }$ of a spherical balloon with radius $r \mathrm {~cm}$ is given by the formula

$$V = \frac { 4 } { 3 } \pi r ^ { 3 }$$

Find $\frac { \mathrm { d } V } { \mathrm {~d} r }$ giving your answer in simplest form. At time $t$ seconds, the volume of the balloon is increasing according to the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 900 } { ( 2 t + 3 ) ^ { 2 } } \quad t \geqslant 0$$ Given that $V = 0$ when $t = 0$
1. solve this differential equation to show that $$V = \frac { 300 t } { 2 t + 3 }$$
2. Hence find the upper limit to the volume of the balloon.
Find the radius of the balloon at $t = 3$, giving your answer in cm to 3 significant figures.
Find the rate of increase of the radius of the balloon at $t = 3$, giving your answer to 2 significant figures. Show your working and state the units of your answer.

Edexcel P4 2023 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-26_582_773_255_648} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} A curve $C$ has parametric equations $$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point $P$ with parameter $t = \frac { \pi } { 4 }$ lies on $C$.
The line $l$ is the normal to $C$ at $P$, as shown in Figure 3.

Show, using calculus, that an equation for $l$ is $$8 y + 2 x = 17$$ The region $S$, shown shaded in Figure 3, is bounded by $C , l$ and the $x$-axis.
Find, using calculus, the exact area of $S$.

Edexcel P4 2023 January Q9

A student was asked to prove, for $p \in \mathbb { N }$, that
"if $p ^ { 3 }$ is a multiple of 3 , then $p$ must be a multiple of 3 "

The start of the student's proof by contradiction is shown in the box below. Assumption:
There exists a number $p , p \in \mathbb { N }$, such that $p ^ { 3 }$ is a multiple of 3 , and $p$ is NOT a multiple of 3 Let $p = 3 k + 1 , k \in \mathbb { N }$. $$\text { Consider } \begin{aligned} p ^ { 3 } = ( 3 k + 1 ) ^ { 3 } & = 27 k ^ { 3 } + 27 k ^ { 2 } + 9 k + 1
& = 3 \left( 9 k ^ { 3 } + 9 k ^ { 2 } + 3 k \right) + 1 \quad \text { which is not a multiple of } 3 \end{aligned}$$

Show the calculations and statements that are required to complete the proof.
Hence prove, by contradiction, that $\sqrt [ 3 ] { 3 }$ is an irrational number.

Edexcel P4 2024 January Q1

Find, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, the binomial expansion of

$$( 1 - 4 x ) ^ { - 3 } \quad | x | < \frac { 1 } { 4 }$$ fully simplifying each term.

Edexcel P4 2024 January Q2

Given that

$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$

find the values of the constants $A , B$ and $C$.
Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form $p \ln q + r$ where $p$, $q$ and $r$ are rational numbers.

Edexcel P4 2024 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-08_815_849_248_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The curve $C$, shown in Figure 1, has equation $$y ^ { 2 } x + 3 y = 4 x ^ { 2 } + k \quad y > 0$$ where $k$ is a constant.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$ The point $P ( p , 2 )$, where $p$ is a constant, lies on $C$.
Given that $P$ is the minimum turning point on $C$,
find
1. the value of $p$
2. the value of $k$

Edexcel P4 2024 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-12_595_588_248_740} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A cone, shown in Figure 2, has

fixed height 5 cm
base radius $r \mathrm {~cm}$
slant height $l \mathrm {~cm}$
1. Find an expression for $l$ in terms of $r$

Given that the base radius is increasing at a constant rate of 3 cm per minute,

find the rate at which the total surface area of the cone is changing when the radius of the cone is 1.5 cm . Give your answer in $\mathrm { cm } ^ { 2 }$ per minute to one decimal place.
[0pt] [The total surface area, $S$, of a cone is given by the formula $S = \pi r ^ { 2 } + \pi r l$ ]

Edexcel P4 2024 January Q5

(a) Find $\int x ^ { 2 } \cos 2 x d x$
(b) Hence solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( \frac { t \cos t } { y } \right) ^ { 2 }$$ giving your answer in the form $y ^ { n } = \mathrm { f } ( t )$ where $n$ is an integer.

