Questions - Edexcel P4 - A-Level Maths

Find, in ascending powers of $x$, the first four non-zero terms of the binomial expansion of $\mathrm { g } ( x )$. Give each coefficient in simplest form.
State the range of values of $x$ for which this expansion is valid.
Use the expansion from part (a) to find a fully simplified rational approximation for $\sqrt { 3 }$ Show your working and make your method clear.

Edexcel P4 2022 October Q5

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]{032d2541-9905-4570-9584-9a144b02fde5-10_741_896_383_587} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = \frac { 12 \sqrt { x } } { \left( 2 x ^ { 2 } + 3 \right) ^ { 1.5 } }$$ The region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x = \frac { 1 } { \sqrt { 2 } }$, the $x$-axis and the line with equation $x = k$. This region is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid of revolution. Given that the volume of this solid is $\frac { 713 } { 648 } \pi$, use algebraic integration to find the exact value of the constant $k$.

Edexcel P4 2022 October Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-14_768_1006_251_532} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 1 + 3 \tan t \quad y = 2 \cos 2 t \quad - \frac { \pi } { 6 } \leqslant t \leqslant \frac { \pi } { 3 }$$ The curve crosses the $x$-axis at point $P$, as shown in Figure 3.

Find the equation of the tangent to $C$ at $P$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found. The curve $C$ has equation $y = \mathrm { f } ( x )$, where f is a function with domain $[ k , 1 + 3 \sqrt { 3 } ]$
Find the exact value of the constant $k$.
Find the range of f.

Edexcel P4 2022 October Q7

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

Solutions relying entirely on calculator technology are not acceptable.

Use the substitution $u = \mathrm { e } ^ { x } - 3$ to show that $$\int _ { \ln 5 } ^ { \ln 7 } \frac { 4 \mathrm { e } ^ { 3 x } } { \mathrm { e } ^ { x } - 3 } \mathrm {~d} x = a + b \ln 2$$ where $a$ and $b$ are constants to be found.
Show, by integration, that $$\int 3 \mathrm { e } ^ { x } \cos 2 x \mathrm {~d} x = p \mathrm { e } ^ { x } \sin 2 x + q \mathrm { e } ^ { x } \cos 2 x + c$$ where $p$ and $q$ are constants to be found and $c$ is an arbitrary constant.

Edexcel P4 2022 October Q8

A student was asked to prove by contradiction that
"there are no positive integers $x$ and $y$ such that $3 x ^ { 2 } + 2 x y - y ^ { 2 } = 25$ "
The start of the student's proof is shown in the box below.

Assume that integers $x$ and $y$ exist such that $3 x ^ { 2 } + 2 x y - y ^ { 2 } = 25$ $$\Rightarrow ( 3 x - y ) ( x + y ) = 25$$ $$\begin{aligned} & \text { If } \quad ( 3 x - y ) = 1 \quad \text { and } ( x + y ) = 25
& \left. \begin{array} { l } 3 x - y = 1
x + y = 25 \end{array} \right\} \Rightarrow 4 x = 26 \Rightarrow x = 6.5 , y = 18.5 \quad \text { Not integers } \end{aligned}$$ Show the calculations and statements that are needed to complete the proof.

Edexcel P4 2022 October Q9

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

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2
8

Edexcel P4 2022 October Q10

10 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 4
- 1
2 \end{array} \right) + \mu \left( \begin{array} { l } 5
4
8 \end{array} \right) \end{aligned}$$ where $\lambda$ and $\mu$ are scalar parameters.
Prove that lines $l _ { 1 }$ and $l _ { 2 }$ are skew.

A spherical ball of ice of radius 12 cm is placed in a bucket of water.

In a model of the situation,

the ball remains spherical as it melts
$t$ minutes after the ball of ice is placed in the bucket, its radius is $r \mathrm {~cm}$
the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius
the radius of the ball of ice is 6 cm after 15 minutes

Using the model and the information given,

find an equation linking $r$ and $t$,
find the time taken for the ball of ice to melt completely.
On Diagram 1 on page 27, sketch a graph of $r$ against $t$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-27_662_728_1959_671} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure}

Edexcel P4 2022 October Q11

11. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-30_766_853_242_607} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the closed curve with equation $$( x + y ) ^ { 3 } + 10 y ^ { 2 } = 108 x$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 108 - 3 ( x + y ) ^ { 2 } } { 20 y + 3 ( x + y ) ^ { 2 } }$$ The curve is used to model the shape of a cycle track with both $x$ and $y$ measured in km .
The points $P$ and $Q$ represent points that are furthest north and furthest south of the origin $O$, as shown in Figure 4. Using the result given in part (a),
find how far the point $Q$ is south of $O$. Give your answer to the nearest 100 m .

