Questions P4 - A-Level Maths

Edexcel P4 2024 June Q5

5
- 1 \end{array} \right)$$ where $a$ is a constant.
Given that $\overrightarrow { O C }$ is perpendicular to $\overrightarrow { B C }$
(b) find the possible values of $a$.

The curve $C$ is defined by the equation

$$8 x ^ { 3 } - 3 y ^ { 2 } + 2 x y = 9$$ Find an equation of the normal to $C$ at the point ( 2,5 ), giving your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers. 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-08_558_542_258_749} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a segment $P Q R P$ of a circle with centre $O$ and radius 5 cm .
Given that \begin{itemize} \item angle $P O R$ is $\theta$ radians \item $\theta$ is increasing, from 0 to $\pi$, at a constant rate of 0.1 radians per second \item the area of the segment $P Q R P$ is $A \mathrm {~cm} ^ { 2 }$

Edexcel P4 2024 June Q7

7
2
- 5 \end{array} \right)$$ Given that $$\overrightarrow { A B } = \left( \begin{array} { r } - 2
4
3 \end{array} \right)$$

find the coordinates of the point $B$. The point $C$ has position vector $$\overrightarrow { O C } = \left( \begin{array} { r } a
5
- 1 \end{array} \right)$$ where $a$ is a constant.
Given that $\overrightarrow { O C }$ is perpendicular to $\overrightarrow { B C }$
find the possible values of $a$.
1. The curve $C$ is defined by the equation
$$8 x ^ { 3 } - 3 y ^ { 2 } + 2 x y = 9$$ Find an equation of the normal to $C$ at the point ( 2,5 ), giving your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-08_558_542_258_749} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a segment $P Q R P$ of a circle with centre $O$ and radius 5 cm .
Given that
- angle $P O R$ is $\theta$ radians
- $\theta$ is increasing, from 0 to $\pi$, at a constant rate of 0.1 radians per second
- the area of the segment $P Q R P$ is $A \mathrm {~cm} ^ { 2 }$
- show that
$$\frac { \mathrm { d } A } { \mathrm {~d} \theta } = K ( 1 - \cos \theta )$$ where $K$ is a constant to be found.
Find, in $\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }$, the rate of increase of the area of the segment when $\theta = \frac { \pi } { 3 }$ 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-10_803_1086_248_493} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t ^ { 2 } + 2 t \quad y = \frac { 2 } { t ( 3 - t ) } \quad a \leqslant t \leqslant b$$ where $a$ and $b$ are constants.
The ends of the curve lie on the line with equation $y = 1$
Find the value of $a$ and the value of $b$ The region $R$, shown shaded in Figure 2, is bounded by the curve and the line with equation $y = 1$
Show that the area of region $R$ is given by $$M - k \int _ { a } ^ { b } \frac { t + 1 } { t ( 3 - t ) } \mathrm { d } t$$ where $M$ and $k$ are constants to be found.
1. Write $\frac { t + 1 } { t ( 3 - t ) }$ in partial fractions.
2. Use algebraic integration to find the exact area of $R$, giving your answer in simplest form.
  1. With respect to a fixed origin $O$, the line $l _ { 1 }$ is given by the equation
$$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + \lambda ( 8 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } )$$ where $\lambda$ is a scalar parameter.
The point $A$ lies on $l _ { 1 }$
Given that $| \overrightarrow { O A } | = 5 \sqrt { 10 }$
show that at $A$ the parameter $\lambda$ satisfies $$81 \lambda ^ { 2 } + 52 \lambda - 220 = 0$$ Hence
1. show that one possible position vector for $A$ is $- 15 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }$
2. find the other possible position vector for $A$. The line $l _ { 2 }$ is parallel to $l _ { 1 }$ and passes through $O$.
  Given that
  - $\overrightarrow { O A } = - 15 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }$
point $B$ lies on $l _ { 2 }$ where $| \overrightarrow { O B } | = 4 \sqrt { 10 }$
find the area of triangle $O A B$, giving your answer to one decimal place.

The current, $x$ amps, at time $t$ seconds after a switch is closed in a particular electric circuit is modelled by the equation

Solve the differential equation to find an equation, in terms of $k$, for the current in the circuit at time $t$ seconds.
Give your answer in the form $x = \mathrm { f } ( t )$. Given that in the long term the current in the circuit approaches 7 amps,
find the value of $k$.
Hence find the time in seconds it takes for the current to reach 5 amps, giving your answer to 2 significant figures.

