Questions - Edexcel CP1 - A-Level Maths

Edexcel CP1 2022 June Q3

(a) Determine the general solution of the differential equation

$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = \mathrm { e } ^ { 2 x } \cos ^ { 2 } x$$ giving your answer in the form $y = \mathrm { f } ( x )$ Given that $y = 3$ when $x = 0$
(b) determine the smallest positive value of $x$ for which $y = 0$

Edexcel CP1 2022 June Q4

(a) Use the method of differences to prove that for $n > 2$

$$\sum _ { r = 2 } ^ { n } \ln \left( \frac { r + 1 } { r - 1 } \right) \equiv \ln \left( \frac { n ( n + 1 ) } { 2 } \right)$$ (4)
(b) Hence find the exact value of $$\sum _ { r = 51 } ^ { 100 } \ln \left( \frac { r + 1 } { r - 1 } \right) ^ { 35 }$$ Give your answer in the form $a \ln \left( \frac { b } { c } \right)$ where $a , b$ and $c$ are integers to
be determined.

Edexcel CP1 2022 June Q5

5. $$\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 3
2 & 3 & 0
4 & a & 2 \end{array} \right) \quad \text { where } a \text { is a constant }$$

Show that $\mathbf { M }$ is non-singular for all values of $a$.
Determine, in terms of $a , \mathbf { M } ^ { - 1 }$

Edexcel CP1 2022 June Q6

(a) Express as partial fractions

$$\frac { 2 x ^ { 2 } + 3 x + 6 } { ( x + 1 ) \left( x ^ { 2 } + 4 \right) }$$ (b) Hence, show that $$\int _ { 0 } ^ { 2 } \frac { 2 x ^ { 2 } + 3 x + 6 } { ( x + 1 ) \left( x ^ { 2 } + 4 \right) } d x = \ln ( a \sqrt { 2 } ) + b \pi$$ where $a$ and $b$ are constants to be determined.

Edexcel CP1 2022 June Q7

Given that $z = a + b \mathrm { i }$ is a complex number where $a$ and $b$ are real constants,
1. show that $z z ^ { * }$ is a real number.
Given that
- $z z ^ { * } = 18$
- $\frac { z } { z ^ { * } } = \frac { 7 } { 9 } + \frac { 4 \sqrt { 2 } } { 9 } \mathrm { i }$
- determine the possible complex numbers $z$

Edexcel CP1 2022 June Q8

(a) Given

$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad n \in \mathbb { N }$$ show that $$32 \cos ^ { 6 } \theta \equiv \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10$$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f237de57-ed6d-4bea-8bb0-1b4e5b66d7da-22_218_357_653_331} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{f237de57-ed6d-4bea-8bb0-1b4e5b66d7da-22_307_824_621_897} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 1 shows a solid paperweight with a flat base.
Figure 2 shows the curve with equation $$y = H \cos ^ { 3 } \left( \frac { x } { 4 } \right) \quad - 4 \leqslant x \leqslant 4$$ where $H$ is a positive constant and $x$ is in radians.
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x = - 4$, the line with equation $x = 4$ and the $x$-axis. The paperweight is modelled by the solid of revolution formed when $R$ is rotated $\mathbf { 1 8 0 } ^ { \circ }$ about the $x$-axis. Given that the maximum height of the paperweight is 2 cm ,
(b) write down the value of $H$.
(c) Using algebraic integration and the result in part (a), determine, in $\mathrm { cm } ^ { 3 }$, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.
[0pt] [Solutions based entirely on calculator technology are not acceptable.]
(d) State a limitation of the model.

Edexcel CP1 2022 June Q9

(i) (a) Explain why $\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x$ is an improper integral.
(b) Show that $\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x$ is divergent.
(ii)

$$4 \sinh x = p \cosh x \quad \text { where } p \text { is a real constant }$$ Given that this equation has real solutions, determine the range of possible values for $p$

Edexcel CP1 2022 June Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f237de57-ed6d-4bea-8bb0-1b4e5b66d7da-28_428_301_246_881} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The motion of a pendulum, shown in Figure 3, is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ where $\theta$ is the angle, in radians, that the pendulum makes with the downward vertical, $t$ seconds after it begins to move.

