Questions P4 - A-Level Maths

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { - 2 x } + 3 } { \mathrm { e } ^ { - 2 x } - 2 y }$$ The curve crosses the $y$-axis at the origin and at the point $P$.
The tangent to the curve at the origin and the tangent to the curve at $P$ meet at the point $R$.
Find the coordinates of $R$.
\includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-17_2644_1838_121_116}

Edexcel P4 2021 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{960fe82f-c180-422c-b409-a5cdc5fae924-18_563_844_255_552} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 2 \cos 2 t \quad y = 4 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region $R$, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and the $y$-axis.

1. Show, making your working clear, that the area of $R = \int _ { 0 } ^ { \frac { \pi } { 4 } } 32 \sin ^ { 2 } t \cos t d t$
2. Hence find, by algebraic integration, the exact value of the area of $R$.
Show that all points on $C$ satisfy $y = \sqrt { a x + b }$, where $a$ and $b$ are constants to be found. The curve $C$ has equation $y = \mathrm { f } ( x )$ where f is the function $$f ( x ) = \sqrt { a x + b } \quad - 2 \leqslant x \leqslant 2$$ and $a$ and $b$ are the constants found in part (b).
State the range of f.

Edexcel P4 2021 June Q7

Relative to a fixed origin $O$, the line $l$ has equation

$$\mathbf { r } = \left( \begin{array} { r } 1
- 10
- 9 \end{array} \right) + \lambda \left( \begin{array} { l } 4
4
2 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }$$ Given that $\overrightarrow { O A }$ is a unit vector parallel to $l$,

find $\overrightarrow { O A }$ The point $X$ lies on $l$.
Given that $X$ is the point on $l$ that is closest to the origin,
find the coordinates of $X$. The points $O , X$ and $A$ form the triangle $O X A$.
Find the exact area of triangle $O X A$.

Edexcel P4 2021 June Q8

8. (a) Given that $y = 1$ at $x = 0$, solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y ^ { \frac { 1 } { 3 } } } { \mathrm { e } ^ { 2 x } } \quad y \geqslant 0$$ giving your answer in the form $y ^ { 2 } = \mathrm { g } ( x )$.
(b) Hence find the equation of the horizontal asymptote to the curve with equation $y ^ { 2 } = \mathrm { g } ( x )$.
\includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-27_2644_1840_118_111} \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-29_2646_1838_121_116}

Edexcel P4 2021 June Q9

9. (i) Relative to a fixed origin $O$, the points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ respectively. Points $A , B$ and $C$ lie in a straight line, with $B$ lying between $A$ and $C$.
Given $A B : A C = 1 : 3$ show that $$\mathbf { c } = 3 \mathbf { b } - 2 \mathbf { a }$$ (ii) Given that $n \in \mathbb { N }$, prove by contradiction that if $n ^ { 2 }$ is a multiple of 3 then $n$ is a multiple of 3

\includegraphics[max width=\textwidth, alt={}]{960fe82f-c180-422c-b409-a5cdc5fae924-32_2644_1837_118_114}

Edexcel P4 2022 June Q1

The binomial expansion of

$$( 3 + k x ) ^ { - 2 } \quad | k x | < 3$$ where $k$ is a non-zero constant, may be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } + \ldots$$ where $A$, $B$, $C$ and $D$ are constants.

Find the value of $A$ Given that $C = 3 B$
show that $$k ^ { 2 } + 6 k = 0$$
Hence (i) find the value of $k$
(ii) find the value of $D$

Edexcel P4 2022 June Q2

(a) Express $\frac { 1 } { ( 1 + 3 x ) ( 1 - x ) }$ in partial fractions.
(b) Hence find the solution of the differential equation

$$( 1 + 3 x ) ( 1 - x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = \tan y \quad - \frac { 1 } { 3 } < x \leqslant \frac { 1 } { 2 }$$ for which $x = \frac { 1 } { 2 }$ when $y = \frac { \pi } { 2 }$
Give your answer in the form $\sin ^ { n } y = \mathrm { f } ( x )$ where $n$ is an integer to be found.

Edexcel P4 2022 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2dffe245-b18a-4486-af8e-bad598ceb6e8-08_401_652_246_708} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A tablet is dissolving in water.
The tablet is modelled as a cylinder, shown in Figure 1.
At $t$ seconds after the tablet is dropped into the water, the radius of the tablet is $x \mathrm {~mm}$ and the length of the tablet is $3 x \mathrm {~mm}$. The cross-sectional area of the tablet is decreasing at a constant rate of $0.5 \mathrm {~mm} ^ { 2 } \mathrm {~s} ^ { - 1 }$

Find $\frac { \mathrm { d } x } { \mathrm {~d} t }$ when $x = 7$
Find, according to the model, the rate of decrease of the volume of the tablet when $x = 4$

Edexcel P4 2022 June Q4

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

\section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$16 x ^ { 3 } - 9 k x ^ { 2 } y + 8 y ^ { 3 } = 875$$ where $k$ is a constant.

