Questions - Edexcel CP AS

Find $\mathbf { M } ^ { - 1 }$ giving each element in exact form.
Solve the simultaneous equations $$\begin{array} { r } 2 x + y - 3 z = - 4
4 x - 2 y + z = 9
3 x + 5 y - 2 z = 5 \end{array}$$
Interpret the answer to part (b) geometrically.
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Edexcel CP AS 2018 June Q2

The cubic equation

$$z ^ { 3 } - 3 z ^ { 2 } + z + 5 = 0$$ has roots $\alpha , \beta$ and $\gamma$.
Without solving the equation, find the cubic equation whose roots are ( $2 \alpha + 1$ ), ( $2 \beta + 1$ ) and ( $2 \gamma + 1$ ), 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.

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Edexcel CP AS 2018 June Q3

(a) Shade on an Argand diagram the set of points

$$\{ z \in \mathbb { C } : | z - 1 - \mathrm { i } | \leqslant 3 \} \cap \quad z \in \mathbb { C } : \frac { \pi } { 4 } \leqslant \arg ( z - 2 ) \leqslant \frac { 3 \pi } { 4 }$$ The complex number $w$ satisfies $$| w - 1 - \mathrm { i } | = 3 \text { and } \arg ( w - 2 ) = \frac { \pi } { 4 }$$ (b) Find, in simplest form, the exact value of $| w | ^ { 2 }$

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Edexcel CP AS 2018 June Q4

Part of the mains water system for a housing estate consists of water pipes buried beneath the ground surface. The water pipes are modelled as straight line segments. One water pipe, $W$, is buried beneath a particular road. With respect to a fixed origin $O$, the road surface is modelled as a plane with equation $3 x - 5 y - 18 z = 7$, and $W$ passes through the points $A ( - 1 , - 1 , - 3 )$ and $B ( 1,2 , - 3 )$. The units are in metres.
1. Use the model to calculate the acute angle between $W$ and the road surface.
A point $C ( - 1 , - 2,0 )$ lies on the road. A section of water pipe needs to be connected to $W$ from $C$.
Using the model, find, to the nearest cm, the shortest length of pipe needed to connect $C$ to $W$.
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Edexcel CP AS 2018 June Q5

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

Describe fully the single geometrical transformation $U$ represented by the matrix $\mathbf { A }$. The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { B }$, is a reflection in the line $y = - x$
Write down the matrix $\mathbf { B }$. Given that $U$ followed by $V$ is the transformation $T$, which is represented by the matrix $\mathbf { C }$, (c) find the matrix $\mathbf { C }$.
Show that there is a real number $k$ for which the point $( 1 , k )$ is invariant under $T$.
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Edexcel CP AS 2018 June Q6

(a) Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that

$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { 1 } { 2 } n \left[ 6 n ^ { 2 } - 3 n - 1 \right]$$ for all positive integers $n$.
(b) Hence find any values of $n$ for which $$\sum _ { r = 5 } ^ { n } ( 3 r - 2 ) ^ { 2 } + 103 \sum _ { r = 1 } ^ { 28 } r \cos \left( \frac { r \pi } { 2 } \right) = 3 n ^ { 3 }$$

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Edexcel CP AS 2018 June Q7

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where $p$ and $q$ are real constants.
The equation $f ( z ) = 0$ has roots $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$
When plotted on an Argand diagram, the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ form the vertices of a triangle of area 35 Given that $z _ { 1 } = 3$, find the values of $p$ and $q$.

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Edexcel CP AS 2018 June Q8

(i) Prove by induction that for $n \in \mathbb { Z } ^ { + }$

$$\left( \begin{array} { l l } 5 & - 8
2 & - 3 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 4 n + 1 & - 8 n
2 n & 1 - 4 n \end{array} \right)$$ (ii) Prove by induction that for $n \in \mathbb { Z } ^ { + }$ $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21

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Edexcel CP AS 2018 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e48fac26-15a2-4a5e-9204-9d49db8a998a-32_789_452_331_497} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{e48fac26-15a2-4a5e-9204-9d49db8a998a-32_681_523_424_1248} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section $A B C D E F G H A$ of the bottle with the measurements taken by the student. The horizontal cross-section between $C F$ and $D E$ is a circle of diameter 8 cm and the horizontal cross-section between $B G$ and $A H$ is a circle of diameter 2 cm . The student thinks that the curve $G F$ could be modelled as a curve with equation $$y = a x ^ { 2 } + b \quad 1 \leqslant x \leqslant 4$$ where $a$ and $b$ are constants and $O$ is the fixed origin, as shown in Figure 2.