1. Show that a particular solution of the differential equation is $$\theta = \frac { 1 } { 12 } t \sin 3 t$$
2. Hence, find the general solution of the differential equation. Initially, the pendulum
  - makes an angle of $\frac { \pi } { 3 }$ radians with the downward vertical
is at rest
determine, according to the model, the value of $\alpha$ to 3 significant figures. Given that the true value of $\alpha$ is 0.62
evaluate the model. The differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ models the motion of the pendulum as moving with forced harmonic motion.
Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

Edexcel CP1 2023 June Q1

The cubic equation

$$x ^ { 3 } - 7 x ^ { 2 } - 12 x + 6 = 0$$ has roots $\alpha , \beta$ and $\gamma$.
Without solving the equation, determine a cubic equation whose roots are ( $\alpha + 2$ ), $( \beta + 2 )$ and $( \gamma + 2 )$, giving your answer in the form $w ^ { 3 } + p w ^ { 2 } + q w + r = 0$, where $p , q$ and $r$ are integers to be found.

Edexcel CP1 2023 June Q2

(a) Write $x ^ { 2 } + 4 x - 5$ in the form $( x + p ) ^ { 2 } + q$ where $p$ and $q$ are integers.
(b) Hence use a standard integral from the formula book to find

$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form $A \ln B$ where $A$ and $B$ are constants in simplest form.

Edexcel CP1 2023 June Q3

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

$$z _ { 1 } = - 4 + 4 i$$

Express $\mathrm { z } _ { 1 }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r \in \mathbb { R } , r > 0$ and $0 \leqslant \theta < 2 \pi$ $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
Determine in the form $a + \mathrm { i } b$, where $a$ and $b$ are exact real numbers,
1. $\frac { Z _ { 1 } } { Z _ { 2 } }$
2. $\left( z _ { 2 } \right) ^ { 4 }$
Show on a single Argand diagram
1. the complex numbers $z _ { 1 } , z _ { 2 }$ and $\frac { z _ { 1 } } { z _ { 2 } }$
2. the region defined by $\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}$

Edexcel CP1 2023 June Q4

Prove by induction that for $n \in \mathbb { N }$

$$\left( \begin{array} { c c } 1 & - 2
0 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & - 2 n
0 & 1 \end{array} \right)$$

Edexcel CP1 2023 June Q5

The line $l _ { 1 }$ has equation $\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }$

The plane $\Pi _ { 1 }$ has equation $2 x + 3 y - 2 z = 6$

Find the point of intersection of $l _ { 1 }$ and $\Pi _ { 1 }$ The line $l _ { 2 }$ is the reflection of the line $l _ { 1 }$ in the plane $\Pi _ { 1 }$
Show that a vector equation for the line $l _ { 2 }$ is $$\mathbf { r } = \left( \begin{array} { r } - 7
2
- 7 \end{array} \right) + \mu \left( \begin{array} { c } 10

Edexcel CP1 2023 June Q6

6
2 \end{array} \right)$$ where $\mu$ is a scalar parameter. The plane $\Pi _ { 2 }$ contains the line $l _ { 1 }$ and the line $l _ { 2 }$
(c) Determine a vector equation for the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$ The plane $\Pi _ { 3 }$ has equation r. $\left( \begin{array} { l } 1
1
a \end{array} \right) = b$ where $a$ and $b$ are constants.
Given that the planes $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$ form a sheaf,
(d) determine the value of $a$ and the value of $b$.

Water is flowing into and out of a large tank.

Initially the tank contains 10 litres of water.
The rate of flow of the water is modelled so that

there are $V$ litres of water in the tank at time $t$ minutes after the water begins to flow
water enters the tank at a rate of $\left( 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } \right)$ litres per minute
water leaves the tank at a rate proportional to the volume of water remaining in the tank

Given that when $t = 0$ the volume of water in the tank is decreasing at a rate of 3 litres per minute, use the model to
(a) show that the volume of water in the tank at time $t$ satisfies $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } - 0.4 V$$ (b) Determine $\frac { \mathrm { d } } { \mathrm { d } t } \left( \arctan \mathrm { e } ^ { 0.4 t } \right)$ Hence, by solving the differential equation from part (a),
(c) determine an equation for the volume of water in the tank at time $t$. Give your answer in simplest form as $V = \mathrm { f } ( t )$ After 10 minutes, the volume of water in the tank was 8 litres.
(d) Evaluate the model in light of this information.