Edexcel P4 2024 June Q8

8. $$f ( x ) = ( 8 - 3 x ) ^ { \frac { 4 } { 3 } } \quad 0 < x < \frac { 8 } { 3 }$$

Show that the binomial expansion of $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$ is $$A - 8 x + \frac { x ^ { 2 } } { 2 } + B x ^ { 3 } + \ldots$$ where $A$ and $B$ are constants to be found.
Use proof by contradiction to prove that the curve with equation $$y = 8 + 8 x - \frac { 15 } { 2 } x ^ { 2 }$$ does not intersect the curve with equation $$y = A - 8 x + \frac { x ^ { 2 } } { 2 } + B x ^ { 3 } \quad 0 < x < \frac { 8 } { 3 }$$ where $A$ and $B$ are the constants found in part (a).
(Solutions relying on calculator technology are not acceptable.)

Edexcel P4 2024 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-26_543_604_255_733} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The curve $C$, shown in Figure 3, has equation $$y = \frac { x ^ { - \frac { 1 } { 4 } } } { \sqrt { 1 + x } ( \arctan \sqrt { x } ) }$$ The region $R$, shown shaded in Figure 3, is bounded by $C$, the line with equation $x = 3$, the $x$-axis and the line with equation $x = \frac { 1 } { 3 }$ The region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid.
Using the substitution $\tan u = \sqrt { x }$

show that the volume $V$ of the solid formed is given by $$k \int _ { a } ^ { b } \frac { 1 } { u ^ { 2 } } \mathrm {~d} u$$ where $k , a$ and $b$ are constants to be found.
Hence, using algebraic integration, find the value of $V$ in simplest form.

Edexcel P4 2020 October Q1

Given that $n$ is an integer, use algebra, to prove by contradiction, that if $n ^ { 3 }$ is even then $n$ is even.

Edexcel P4 2020 October Q2

(a) Use the binomial expansion to expand

$$( 4 - 5 x ) ^ { - \frac { 1 } { 2 } } \quad | x | < \frac { 4 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$ giving each coefficient as a fully simplified fraction. $$f ( x ) = \frac { 2 + k x } { \sqrt { 4 - 5 x } } \quad \text { where } k \text { is a constant and } | x | < \frac { 4 } { 5 }$$ Given that the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, is $$1 + \frac { 3 } { 10 } x + m x ^ { 2 } + \ldots \quad \text { where } m \text { is a constant }$$ (b) find the value of $k$,
(c) find the value of $m$.

Edexcel P4 2020 October Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{79ac81c3-cd05-4f28-8840-3c8a6960e7b7-08_801_679_125_635} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { e } ^ { 0.5 x } - 2$
The region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the $y$-axis. The region $R$ is rotated $360 ^ { \circ }$ about the $x$-axis to form a solid of revolution.
Show that the volume of this solid can be written in the form $a \ln 2 + b$, where $a$ and $b$ are constants to be found.

Edexcel P4 2020 October Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{79ac81c3-cd05-4f28-8840-3c8a6960e7b7-10_833_822_127_561} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve with parametric equations $$x = 2 t ^ { 2 } - 6 t , \quad y = t ^ { 3 } - 4 t , \quad t \in \mathbb { R }$$ The curve cuts the $x$-axis at the origin and at the points $A$ and $B$, as shown in Figure 2.

Find the coordinates of $A$ and show that $B$ has coordinates (20, 0).
Show that the equation of the tangent to the curve at $B$ is $$7 y + 4 x - 80 = 0$$ The tangent to the curve at $B$ cuts the curve again at the point $P$.
Find, using algebra, the $x$ coordinate of $P$.

Edexcel P4 2020 October Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{79ac81c3-cd05-4f28-8840-3c8a6960e7b7-14_600_1022_255_461} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure}

Find $\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$ Figure 3 shows a sketch of part of the curve with equation $$y = \frac { 3 + 2 x + \ln x } { x ^ { 2 } } \quad x > 0.5$$ The finite region $R$, shown shaded in Figure 3, is bounded by the curve, the line with equation $x = 2$, the $x$-axis and the line with equation $x = 4$
Use the answer to part (a) to find the exact area of $R$, writing your answer in simplest form.