Edexcel P4 2024 January Q6

Relative to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 3 \mathbf { i } + p \mathbf { j } + 7 \mathbf { k } ) + \lambda ( 2 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 8 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) + \mu ( 4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where $\lambda$ and $\mu$ are scalar parameters and $p$ is a constant.
Given that $l _ { 1 }$ and $l _ { 2 }$ intersect,

find the value of $p$,
find the position vector of the point of intersection.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$ Give your answer in degrees to one decimal place. The point $A$ lies on $l _ { 1 }$ with parameter $\lambda = 2$
The point $B$ lies on $l _ { 2 }$ with $\overrightarrow { A B }$ perpendicular to $l _ { 2 }$
Find the coordinates of $B$

Edexcel P4 2024 January Q7

(a) Using the substitution $u = 4 x + 2 \sin 2 x$, or otherwise, show that

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 4 x + 2 \sin 2 x } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 8 } \left( \mathrm { e } ^ { 2 \pi } - 1 \right)$$ Figure 3 The curve shown in Figure 3, has equation $$y = 6 \mathrm { e } ^ { 2 x + \sin 2 x } \cos x$$ The region $R$, shown shaded in Figure 3, is bounded by the positive $x$-axis, the positive $y$-axis and the curve. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid.
(b) Use the answer to part (a) to find the volume of the solid formed, giving the answer in simplest form.

Edexcel P4 2024 January Q8

Use proof by contradiction to prove that the curve with equation

$$y = 2 x + x ^ { 3 } + \cos x$$ has no stationary points.

Edexcel P4 2024 January Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-28_597_1020_251_525} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the curve $C$ with parametric equations $$x = \sec t \quad y = \sqrt { 3 } \tan \left( t + \frac { \pi } { 3 } \right) \quad \frac { \pi } { 6 } < t < \frac { \pi } { 2 }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$
Find an equation for the tangent to $C$ at the point where $t = \frac { \pi } { 3 }$ Give your answer in the form $y = m x + c$, where $m$ and $c$ are constants.
Show that all points on $C$ satisfy the equation $$y = \frac { A x ^ { 2 } + B \sqrt { 3 x ^ { 2 } - 3 } } { 4 - 3 x ^ { 2 } }$$ where $A$ and $B$ are constants to be found.

Edexcel P4 2021 June Q1

Given that $k$ is a constant and the binomial expansion of

$$\sqrt { 1 + k x } \quad | k x | < 1$$ in ascending powers of $x$ up to the term in $x ^ { 3 }$ is $$1 + \frac { 1 } { 8 } x + A x ^ { 2 } + B x ^ { 3 }$$

1. find the value of $k$,
2. find the value of the constant $A$ and the constant $B$.
Use the expansion to find an approximate value to $\sqrt { 1.15 }$ Show your working and give your answer to 6 decimal places.

Edexcel P4 2021 June Q2

2.
\includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-06_974_1088_116_548} \section*{Figure 1} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { 9 } { ( 2 x - 3 ) ^ { 1.25 } } \quad x > \frac { 3 } { 2 }$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the line with equation $y = 9$ and the line with equation $x = 6$ This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution. Find, by algebraic integration, the exact volume of the solid generated.

Edexcel P4 2021 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{960fe82f-c180-422c-b409-a5cdc5fae924-08_524_878_255_532} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A bowl with circular cross section and height 20 cm is shown in Figure 2.
The bowl is initially empty and water starts flowing into the bowl.
When the depth of water is $h \mathrm {~cm}$, the volume of water in the bowl, $V \mathrm {~cm} ^ { 3 }$, is modelled by the equation $$V = \frac { 1 } { 3 } h ^ { 2 } ( h + 4 ) \quad 0 \leqslant h \leqslant 20$$ Given that the water flows into the bowl at a constant rate of $160 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$, find, according to the model,

the time taken to fill the bowl,
the rate of change of the depth of the water, in $\mathrm { cm } \mathrm { s } ^ { - 1 }$, when $h = 5$

Edexcel P4 2021 June Q4

4. Use algebraic integration and the substitution $u = \sqrt { x }$ to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 10 } { 5 x + 2 x \sqrt { x } } \mathrm {~d} x$$ Write your answer in the form $4 \ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers to be found.
(Solutions relying entirely on calculator technology are not acceptable.)

Questions — Edexcel P4 (127 questions)

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