Edexcel CP1 2023 June Q7

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
1. Explain why, for $n \in \mathbb { N }$
$$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$ for any function $\mathrm { f } ( r )$.
Use the standard summation formulae to show that, for $n \in \mathbb { N }$ $$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$
Hence evaluate $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$

Edexcel CP1 2023 June Q10

10
6
2 \end{array} \right)$$ where $\mu$ is a scalar parameter. The plane $\Pi _ { 2 }$ contains the line $l _ { 1 }$ and the line $l _ { 2 }$
(c) Determine a vector equation for the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$ The plane $\Pi _ { 3 }$ has equation r. $\left( \begin{array} { l } 1
1
a \end{array} \right) = b$ where $a$ and $b$ are constants.
Given that the planes $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$ form a sheaf,
(d) determine the value of $a$ and the value of $b$.

Water is flowing into and out of a large tank.

Initially the tank contains 10 litres of water.
The rate of flow of the water is modelled so that

there are $V$ litres of water in the tank at time $t$ minutes after the water begins to flow
water enters the tank at a rate of $\left( 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } \right)$ litres per minute
water leaves the tank at a rate proportional to the volume of water remaining in the tank

Given that when $t = 0$ the volume of water in the tank is decreasing at a rate of 3 litres per minute, use the model to
(a) show that the volume of water in the tank at time $t$ satisfies $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } - 0.4 V$$ (b) Determine $\frac { \mathrm { d } } { \mathrm { d } t } \left( \arctan \mathrm { e } ^ { 0.4 t } \right)$ Hence, by solving the differential equation from part (a),
(c) determine an equation for the volume of water in the tank at time $t$. Give your answer in simplest form as $V = \mathrm { f } ( t )$ After 10 minutes, the volume of water in the tank was 8 litres.
(d) Evaluate the model in light of this information.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
(a) Explain why, for $n \in \mathbb { N }$

$$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$ for any function $\mathrm { f } ( r )$.
(b) Use the standard summation formulae to show that, for $n \in \mathbb { N }$ $$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$ (c) Hence evaluate $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$

A colony of small mammals is being studied.

In the study, the mammals are divided into 3 categories

$N$ (newborns)	0 to less than 1 month old
$J$ (juveniles)	1 to 3 months old
$B$ (breeders)	over 3 months old

(a) State one limitation of the model regarding the division into these categories. A model for the population of the colony is given by the matrix equation $$\left( \begin{array} { l } N _ { n + 1 }
J _ { n + 1 }
B _ { n + 1 } \end{array} \right) = \left( \begin{array} { c c c } 0 & 0 & 2
a & b & 0
0 & 0.48 & 0.96 \end{array} \right) \left( \begin{array} { l } N _ { n }
J _ { n }
B _ { n } \end{array} \right)$$ where $a$ and $b$ are constants, and $N _ { n } , J _ { n }$ and $B _ { n }$ are the respective numbers of the mammals in each category $n$ months after the start of the study. At the start of the study the colony has breeders only, with no newborns or juveniles.
According to the model, after 2 months the number of newborns is 48 and the number of juveniles is 40
(b) (i) Determine the number of mammals in the colony at the start of the study.
(ii) Show that $a = 0.8$
(c) Determine, in terms of $b$, $$\left( \begin{array} { c c c } 0 & 0 & 2
0.8 & b & 0
0 & 0.48 & 0.96 \end{array} \right) ^ { - 1 }$$ Given that the model predicts approximately 1015 mammals in total at the start of a particular month, and approximately 596 newborns, 464 juveniles and 437 breeders at the start of the next month,
(d) determine the value of $b$, giving your answer to 2 decimal places. It is decided to monitor the number of newborn males and females as a part of the study. Assuming that $42 \%$ of newborns are male,
(e) refine the matrix equation for the model to reflect this information, giving a reason for your answer.
(There is no need to estimate any unknown values for the refined model, but any known values should be made clear.)