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 k x y - 16 x ^ { 2 } } { 8 y ^ { 2 } - 3 k x ^ { 2 } }$$ Given that the curve has a turning point at $x = \frac { 5 } { 2 }$
find the value of $k$

Edexcel P4 2022 June Q5

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

\section*{Solutions relying on calculator technology are not acceptable.}

Use the substitution $x = 2 \sin u$ to show that $$\int _ { 0 } ^ { 1 } \frac { 3 x + 2 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x = \int _ { 0 } ^ { p } \left( \frac { 3 } { 2 } \operatorname { secutanu } + \frac { 1 } { 2 } \sec ^ { 2 } u \right) d u$$ where $p$ is a constant to be found.
Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x + 2 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$

Edexcel P4 2022 June Q6

Relative to a fixed origin $O$,

the point $A$ has position vector $\quad \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k }$
the point $B$ has position vector $5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$
the point $C$ has position vector $3 \mathbf { i } + p \mathbf { j } - \mathbf { k }$
where $p$ is a constant.
The line $l$ passes through $A$ and $B$.
1. Find a vector equation for the line $l$

Given that $\overrightarrow { A C }$ is perpendicular to $l$

find the value of $p$

Hence find the area of triangle $A B C$, giving your answer as a surd in simplest form.

Edexcel P4 2022 June Q7

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} The curve $C$ has parametric equations $$x = \sin t - 3 \cos ^ { 2 } t \quad y = 3 \sin t + 2 \cos t \quad 0 \leqslant t \leqslant 5$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3$ where $t = \pi$ The point $P$ lies on $C$ where $t = \pi$
Find the equation of the tangent to the curve at $P$ in the form $y = m x + c$ where $m$ and $c$ are constants to be found. Given that the tangent to the curve at $P$ cuts $C$ at the point $Q$
show that the value of $t$ at point $Q$ satisfies the equation $$9 \cos ^ { 2 } t + 2 \cos t - 7 = 0$$
Hence find the exact value of the $y$ coordinate of $Q$

Edexcel P4 2022 June Q8

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]{2dffe245-b18a-4486-af8e-bad598ceb6e8-26_446_492_434_447} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2dffe245-b18a-4486-af8e-bad598ceb6e8-26_441_495_402_1139} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 2 shows the curve with equation $$y = 10 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad 0 \leqslant x \leqslant 10$$ The finite region $R$, shown shaded in Figure 2, is bounded by the curve, the $x$-axis and the line with equation $x = 10$ The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.

Show that the volume, $V$, of this solid is given by $$V = k \int _ { 0 } ^ { 10 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$ where $k$ is a constant to be found.
Find $\int x ^ { 2 } e ^ { - x } d x$ Figure 3 represents an exercise weight formed by joining two of these solids together.
The exercise weight has mass 5 kg and is 20 cm long.
Given that $$\text { density } = \frac { \text { mass } } { \text { volume } }$$ and using your answers to part (a) and part (b),
find the density of this exercise weight. Give your answer in grams per $\mathrm { cm } ^ { 3 }$ to 3 significant figures.

Edexcel P4 2022 June Q9

Use proof by contradiction to show that, when $n$ is an integer,

$$n ^ { 2 } - 2$$ is never divisible by 4

Edexcel P4 2023 June Q1

(a) Find the first 4 terms of the binomial expansion, in ascending powers of $x$, of

$$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form. Given that $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { n } \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } = \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } }$$ (b) write down the value of $n$.
(c) Hence, or otherwise, find the first 3 terms of the binomial expansion, in ascending powers of $x$, of $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form.

Edexcel P4 2023 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2bacec90-3b67-4307-9608-246ecdb6b5e2-06_695_700_251_683} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $$2 ^ { x } - 4 x y + y ^ { 2 } = 13 \quad y \geqslant 0$$ The point $P$ lies on $C$ and has $x$ coordinate 2

Find the $y$ coordinate of $P$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The tangent to $C$ at $P$ crosses the $x$-axis at the point $Q$.
Find the $x$ coordinate of $Q$, giving your answer in the form $\frac { a \ln 2 + b } { c \ln 2 + d }$ where $a , b , c$ and $d$ are integers to be found.