Edexcel P4 2020 October Q6

6. A curve $C$ has equation $$y = x ^ { \sin x } \quad x > 0 \quad y > 0$$

Find, by firstly taking natural logarithms, an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Hence show that the $x$ coordinates of the stationary points of $C$ are solutions of the equation $$\tan x + x \ln x = 0$$

Edexcel P4 2020 October Q7

7. (i) Using a suitable substitution, find, using calculus, the value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { 2 x - 1 } } \mathrm {~d} x$$ (Solutions relying entirely on calculator technology are not acceptable.)
(ii) Find $$\int \frac { 6 x ^ { 2 } - 16 } { ( x + 1 ) ( 2 x - 3 ) } d x$$

Edexcel P4 2020 October Q8

8. Relative to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$\begin{aligned} & l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 4
- 3
2 \end{array} \right) + \lambda \left( \begin{array} { r } 3
- 2
- 1 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }
& l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { r } 2
0
- 9 \end{array} \right) + \mu \left( \begin{array} { r } 2
- 1
- 3 \end{array} \right) \quad \text { where } \mu \text { is a scalar parameter } \end{aligned}$$ Given that $l _ { 1 }$ and $l _ { 2 }$ meet at the point $X$,

find the position vector of $X$. The point $P ( 10 , - 7,0 )$ lies on $l _ { 1 }$
The point $Q$ lies on $l _ { 2 }$
Given that $\overrightarrow { P Q }$ is perpendicular to $l _ { 2 }$
calculate the coordinates of $Q$.

Edexcel P4 2020 October Q9

9. Bacteria are growing on the surface of a dish in a laboratory. The area of the dish, $A \mathrm {~cm} ^ { 2 }$, covered by the bacteria, $t$ days after the bacteria start to grow, is modelled by the differential equation $$\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { A ^ { \frac { 3 } { 2 } } } { 5 t ^ { 2 } } \quad t > 0$$ Given that $A = 2.25$ when $t = 3$

show that $$A = \left( \frac { p t } { q t + r } \right) ^ { 2 }$$ where $p , q$ and $r$ are integers to be found. According to the model, there is a limit to the area that will be covered by the bacteria.
Find the value of this limit.
\includegraphics[max width=\textwidth, alt={}, center]{79ac81c3-cd05-4f28-8840-3c8a6960e7b7-31_2255_50_314_34}

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Edexcel P4 2021 October Q1

The curve $C$ has equation

$$2 x - 4 y ^ { 2 } + 3 x ^ { 2 } y = 4 x ^ { 2 } + 8$$ The point $P ( 3,2 )$ lies on $C$.
Find the equation of the normal to $C$ at the point $P$, writing your answer in the form $a x + b y + c = 0$ where $a$, $b$ and $c$ are integers to be found.

Edexcel P4 2021 October Q2

2. Find the particular solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y ^ { 2 } } { \sqrt { 4 x + 5 } } \quad x > - \frac { 5 } { 4 }$$ for which $y = \frac { 1 } { 3 }$ at $x = - \frac { 1 } { 4 }$ giving your answer in the form $y = \mathrm { f } ( x )$
(6)

Edexcel P4 2021 October Q3

3. $$\mathrm { g } ( x ) = \frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 6 } { x ( x + 3 ) } \equiv A x + B + \frac { C } { x } + \frac { D } { x + 3 }$$

Find the values of the constants $A , B , C$ and $D$. A curve has equation $$y = g ( x ) \quad x > 0$$ Using the answer to part (a),
find $\mathrm { g } ^ { \prime } ( x )$.
Hence, explain why $\mathrm { g } ^ { \prime } ( x ) > 3$ for all values of $x$ in the domain of g .

Edexcel P4 2021 October Q4

4. $$\mathrm { f } ( x ) = \sqrt { 1 - 4 x ^ { 2 } } \quad | x | < \frac { 1 } { 2 }$$

Find, in ascending powers of $x$, the first four non-zero terms of the binomial expansion of $\mathrm { f } ( x )$. Give each coefficient in simplest form.
By substituting $x = \frac { 1 } { 4 }$ into the binomial expansion of $\mathrm { f } ( x )$, obtain an approximation for $\sqrt { 3 }$ Give your answer to 4 decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{08756c4b-6619-42da-ac8a-2bf065c01de8-13_42_63_2606_1852}

Edexcel P4 2021 October Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-14_787_638_251_653} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with parametric equations $$x = 5 + 2 \tan t \quad y = 8 \sec ^ { 2 } t \quad - \frac { \pi } { 3 } \leqslant t \leqslant \frac { \pi } { 4 }$$

Use parametric differentiation to find the gradient of $C$ at $x = 3$ The curve $C$ has equation $y = \mathrm { f } ( x )$, where f is a quadratic function.
Find $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a$, $b$ and $c$ are constants to be found.
Find the range of f.

