Questions - Edexcel CP AS

Edexcel CP AS 2020 June Q8

Prove by induction that, for $n \in \mathbb { Z } ^ { + }$

$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7

Edexcel CP AS 2020 June Q9

The cubic equation

$$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$ has roots $\alpha , \beta$, and $\gamma$.
Without solving the cubic equation,

determine the value of $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$
find a cubic equation that has roots $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$, giving your answer in the form $x ^ { 3 } + a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers to be determined.

Edexcel CP AS 2020 June Q10

Given that there are two distinct complex numbers $z$ that satisfy

$$\{ z : | z - 3 - 5 i | = 2 r \} \cap \quad z : \arg ( z - 2 ) = \frac { 3 \pi } { 4 }$$ determine the exact range of values for the real constant $r$.

Edexcel CP AS 2021 June Q1

1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1
1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0
0 & 3 \end{array} \right)$$

1. Describe fully the single geometrical transformation $P$ represented by the matrix $\mathbf { P }$.
2. Describe fully the single geometrical transformation $Q$ represented by the matrix $\mathbf { Q }$. The transformation $P$ followed by the transformation $Q$ is the transformation $R$, which is represented by the matrix $\mathbf { R }$.
Determine $\mathbf { R }$.
1. Evaluate the determinant of $\mathbf { R }$.
2. Explain how the value obtained in (c)(i) relates to the transformation $R$.

Edexcel CP AS 2021 June Q2

The cubic equation

$$9 x ^ { 3 } - 5 x ^ { 2 } + 4 x + 7 = 0$$ has roots $\alpha , \beta$ and $\gamma$.
Without solving the equation, find the cubic equation whose roots are ( $3 \alpha - 2$ ), ( $3 \beta - 2$ ) and ( $3 \gamma - 2$ ), giving your answer in the form $a w ^ { 3 } + b w ^ { 2 } + c w + d = 0$, where $a , b , c$ and $d$ are integers to be determined.

Edexcel CP AS 2021 June Q3

(a) Use the standard results for summations to show that for all positive integers $n$

$$\sum _ { r = 1 } ^ { n } ( 5 r - 2 ) ^ { 2 } = \frac { 1 } { 6 } n \left( a n ^ { 2 } + b n + c \right)$$ where $a$, $b$ and $c$ are integers to be determined.
(b) Hence determine the value of $k$ for which $$\sum _ { r = 1 } ^ { k } ( 5 r - 2 ) ^ { 2 } = 94 k ^ { 2 }$$

Edexcel CP AS 2021 June Q4

4. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 4
k & 2 & - 2
4 & 1 & - 2 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r } k - 7 & 6 & - 10
2 & - 20 & 24
- 3 & 2 & - 1 \end{array} \right)$$ where $k$ is a constant.

Determine, in simplest form in terms of $k$, the matrix $\mathbf { M N }$.
Given that $k = 5$
1. write down $\mathbf { M N }$
2. hence write down $\mathbf { M } ^ { - 1 }$
Solve the simultaneous equations $$\begin{aligned} & 2 x + y + 4 z = 2
& 5 x + 2 y - 2 z = 3
& 4 x + y - 2 z = - 1 \end{aligned}$$
Interpret the answer to part (c) geometrically.

Edexcel CP AS 2021 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-12_584_830_246_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows an Argand diagram.
The set $P$, of points that lie within the shaded region including its boundaries, is defined by $$P = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \}$$ where $a$, $b$, $c$ and $d$ are integers.

Write down the values of $a , b , c$ and $d$. The set $Q$ is defined by $$Q = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \} \cap \{ z \in \mathbb { C } : | z - \mathrm { i } | \leqslant | z - 3 \mathrm { i } | \}$$
Determine the exact area of the region defined by $Q$, giving your answer in simplest form.

Edexcel CP AS 2021 June Q6

A mining company has identified a mineral layer below ground.