Edexcel P4 2023 June Q3

3. $$\mathrm { f } ( x ) = \frac { 8 x - 5 } { ( 2 x - 1 ) ( 4 x - 3 ) } \quad x > 1$$

Express $\mathrm { f } ( x )$ in partial fractions.
Hence find $\int \mathrm { f } ( x ) \mathrm { d } x$
Use the answer to part (b) to find the value of $k$ for which $$\int _ { k } ^ { 3 k } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln 20$$

Edexcel P4 2023 June Q4

Relative to a fixed origin $O$,

the point $A$ has position vector $4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }$
the point $B$ has position vector $5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }$
the point $P$ has position vector $2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$

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

Find a vector equation for $l$. The point $C$ lies on $l$ so that $P C$ is perpendicular to $l$.
Find the coordinates of $C$. The point $P ^ { \prime }$ is the reflection of $P$ in the line $l$.
Find the coordinates of $P ^ { \prime }$
Hence find $\left| \overrightarrow { P P ^ { \prime } } \right|$, giving your answer as a simplified surd.

Edexcel P4 2023 June Q5

(i) Find

$$\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$$ (4)
(ii) Use the substitution $u = \sqrt { 1 - 3 x }$ to show that $$\int \frac { 27 x } { \sqrt { 1 - 3 x } } \mathrm {~d} x = - 2 ( 1 - 3 x ) ^ { \frac { 1 } { 2 } } ( A x + B ) + k$$ where $A$ and $B$ are integers to be found and $k$ is an arbitrary constant.

Edexcel P4 2023 June Q6

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

The temperature, $\theta ^ { \circ } \mathrm { C }$, of a car engine, $t$ minutes after the engine is turned off, is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 15 ) ^ { 2 }$$ where $k$ is a constant.
Given that the temperature of the car engine

is $85 ^ { \circ } \mathrm { C }$ at the instant the engine is turned off
is $40 ^ { \circ } \mathrm { C }$ exactly 10 minutes after the engine is turned off
1. solve the differential equation to show that, according to the model

$$\theta = \frac { a t + b } { c t + d }$$ where $a , b , c$ and $d$ are integers to be found.

Hence find, according to the model, the time taken for the temperature of the car engine to reach $20 ^ { \circ } \mathrm { C }$. Give your answer to the nearest minute.

Edexcel P4 2023 June Q7

Use proof by contradiction to prove that $\sqrt { 7 }$ is irrational.
(You may assume that if $k$ is an integer and $k ^ { 2 }$ is a multiple of 7 then $k$ is a multiple of 7 )

Edexcel P4 2023 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2bacec90-3b67-4307-9608-246ecdb6b5e2-28_664_844_255_612} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve $C$ with parametric equations $$x = t + \frac { 1 } { t } \quad y = t - \frac { 1 } { t } \quad t > 0.7$$ The curve $C$ intersects the $x$-axis at the point $Q$.

Find the $x$ coordinate of $Q$. The line $l$ is the normal to $C$ at the point $P$ as shown in Figure 2.
Given that $t = 2$ at $P$
write down the coordinates of $P$
Using calculus, show that an equation of $l$ is $$3 x + 5 y = 15$$ The region, $R$, shown shaded in Figure 2 is bounded by the curve $C$, the line $l$ and the $x$-axis.
Using algebraic integration, find the exact volume of the solid of revolution formed when the region $R$ is rotated through $2 \pi$ radians about the $x$-axis.

Edexcel P4 2024 June Q1

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

Find $$\int _ { 0 } ^ { \frac { \pi } { 6 } } x \cos 3 x d x$$ giving your answer in simplest form.

Edexcel P4 2024 June Q2

With respect to a fixed origin, $O$, the point $A$ has position vector

$$\overrightarrow { O A } = \left( \begin{array} { r }

Edexcel P4 2024 June Q4

4
3 \end{array} \right)$$

find the coordinates of the point $B$. The point $C$ has position vector $$\overrightarrow { O C } = \left( \begin{array} { r } a

Questions P4 (127 questions)

Browse by module

Browse by board

Edexcel P4 2021 June Q5

Edexcel P4 2021 June Q6

Edexcel P4 2021 June Q7

Edexcel P4 2021 June Q8

Edexcel P4 2021 June Q9

Edexcel P4 2022 June Q1

Edexcel P4 2022 June Q2

Edexcel P4 2022 June Q3

Edexcel P4 2022 June Q4

Edexcel P4 2022 June Q5

Edexcel P4 2022 June Q6

Edexcel P4 2022 June Q7

Edexcel P4 2022 June Q8

Edexcel P4 2022 June Q9

Edexcel P4 2023 June Q1

Edexcel P4 2023 June Q2

Edexcel P4 2023 June Q3

Edexcel P4 2023 June Q4

Edexcel P4 2023 June Q5

Edexcel P4 2023 June Q6

Edexcel P4 2023 June Q7

Edexcel P4 2023 June Q8

Edexcel P4 2024 June Q1

Edexcel P4 2024 June Q2

Edexcel P4 2024 June Q4