Questions P4 - A-Level Maths

Edexcel P4 2021 January Q1

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

$$\left( \frac { 1 } { 4 } - 5 x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 20 }$$ giving each coefficient in its simplest form. By substituting $x = \frac { 1 } { 100 }$ into the answer for (a),
(b) find an approximation for $\sqrt { 5 }$ Give your answer in the form $\frac { a } { b }$ where $a$ and $b$ are integers to be found.

Edexcel P4 2021 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-04_511_506_264_721} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of parallelogram $A B C D$.
Given that $\overrightarrow { A B } = 6 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }$ and $\overrightarrow { B C } = 2 \mathbf { i } + 5 \mathbf { j } + 8 \mathbf { k }$

find the size of angle $A B C$, giving your answer in degrees, to 2 decimal places.
Find the area of parallelogram $A B C D$, giving your answer to one decimal place.

Edexcel P4 2021 January Q3

3. Prove by contradiction that there is no greatest odd integer.

Edexcel P4 2021 January Q4

4. The curve $C$ is defined by the parametric equations $$x = \frac { 1 } { t } + 2 \quad y = \frac { 1 - 2 t } { 3 + t } \quad t > 0$$

Show that the equation of $C$ can be written in the form $y = \mathrm { g } ( x )$ where g is the function $$\mathrm { g } ( x ) = \frac { a x + b } { c x + d } \quad x > k$$ where $a , b , c , d$ and $k$ are integers to be found.
Hence, or otherwise, state the range of g .
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Edexcel P4 2021 January Q5

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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Edexcel P4 2021 January Q6

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. \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

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

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

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

(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

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

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

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

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

Edexcel P4 2022 January Q7

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

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

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

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

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

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

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

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

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.

Questions P4 (127 questions)

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Edexcel P4 2021 January Q1

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