Edexcel P4 2021 October Q6

6. 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]{08756c4b-6619-42da-ac8a-2bf065c01de8-18_650_938_413_504} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 16 \sin 2 x } { ( 3 + 4 \sin x ) ^ { 2 } } \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The region $R$, shown shaded in Figure 2, is bounded by the curve, the $x$-axis and the line with equation $x = \frac { \pi } { 6 }$ Using the substitution $u = 3 + 4 \sin x$, show that the area of $R$ can be written in the form $a + \ln b$, where $a$ and $b$ are rational constants to be found.

Edexcel P4 2021 October Q7

7. With respect to a fixed origin $O$,

the line $l$ has equation $\mathbf { r } = \left( \begin{array} { r } 4
2
- 3 \end{array} \right) + \lambda \left( \begin{array} { r } - 4
- 3
5 \end{array} \right)$ where $\lambda$ is a scalar constant
the point $A$ has position vector $9 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$

Given that $X$ is the point on $l$ nearest to $A$,

find
1. the coordinates of $X$
2. the shortest distance from $A$ to $l$. Give your answer in the form $\sqrt { d }$, where $d$ is an integer. The point $B$ is the image of $A$ after reflection in $l$.
Find the position vector of $B$. Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-26_668_661_408_644} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}

Edexcel P4 2021 October Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-30_528_1031_242_452} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a cylindrical tank that contains some water. The tank has an internal diameter of 8 m and an internal height of 4.2 m .
Water is flowing into the tank at a constant rate of $( 0.6 \pi ) \mathrm { m } ^ { 3 }$ per minute. There is a tap at point $T$ at the bottom of the tank. At time $t$ minutes after the tap has been opened,

the depth of the water is $h$ metres
the water is leaving the tank at a rate of $( 0.15 \pi h ) \mathrm { m } ^ { 3 }$ per minute
1. Show that

$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 12 - 3 h } { 320 }$$ Given that the depth of the water in the tank is 0.5 m when the tap is opened,

find the time taken for the depth of water in the tank to reach 3.5 m .

Edexcel P4 2021 October Q10

10. (a) A student's attempt to answer the question
"Prove by contradiction that if $n ^ { 3 }$ is even, then $n$ is even" is shown below. Line 5 of the proof is missing. Assume that there exists a number $n$ such that $n ^ { 3 }$ is even, but $n$ is odd. If $n$ is odd then $n = 2 p + 1$ where $p \in \mathbb { Z }$
So $n ^ { 3 } = ( 2 p + 1 ) ^ { 3 }$ $$\begin{aligned} & = 8 p ^ { 3 } + 12 p ^ { 2 } + 6 p + 1
& = \end{aligned}$$ This contradicts our initial assumption, so if $n ^ { 3 }$ is even, then $n$ is even. Complete this proof by filling in line 5.
(b) Hence, prove by contradiction that $\sqrt [ 3 ] { 2 }$ is irrational.

Edexcel P4 2022 October Q1

A curve $C$ has parametric equations

$$x = \frac { t } { t - 3 } \quad y = \frac { 1 } { t } + 2 \quad t \in \mathbb { R } \quad t > 3$$ Show that all points on $C$ lie on the curve with Cartesian equation $$y = \frac { a x - 1 } { b x }$$ where $a$ and $b$ are constants to be found.

Edexcel P4 2022 October Q2

(a) Express $\frac { 3 x } { ( 2 x - 1 ) ( x - 2 ) }$ in partial fraction form.
(b) Hence show that

$$\int _ { 5 } ^ { 25 } \frac { 3 x } { ( 2 x - 1 ) ( x - 2 ) } d x = \ln k$$ where $k$ is a fully simplified fraction to be found.
(Solutions relying entirely on calculator technology are not acceptable.)

Edexcel P4 2022 October Q3