Edexcel P4 2023 October Q1

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

$$\frac { 8 } { ( 2 - 5 x ) ^ { 2 } }$$ writing each term in simplest form.
(b) Find the range of values of $x$ for which this expansion is valid.

Edexcel P4 2023 October Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-04_271_223_246_922} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a cube which is increasing in size.
At time $t$ seconds,

the length of each edge of the cube is $x \mathrm {~cm}$
the surface area of the cube is $S \mathrm {~cm} ^ { 2 }$
the volume of the cube is $V \mathrm {~cm} ^ { 3 }$

Given that the surface area of the cube is increasing at a constant rate of $4 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }$

show that $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { k } { x }$ where $k$ is a constant to be found,
show that $\frac { \mathrm { d } V } { \mathrm {~d} t } = V ^ { p }$ where $p$ is a constant to be found.

Edexcel P4 2023 October Q3

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

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

Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ giving your answer in simplest form.
Use integration by substitution to show that $$\int _ { 3 } ^ { \frac { 21 } { 2 } } \frac { 4 x } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where $a$ and $b$ are constants to be found.

Edexcel P4 2023 October Q4

(a) Prove by contradiction that for all positive numbers $k$

$$k + \frac { 9 } { k } \geqslant 6$$ (b) Show that the result in part (a) is not true for all real numbers.

Edexcel P4 2023 October Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-12_678_987_248_539} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with equation $$y ^ { 3 } - x ^ { 2 } + 4 x ^ { 2 } y = k$$ where $k$ is a positive constant greater than 1

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P$ lies on $C$.
Given that the normal to $C$ at $P$ has equation $y = x$, as shown in Figure 2,
find the value of $k$.

Edexcel P4 2023 October Q6

The line $l _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2
3
- 7 \end{array} \right) + \lambda \left( \begin{array} { l } 1
2
2 \end{array} \right)$ where $\lambda$ is a scalar parameter.

The line $l _ { 2 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2
3
- 7 \end{array} \right) + \mu \left( \begin{array} { r } 4
- 1
8 \end{array} \right)$ where $\mu$ is a scalar parameter.
Given that $l _ { 1 }$ and $l _ { 2 }$ meet at the point $P$

state the coordinates of $P$ Given that the angle between lines $l _ { 1 }$ and $l _ { 2 }$ is $\theta$
find the value of $\cos \theta$, giving the answer as a fully simplified fraction. The point $Q$ lies on $l _ { 1 }$ where $\lambda = 6$
Given that point $R$ lies on $l _ { 2 }$ such that triangle $Q P R$ is an isosceles triangle with $P Q = P R$
find the exact area of triangle $Q P R$
find the coordinates of the possible positions of point $R$

Edexcel P4 2023 October Q7

The number of goats on an island is being monitored.

When monitoring began there were 3000 goats on the island.
In a simple model, the number of goats, $x$, in thousands, is modelled by the equation $$x = \frac { k ( 9 t + 5 ) } { 4 t + 3 }$$ where $k$ is a constant and $t$ is the number of years after monitoring began.

Show that $k = 1.8$
Hence calculate the long-term population of goats predicted by this model. In a second model, the number of goats, $x$, in thousands, is modelled by the differential equation $$3 \frac { \mathrm {~d} x } { \mathrm {~d} t } = x ( 9 - 2 x )$$
Write $\frac { 3 } { x ( 9 - 2 x ) }$ in partial fraction form.
Solve the differential equation with the initial condition to show that $$x = \frac { 9 } { 2 + \mathrm { e } ^ { - 3 t } }$$
Find the long-term population of goats predicted by this second model.

Edexcel P4 2023 October Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-24_579_642_251_715} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 6 t - 3 \sin 2 t \quad y = 2 \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The curve meets the $y$-axis at 2 and the $x$-axis at $k$, where $k$ is a constant.

State the value of $k$.
Use parametric differentiation to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t$$ where $\lambda$ is a constant to be found. The point $P$ with parameter $\mathrm { t } = \frac { \pi } { 4 }$ lies on $C$.
The tangent to $C$ at the point $P$ cuts the $y$-axis at the point $N$.
Find the exact $y$ coordinate of $N$, giving your answer in simplest form. The region bounded by the curve, the $x$-axis and the $y$-axis is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
1. Show that the volume of this solid is given by $$\int _ { 0 } ^ { \alpha } \beta ( 1 - \cos 4 t ) d t$$ where $\alpha$ and $\beta$ are constants to be found.
2. Hence, using algebraic integration, find the exact volume of this solid.