Edexcel CP1 2024 June Q1

1. $$\mathrm { f } ( z ) = z ^ { 4 } - 6 z ^ { 3 } + a z ^ { 2 } + b z + 145$$ where $a$ and $b$ are real constants.
Given that $2 + 5 \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

determine the other roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
Show all the roots of $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.

Edexcel CP1 2024 June Q2

The roots of the equation

$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$ are $\alpha , \beta$ and $\gamma$
Without solving the equation,

write down the value of each of $$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
Use the answers to part (a) to determine the value of
1. $\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }$
2. $( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )$
3. $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$

Edexcel CP1 2024 June Q3

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dc3e8e46-c60b-4263-9652-d7c2a322cfae-10_563_561_395_753} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the design for a bathing pool.
The pool, $P$, shown unshaded in Figure 1, is surrounded by a tiled area, $T$, shown shaded in Figure 1. The tiled area is bounded by the edge of the pool and by a circle, $C$, with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation $$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ where $a$ is a positive constant.
Given that the shortest distance between the edge of the pool and the circle $C$ is 0.5 m ,

determine the value of $a$.
Hence, using algebraic integration, determine, according to the model, the exact area of $T$.

Edexcel CP1 2024 June Q4

The complex number $z = \mathrm { e } ^ { \mathrm { i } \theta }$, where $\theta$ is real.
1. Show that
$$z ^ { n } + \frac { 1 } { z ^ { n } } \equiv 2 \cos n \theta$$ where $n$ is a positive integer.
Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
Hence, making your reasoning clear, determine all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval $0 \leqslant \theta < 2 \pi$

Edexcel CP1 2024 June Q5

A raindrop falls from rest from a cloud. The velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically downwards, of the raindrop, $t$ seconds after the raindrop starts to fall, is modelled by the differential equation

$$( t + 2 ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 v = k ( t + 2 ) - 3 \quad t \geqslant 0$$ where $k$ is a positive constant.

Solve the differential equation to show that $$v = \frac { k } { 4 } ( t + 2 ) - 1 + \frac { 4 ( 2 - k ) } { ( t + 2 ) ^ { 3 } }$$ Given that $v = 4$ when $t = 2$
determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.
Comment on the validity of the model for very large values of $t$

Edexcel CP1 2024 June Q6

Prove by induction that, for all positive integers $n$,

$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

Edexcel CP1 2024 June Q7

The line $l _ { 1 }$ has equation

$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line $l _ { 2 }$ has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where $\lambda$ and $\mu$ are scalar parameters and $p$ is a constant.
The plane $\Pi$ contains $l _ { 1 }$ and $l _ { 2 }$

Show that the vector $3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }$ is perpendicular to $\Pi$
Hence determine a Cartesian equation of $\Pi$
Hence determine the value of $p$ Given that
- the lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $A$
- the point $B$ has coordinates $( 12 , - 11,6 )$
- determine, to the nearest degree, the acute angle between $A B$ and $\Pi$

Edexcel CP1 2024 June Q8

A scientist is studying the effect of introducing a population of type $A$ bacteria into a population of type $B$ bacteria.

At time $t$ days, the number of type $A$ bacteria, $x$, and the number of type $B$ bacteria, $y$, are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$

Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
Determine a general solution for the number of type $A$ bacteria at time $t$ days.
Determine a general solution for the number of type $B$ bacteria at time $t$ days. The model predicts that, at time $T$ hours, the number of bacteria in the two populations will be equal. Given that $x = 100$ and $y = 275$ when $t = 0$
determine the value of $T$, giving your answer to 2 decimal places.
Suggest a limitation of the model.

Edexcel CP1 Specimen Q1

Prove that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( a n + b ) } { 12 ( n + 2 ) ( n + 3 ) }$$ where $a$ and $b$ are constants to be found.

Questions — Edexcel CP1 (58 questions)

Browse by module

Browse by board