Questions - Edexcel - A-Level Maths

Edexcel P4 2021 January Q5

8 marks Standard +0.3

5. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
Using the substitution $u = 3 + \sqrt { 2 x - 1 }$ find the exact value of $$\int _ { 1 } ^ { 13 } \frac { 4 } { 3 + \sqrt { 2 x - 1 } } d x$$ giving your answer in the form $p + q \ln 2$, where $p$ and $q$ are integers to be found.

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VIIN SIHI NI III M M O N OO

VIIV SIHI NI IIIYM ION OC

Edexcel P4 2021 January Q6

9 marks Standard +0.8

6. A curve has equation $$4 y ^ { 2 } + 3 x = 6 y \mathrm { e } ^ { - 2 x }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The curve crosses the $y$-axis at the origin and at the point $P$.
Find the equation of the normal to the curve at $P$, writing your answer in the form $y = m x + c$ where $m$ and $c$ are constants to be found.

Edexcel P4 2021 January Q7

7 marks Standard +0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-14_620_615_278_662} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure}

Find $\int \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x$ Figure 2 shows a sketch of part of the curve with equation $$y = \mathrm { e } ^ { 2 x } \sin x \quad x \geqslant 0$$ The finite region $R$ is bounded by the curve and the $x$-axis and is shown shaded in Figure 2.
Show that the exact area of $R$ is $\frac { \mathrm { e } ^ { 2 \pi } + 1 } { 5 }$ (Solutions relying on calculator technology are not acceptable.)
Question 7 continue

Edexcel P4 2021 January Q8

6 marks Standard +0.8

8. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 1 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ 5 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 2 \\ - 5 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 3 \\ b \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters and $b$ is a constant.
Prove that for all values of $b \neq 7$, the lines $l _ { 1 }$ and $l _ { 2 }$ are skew.

Edexcel P4 2021 January Q9

10 marks Challenging +1.2

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-20_714_714_269_616} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = \tan \theta \quad y = 2 \sin 2 \theta \quad \theta \geqslant 0$$ The finite region, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and the line with equation $x = \sqrt { 3 }$ The region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.

Show that the exact volume of this solid of revolution is given by $$\int _ { 0 } ^ { k } p ( 1 - \cos 2 \theta ) d \theta$$ where $p$ and $k$ are constants to be found.
Hence find, by algebraic integration, the exact volume of this solid of revolution.

Edexcel P4 2021 January Q10

14 marks Standard +0.3

10. (a) Write $\frac { 1 } { ( H - 5 ) ( H + 3 ) }$ in partial fraction form. The depth of water in a storage tank is being monitored.
The depth of water in the tank, $H$ metres, is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - \frac { ( H - 5 ) ( H + 3 ) } { 40 }$$ where $t$ is the time, in days, from when monitoring began.
Given that the initial depth of water in the tank was 13 m ,
(b) solve the differential equation to show that $$H = \frac { 10 + 3 \mathrm { e } ^ { - 0.2 t } } { 2 - \mathrm { e } ^ { - 0.2 t } }$$ (c) Hence find the time taken for the depth of water in the tank to fall to 8 m .
(Solutions relying entirely on calculator technology are not acceptable.) According to the model, the depth of water in the tank will eventually fall to $k$ metres.
(d) State the value of the constant $k$.

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Edexcel P4 2022 January Q1

7 marks Moderate -0.3

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

$$\frac { 2 } { \sqrt { 9 - 2 x } } \quad | x | < \frac { 9 } { 2 }$$ giving each coefficient as a simplified fraction. By substituting $x = 1$ into the answer to part (a),
(b) find an approximation for $\sqrt { 7 }$, giving your answer to 4 decimal places.

Edexcel P4 2022 January Q2

4 marks Standard +0.3

2. The curve $C$ has parametric equations $$x = \frac { t ^ { 4 } } { 2 t + 1 } \quad y = \frac { t ^ { 3 } } { 2 t + 1 } \quad t > 0$$

Write down $\frac { x } { y }$ in terms of $t$, giving your answer in simplest form.
Hence show that all points on $C$ satisfy the equation $$x ^ { 3 } - 2 x y ^ { 3 } - y ^ { 4 } = 0$$

Edexcel P4 2022 January Q3

10 marks Standard +0.3

3. The curve $C$ has equation $$3 y ^ { 2 } - 11 x ^ { 2 } + 11 x y = 20 y - 36 x + 28$$

Find, in simplest form, $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. The point $P ( 4 , k )$, where $k$ is a constant, lies on $C$.
Given that $k < 0$
find the value of the gradient of $C$ at $P$

Edexcel P4 2022 January Q4

9 marks Standard +0.3

4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$

Express $\mathrm { f } ( x )$ in partial fractions.
Hence find $\int \mathrm { f } ( x ) \mathrm { d } x$
Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form $a + \ln b$, where $a$ and $b$ are rational numbers to be found.