The mining company wishes to drill down to reach the mineral layer and models the situation as follows. With respect to a fixed origin $O$,

the ground is modelled as a horizontal plane with equation $z = 0$
the mineral layer is modelled as part of the plane containing the points $A ( 10,5 , - 50 ) , B ( 15,30 , - 45 )$ and $C ( - 5,20 , - 60 )$, where the units are in metres
1. Determine an equation for the plane containing $A , B$ and $C$, giving your answer in the form r.n $= d$
2. Determine, according to the model, the acute angle between the ground and the plane containing the mineral layer. Give your answer to the nearest degree.

The mining company plans to drill vertically downwards from the point $( 5,12,0 )$ on the ground to reach the mineral layer.

Using the model, determine, in metres to 1 decimal place, the distance the mining company will need to drill in order to reach the mineral layer.

State a limitation of the assumption that the mineral layer can be modelled as a plane.

Edexcel CP AS 2021 June Q7

7. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + p z ^ { 2 } + q z + r$$ where $p , q$ and $r$ are real constants.
The roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ are $\alpha , \beta , \gamma$ and $\delta$ where $\alpha = 3$ and $\beta = 2 + \mathrm { i }$
Given that $\gamma$ is a complex root of $\mathrm { f } ( \mathrm { z } ) = 0$

1. write down the root $\gamma$,
2. explain why $\delta$ must be real.
Determine the value of $\delta$.
Hence determine the values of $p , q$ and $r$.
Write down the roots of the equation $\mathrm { f } ( - 2 \mathrm { z } ) = 0$

Edexcel CP AS 2021 June Q8

(a) Prove by induction that, for all positive integers $n$,

$$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n + 2 )$$ (b) Hence, show that, for all positive integers $n$, $$\sum _ { r = n } ^ { 2 n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ( a n + b ) ( c n + d )$$ where $a$, $b$, $c$ and $d$ are integers to be determined.

Edexcel CP AS 2021 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-28_639_517_255_774} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the vertical cross-section, $A O B C D E$, through the centre of a wax candle.
In a model, the candle is formed by rotating the region bounded by the $y$-axis, the line $O B$, the curve $B C$, and the curve $C D$ through $360 ^ { \circ }$ about the $y$-axis. The point $B$ has coordinates $( 3,0 )$ and the point $C$ has coordinates $( 5,15 )$.
The units are in centimetres.
The curve $B C$ is represented by the equation $$y = \frac { \sqrt { 225 x ^ { 2 } - 2025 } } { a } \quad 3 \leqslant x < 5$$ where $a$ is a constant.

Determine the value of $a$ according to this model. The curve $C D$ is represented by the equation $$y = 16 - 0.04 x ^ { 2 } \quad 0 \leqslant x < 5$$
Using algebraic integration, determine, according to the model, the exact volume of wax that would be required to make the candle.
State a limitation of the model. When the candle was manufactured, $700 \mathrm {~cm} ^ { 3 }$ of wax were required.
Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.

Edexcel CP AS 2022 June Q1

1. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 1
7 & 2
- 5 & 8 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 3 & 2
- 1 & 6 & 5 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { r r r } - 5 & 2 & 1
4 & 3 & 8
- 6 & 11 & 2 \end{array} \right)$$ Given that $\mathbf { I }$ is the $3 \times 3$ identity matrix,

1. show that there is an integer $k$ for which $$\mathbf { A B } - 3 \mathbf { C } + k \mathbf { I } = \mathbf { 0 }$$ stating the value of $k$
2. explain why there can be no constant $m$ such that $$\mathbf { B A } - 3 \mathbf { C } + m \mathbf { I } = \mathbf { 0 }$$
1. Show how the matrix $\mathbf { C }$ can be used to solve the simultaneous equations $$\begin{aligned} - 5 x + 2 y + z & = - 14
  4 x + 3 y + 8 z & = 3
  - 6 x + 11 y + 2 z & = 7 \end{aligned}$$
2. Hence use your calculator to solve these equations.