Edexcel P4 2018 Specimen Q1

Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } \quad | \boldsymbol { x } | < \frac { 2 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$
Give each coefficient as a fraction in its simplest form.

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Edexcel P4 2018 Specimen Q2

2. A curve $C$ has the equation $$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find an equation of the tangent to $C$ at the point $( 3 , - 2 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
VIII SIHI NI I IIIM I O N OC VIIN SIHI NI JIHMM ION OO VI4V SIHI NI JIIYM ION OO

Edexcel P4 2018 Specimen Q3

3. $$\mathrm { f } ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { B } { ( 3 x - 1 ) } + \frac { C } { ( 3 x - 1 ) ^ { 2 } }$$

Find the values of the constants $A , B$ and $C$
1. Hence find $\int \mathrm { f } ( x ) \mathrm { d } x$
2. Find $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$, giving your answer in the form $a + \ln b$, where $a$ and $b$ are constants.
  (6)

Edexcel P4 2018 Specimen Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-10_899_759_127_621} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with parametric equations $$x = \sqrt { 3 } \sin 2 t \quad y = 4 \cos ^ { 2 } t \quad 0 \leqslant t \leqslant \pi$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 3 } \tan 2 t$, where $k$ is a constant to be found.
Find an equation of the tangent to $C$ at the point where $t = \frac { \pi } { 3 }$ Give your answer in the form $y = a x + b$, where $a$ and $b$ are constants.

Edexcel P4 2018 Specimen Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-14_614_858_303_552} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0$ The curve meets the $x$-axis at the origin $O$ and cuts the $x$-axis at the point $A$ .

Find,in terms of $\ln 2$ ,the $x$ coordinate of the point $A$ .
Find $\int x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$ The finite region $R$ ,shown shaded in Figure 2,is bounded by the $x$-axis and the curve with equation $y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0$
Find,by integration,the exact value for the area of $R$ . Give your answer in terms of $\ln 2$
\includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-18_2655_1943_114_118}

Edexcel P4 2018 Specimen Q6

6. Prove by contradiction that, if $a , b$ are positive real numbers, then $a + b \geqslant 2 \sqrt { a b }$
\includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-20_2655_1943_114_118}

Edexcel P4 2018 Specimen Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-21_664_1244_301_351} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) \quad y = 2 \sin t \quad 0 \leqslant t \leqslant 2 \pi$$

Show that $$x + y = 2 \sqrt { 3 } \cos t$$
Show that a cartesian equation of $C$ is $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where $a$ and $b$ are integers to be found.
\includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-22_2673_1948_107_118}

Edexcel P4 2018 Specimen Q8

8. Water is being heated in a kettle. At time $t$ seconds, the temperature of the water is $\theta ^ { \circ } \mathrm { C }$. The rate of increase of the temperature of the water at time $t$ is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \lambda ( 120 - \theta ) \quad \theta \leqslant 100$$ where $\lambda$ is a positive constant.
Given that $\theta = 20$ when $t = 0$

solve this differential equation to show that $$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t }$$ When the temperature of the water reaches $100 ^ { \circ } \mathrm { C }$, the kettle switches off.
Given that $\lambda = 0.01$, find the time, to the nearest second, when the kettle switches off.
\includegraphics[max width=\textwidth, alt={}]{4de08317-5fb9-4789-8d57-ccf463224c78-26_2642_1833_118_118}

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Edexcel P4 2018 Specimen Q9

With respect to a fixed origin $O$, the line $l _ { 1 }$ is given by the equation

$$\mathbf { r } = \left( \begin{array} { r } 8
1
- 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5
4
3 \end{array} \right)$$ where $\mu$ is a scalar parameter.
The point $A$ lies on $l _ { 1 }$ where $\mu = 1$

Find the coordinates of $A$. The point $P$ has position vector $\left( \begin{array} { l } 1
5
2 \end{array} \right)$
The line $l _ { 2 }$ passes through the point $P$ and is parallel to the line $l _ { 1 }$
Write down a vector equation for the line $l _ { 2 }$
Find the exact value of the distance $A P$. Give your answer in the form $k \sqrt { 2 }$, where $k$ is a constant to be found. The acute angle between $A P$ and $l _ { 2 }$ is $\theta$
Find the value of $\cos \theta$ A point $E$ lies on the line $l _ { 2 }$
Given that $A P = P E$,
find the area of triangle $A P E$,
find the coordinates of the two possible positions of $E$.

Questions — Edexcel P4 (127 questions)

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