Edexcel P4 2022 January Q5

13 marks Standard +0.3

5. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4
4
- 5 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 3

Edexcel P4 2022 January Q6

6 marks Standard +0.3

6 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }

Edexcel P4 2022 January Q7

11 marks Standard +0.3

7. Water is flowing into a large container and is leaking from a hole at the base of the container. At time $t$ seconds after the water starts to flow, the volume, $V \mathrm {~cm} ^ { 3 }$, of water in the container is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 300 - k V$$ where $k$ is a constant.

Solve the differential equation to show that, according to the model, $$V = \frac { 300 } { k } + A \mathrm { e } ^ { - k t }$$ where $A$ is a constant.
(5) Given that the container is initially empty and that when $t = 10$, the volume of water is increasing at a rate of $200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$
find the exact value of $k$.
Hence find, according to the model, the time taken for the volume of water in the container to reach 6 litres. Give your answer to the nearest second.

Edexcel P4 2022 January Q8

4 marks Standard +0.3

8. Use proof by contradiction to prove that, for all positive real numbers $x$ and $y$, $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$

Edexcel P4 2022 January Q13

Standard +0.3

13
- 1
4 \end{array} \right) + \mu \left( \begin{array} { r } 5
1
- 3 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters.

Show that $l _ { 1 }$ and $l _ { 2 }$ meet and find the position vector of their point of intersection $A$.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$, giving your answer in degrees to one decimal place. A circle with centre $A$ and radius 35 cuts the line $l _ { 1 }$ at the points $P$ and $Q$. Given that the $x$ coordinate of $P$ is greater than the $x$ coordinate of $Q$,
find the coordinates of $P$ and the coordinates of $Q$. 6. Use integration by parts to show that $$\int \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x = p \mathrm { e } ^ { 2 x } \sin 3 x + q \mathrm { e } ^ { 2 x } \cos 3 x + k$$ where $p$ and $q$ are rational numbers to be found and $k$ is an arbitrary constant.\\ (6)\\ 7. Water is flowing into a large container and is leaking from a hole at the base of the container. At time $t$ seconds after the water starts to flow, the volume, $V \mathrm {~cm} ^ { 3 }$, of water in the container is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 300 - k V$$ where $k$ is a constant.
Solve the differential equation to show that, according to the model, $$V = \frac { 300 } { k } + A \mathrm { e } ^ { - k t }$$ where $A$ is a constant.\\ (5) Given that the container is initially empty and that when $t = 10$, the volume of water is increasing at a rate of $200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$
find the exact value of $k$.
Hence find, according to the model, the time taken for the volume of water in the container to reach 6 litres. Give your answer to the nearest second.\\ 8. Use proof by contradiction to prove that, for all positive real numbers $x$ and $y$, $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$ 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{594542dd-ee2d-49b6-9fab-77b2d1a44f8c-24_632_734_214_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a closed curve with parametric equations $$x = 5 \cos \theta \quad y = 3 \sin \theta - \sin 2 \theta \quad 0 \leqslant \theta < 2 \pi$$ The region enclosed by the curve is rotated through $\pi$ radians about the $x$-axis to form a solid of revolution.
Show that the volume, $V$, of the solid of revolution is given by $$V = 5 \pi \int _ { \alpha } ^ { \beta } \sin ^ { 3 } \theta ( 3 - 2 \cos \theta ) ^ { 2 } \mathrm {~d} \theta$$ where $\alpha$ and $\beta$ are constants to be found.
Use the substitution $u = \cos \theta$ and algebraic integration to show that $V = k \pi$ where $k$ is a rational number to be found. \includegraphics[max width=\textwidth, alt={}, center]{594542dd-ee2d-49b6-9fab-77b2d1a44f8c-28_2649_1889_109_178}