Edexcel CP AS 2022 June Q2

(a) Express the complex number $w = 4 \sqrt { 3 } - 4 \mathrm { i }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$ where $r > 0$ and $- \pi < \theta \leqslant \pi$
(b) Show, on a single Argand diagram,
1. the point representing $w$
2. the locus of points defined by $\arg ( z + 10 i ) = \frac { \pi } { 3 }$
  (c) Hence determine the minimum distance of $w$ from the locus $\arg ( z + 10 i ) = \frac { \pi } { 3 }$

Edexcel CP AS 2022 June Q3

$\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about the }
y \text {-axis is represented by the matrix } \end{array} \right]$
$\left( \begin{array} { c c c } \cos \theta & 0 & \sin \theta
0 & 1 & 0
- \sin \theta & 0 & \cos \theta \end{array} \right)$

The point $P$ has coordinates (8, 3, 2)
The point $Q$ is the image of $P$ under the transformation reflection in the plane $y = 0$

Write down the coordinates of $Q$ The point $R$ is the image of $P$ under the transformation rotation through $120 ^ { \circ }$ anticlockwise about the $y$-axis, with respect to the right-hand rule.
Determine the exact coordinates of $R$
Hence find $| \overrightarrow { P R } |$ giving your answer as a simplified surd.
Show that $\overrightarrow { P R }$ and $\overrightarrow { P Q }$ are perpendicular.
Hence determine the exact area of triangle $P Q R$, giving your answer as a surd in simplest form.

Edexcel CP AS 2022 June Q4

The roots of the quartic equation

$$3 x ^ { 4 } + 5 x ^ { 3 } - 7 x + 6 = 0$$ are $\alpha , \beta , \gamma$ and $\delta$
Making your method clear and without solving the equation, determine the exact value of

$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }$
$\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma } + \frac { 2 } { \delta }$
$( 3 - \alpha ) ( 3 - \beta ) ( 3 - \gamma ) ( 3 - \delta )$

Edexcel CP AS 2022 June Q5

(a) Use the standard summation formulae to show that, for $n \in \mathbb { N }$,

$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = n \left( n ^ { 2 } - A n - B \right)$$ where $A$ and $B$ are integers to be determined.
(b) Explain why, for $k \in \mathbb { N }$, $$\sum _ { r = 1 } ^ { 3 k } r \tan ( 60 r ) ^ { \circ } = - k \sqrt { 3 }$$ Using the results from part (a) and part (b) and showing all your working,
(c) determine any value of $n$ that satisfies $$\sum _ { r = 5 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = 15 \left[ \sum _ { r = 6 } ^ { 3 n } r \tan ( 60 r ) ^ { \circ } \right] ^ { 2 }$$

Edexcel CP AS 2022 June Q6

The surface of a horizontal tennis court is modelled as part of a horizontal plane, with the origin on the ground at the centre of the court, and

i and j are unit vectors directed across the width and length of the court respectively
$\quad \mathbf { k }$ is a unit vector directed vertically upwards
units are metres

After being hit, a tennis ball, modelled as a particle, moves along the path with equation $$\mathbf { r } = \left( - 4.1 + 9 \lambda - 2.3 \lambda ^ { 2 } \right) \mathbf { i } + ( - 10.25 + 15 \lambda ) \mathbf { j } + \left( 0.84 + 0.8 \lambda - \lambda ^ { 2 } \right) \mathbf { k }$$ where $\lambda$ is a scalar parameter with $\lambda \geqslant 0$
Assuming that the tennis ball continues on this path until it hits the ground,

find the value of $\lambda$ at the point where the ball hits the ground. The direction in which the tennis ball is moving at a general point on its path is given by $$( 9 - 4.6 \lambda ) \mathbf { i } + 15 \mathbf { j } + ( 0.8 - 2 \lambda ) \mathbf { k }$$
Write down the direction in which the tennis ball is moving as it hits the ground.
Hence find the acute angle at which the tennis ball hits the ground, giving your answer in degrees to one decimal place. The net of the tennis court lies in the plane $\mathbf { r } . \mathbf { j } = 0$
Find the position of the tennis ball at the point where it is in the same plane as the net. The maximum height above the court of the top of the net is 0.9 m .
Modelling the top of the net as a horizontal straight line,
state whether the tennis ball will pass over the net according to the model, giving a reason for your answer. With reference to the model,
decide whether the tennis ball will actually pass over the net, giving a reason for your answer.