Edexcel P4 2022 January Q1

6 marks Standard +0.3

The curve $C$ has equation

$$x y ^ { 2 } = x ^ { 2 } y + 6 \quad x \neq 0 \quad y \neq 0$$ Find an equation for the tangent to $C$ at the point $P ( 2,3 )$, giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers.
(6)

Edexcel P4 2022 January Q2

7 marks Standard +0.3

2. (a) Find, in ascending powers of $x$, the first three non-zero terms of the binomial series expansion of $$\sqrt [ 3 ] { 1 + 4 x ^ { 3 } } \quad | x | < \frac { 1 } { \sqrt [ 3 ] { 4 } }$$ giving each coefficient as a simplified fraction.
(b) Use the expansion from part (a) with $x = \frac { 1 } { 3 }$ to find a rational approximation to $\sqrt [ 3 ] { 31 }$ (3)

Edexcel P4 2022 January Q3

9 marks Standard +0.8

3. The curve $C$ has parametric equations $$x = 3 + 2 \sin t \quad y = \frac { 6 } { 7 + \cos 2 t } \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

Show that $C$ has Cartesian equation $$y = \frac { 12 } { ( 7 - x ) ( 1 + x ) } \quad p \leqslant x \leqslant q$$ where $p$ and $q$ are constants to be found.
Hence, find a Cartesian equation for $C$ in the form $$y = \frac { a } { x + b } + \frac { c } { x + d } \quad p \leqslant x \leqslant q$$ where $a , b , c$ and $d$ are constants.

Edexcel P4 2022 January Q4

8 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-10_378_332_246_808} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A regular icosahedron of side length $x \mathrm {~cm}$, shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length $x \mathrm {~cm}$.

Show that the surface area, $A \mathrm {~cm} ^ { 2 }$, of the icosahedron is given by $$A = 5 \sqrt { 3 } x ^ { 2 }$$ Given that the volume, $V \mathrm {~cm} ^ { 3 }$, of the icosahedron is given by $$V = \frac { 5 } { 12 } ( 3 + \sqrt { 5 } ) x ^ { 3 }$$
show that $\frac { \mathrm { d } V } { \mathrm {~d} A } = \frac { ( 3 + \sqrt { 5 } ) x } { 8 \sqrt { 3 } }$ The surface area of the icosahedron is increasing at a constant rate of $0.025 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }$
Find the rate of change of the volume of the icosahedron when $x = 2$, giving your answer to 2 significant figures.

Edexcel P4 2022 January Q5

10 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-14_688_691_251_630} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with parametric equations $$x = \sqrt { 9 - 4 t } \quad y = \frac { t ^ { 3 } } { \sqrt { 9 + 4 t } } \quad 0 \leqslant t \leqslant \frac { 9 } { 4 }$$ The curve touches the $x$-axis when $t = 0$ and meets the $y$-axis when $t = \frac { 9 } { 4 }$ The region $R$, shown shaded in Figure 2, is bounded by the curve, the $x$-axis and the $y$-axis.

Show that the area of $R$ is given by $$K \int _ { 0 } ^ { \frac { 9 } { 4 } } \frac { t ^ { 3 } } { \sqrt { 81 - 16 t ^ { 2 } } } \mathrm {~d} t$$ where $K$ is a constant to be found.
Using the substitution $u = 81 - 16 t ^ { 2 }$, or otherwise, find the exact area of $R$.
(Solutions relying on calculator technology are not acceptable.)

Edexcel P4 2022 January Q6

5 marks Moderate -0.5

Three consecutive terms in a sequence of real numbers are given by

$$k , 1 + 2 k \text { and } 3 + 3 k$$ where $k$ is a constant. Use proof by contradiction to show that this sequence is not a geometric sequence.

Edexcel P4 2022 January Q7

8 marks Standard +0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-20_473_313_244_350} \captionsetup{labelformat=empty} \caption{Figure 3}

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

\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-20_390_627_246_970} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 3 shows the design of a doorknob.
The shape of the doorknob is formed by rotating the curve shown in Figure 4 through $360 ^ { \circ }$ about the $x$-axis, where the units are centimetres. The equation of the curve is given by $$\mathrm { f } ( x ) = \frac { 1 } { 4 } ( 4 - x ) \mathrm { e } ^ { x } \quad 0 \leqslant x \leqslant 4$$

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

11 marks Moderate -0.5

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

11 marks Standard +0.3

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

9 marks Standard +0.3

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.

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