Edexcel CP AS 2022 June Q7

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

$$\left( \begin{array} { l l } - 5 & 9
- 4 & 7 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 - 6 n & 9 n
- 4 n & 1 + 6 n \end{array} \right)$$

Edexcel CP AS 2022 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{545661a6-8d78-488c-b73b-ab2ced60debf-28_663_531_210_258} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{545661a6-8d78-488c-b73b-ab2ced60debf-28_394_903_431_900} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 1 shows a sketch of a 16 cm tall vase which has a flat circular base with diameter 8 cm and a circular opening of diameter 8 cm at the top. A student measures the circular cross-section halfway up the vase to be 8 cm in diameter.
The student models the shape of the vase by rotating a curve, shown in Figure 2, through $360 ^ { \circ }$ about the $x$-axis.

State the value of $a$ that should be used when setting up the model. Two possible equations are suggested for the curve in the model. $$\begin{array} { l l } \text { Model A } & y = a - 2 \sin \left( \frac { 45 } { 2 } x \right) ^ { \circ }
\text { Model B } & y = a + \frac { x ( x - 8 ) ( x + 8 ) } { 100 } \end{array}$$ For each model,
1. find the distance from the base at which the widest part of the vase occurs,
2. find the diameter of the vase at this widest point. The widest part of the vase has diameter 12 cm and is just over 3 cm from the base.
Using this information and making your reasoning clear, suggest which model is more appropriate.
Using algebraic integration, find the volume for the vase predicted by Model B. You must make your method clear. The student pours water from a full one litre jug into the vase and finds that there is 100 ml left in the jug when the vase is full.
Comment on the suitability of Model B in light of this information.

Edexcel CP AS 2023 June Q1

1. $$\left( \begin{array} { l l } x & 9
y & z \end{array} \right) - 3 \left( \begin{array} { l l } z & y
z & y \end{array} \right) = k \mathbf { I }$$ where $x , y , z$ and $k$ are constants.
Determine the value of $x$, the value of $y$ and the value of $z$.

Edexcel CP AS 2023 June Q2

$\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad$ where $a$ and $b$ are real constants

Given that $- 3 + 4 \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

determine the value of $a$ and the value of $b$.
Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.
Write down the roots of the equation $\mathrm { f } ( \mathrm { z } + 2 ) = 0$

Edexcel CP AS 2023 June Q3

3. $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

Describe fully the single geometric transformation $A$ represented by the matrix $\mathbf { A }$. $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & 0
\sqrt { 3 } & 0 & 5 \sqrt { 3 }
1 & 2 & 0 \end{array} \right)$$ The transformation $B$ is represented by the matrix $\mathbf { B }$.
The transformation $A$ followed by the transformation $B$ is the transformation $C$, which is represented by the matrix $\mathbf { C }$. To determine matrix $\mathbf { C }$, a student attempts the following matrix multiplication. $$\left( \begin{array} { c c c } 1 & 0 & 0
0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right) \left( \begin{array} { c c c } 1 & 3 & 0
\sqrt { 3 } & 0 & 5 \sqrt { 3 }
1 & 2 & 0 \end{array} \right)$$
State the error made by the student.
Determine the correct matrix $\mathbf { C }$.

Edexcel CP AS 2023 June Q4

(i) (a) Show that

$$\frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } = k ( 1 + \mathrm { i } )$$ where $k$ is a constant to be determined.
(Solutions relying on calculator technology are not acceptable.) Given that

$n$ is a positive integer
$\left( \frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } \right) ^ { n }$ is a real number
(b) use the answer to part (a) to write down the smallest possible value of $n$.
(ii) The complex number $z = a + b \mathrm { i }$ where $a$ and $b$ are real constants.

Given that

$\left| z ^ { 10 } \right| = 59049$
$\arg \left( z ^ { 10 } \right) = - \frac { 5 \pi } { 3 }$
determine the value of $a$ and the value of $b$.

Questions — Edexcel CP AS (72 questions)

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