Find the value of $a$ and the value of $b$ according to the model.
Use the model to find the volume of water that the bottle can contain.
State a limitation of the model. The label on the bottle states that the bottle holds approximately $750 \mathrm {~cm} ^ { 3 }$ of water.
Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.

Edexcel CP AS 2019 June Q1

1. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5
2 & - 7 \end{array} \right)$$

Show that the matrix $\mathbf { M }$ is non-singular. The transformation $T$ of the plane is represented by the matrix $\mathbf { M }$.
The triangle $R$ is transformed to the triangle $S$ by the transformation $T$.
Given that the area of $S$ is 63 square units,
find the area of $R$.
Show that the line $y = 2 x$ is invariant under the transformation $T$.

Edexcel CP AS 2019 June Q2

The cubic equation

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

Edexcel CP AS 2019 June Q3

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

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } = \frac { n } { 2 n + 1 }$$

Edexcel CP AS 2019 June Q4

The line $l$ has equation

$$\frac { x + 2 } { 1 } = \frac { y - 5 } { - 1 } = \frac { z - 4 } { - 3 }$$ The plane $\Pi$ has equation $$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) = - 7$$ Determine whether the line $l$ intersects $\Pi$ at a single point, or lies in $\Pi$, or is parallel to $\Pi$ without intersecting it.
(5)

Edexcel CP AS 2019 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-10_483_528_260_772} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The complex numbers $z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }$ and $z _ { 3 } = 1 + \mathrm { i }$ are plotted in Figure 1, on an Argand diagram for the complex plane with $z = x + \mathrm { i } y$

Explain why $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ cannot all be roots of a quartic polynomial equation with real coefficients.
Show that $\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }$
Hence show that $\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }$ A copy of Figure 1, labelled Diagram 1, is given on page 12.
Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality $$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
\includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
\section*{Diagram 1}

Edexcel CP AS 2019 June Q6

An art display consists of an arrangement of $n$ marbles.

When arranged in ascending order of mass, the mass of the first marble is 10 grams. The mass of each subsequent marble is 3 grams more than the mass of the previous one, so that the $r$ th marble has mass $( 7 + 3 r )$ grams.

Show that the mean mass, in grams, of the marbles in the display is given by $$\frac { 1 } { 2 } ( 3 n + 17 )$$ Given that there are 85 marbles in the display,
use the standard summation formulae to find the standard deviation of the mass of the marbles in the display, giving your answer, in grams, to one decimal place.

Edexcel CP AS 2019 June Q7

7. $$\mathrm { f } ( z ) = z ^ { 3 } - 8 z ^ { 2 } + p z - 24$$ where $p$ is a real constant.
Given that the equation $\mathrm { f } ( z ) = 0$ has distinct roots $$\alpha , \beta \text { and } \left( \alpha + \frac { 12 } { \alpha } - \beta \right)$$

solve completely the equation $\mathrm { f } ( z ) = 0$
Hence find the value of $p$.

Edexcel CP AS 2019 June Q8

A gas company maintains a straight pipeline that passes under a mountain.

The pipeline is modelled as a straight line and one side of the mountain is modelled as a plane. There are accessways from a control centre to two access points on the pipeline.
Modelling the control centre as the origin $O$, the two access points on the pipeline have coordinates $P ( - 300,400 , - 150 )$ and $Q ( 300,300 , - 50 )$, where the units are metres.

Find a vector equation for the line $P Q$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$, where $\lambda$ is a scalar parameter. The equation of the plane modelling the side of the mountain is $2 x + 3 y - 5 z = 300$
The company wants to create a new accessway from this side of the mountain to the pipeline. The accessway will consist of a tunnel of shortest possible length between the pipeline and the point $M ( 100 , k , 100 )$ on this side of the mountain, where $k$ is a constant.
Using the model, find
1. the coordinates of the point at which this tunnel will meet the pipeline,
2. the length of this tunnel. It is only practical to construct the new accessway if it will be significantly shorter than both of the existing accessways, $O P$ and $O Q$.
Determine whether the company should build the new accessway.
Suggest one limitation of the model.

Edexcel CP AS 2019 June Q9

9. $$\mathrm { f } ( x ) = 2 x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 2 } { 3 } } \quad x > 0$$ The finite region bounded by the curve $y = \mathrm { f } ( x )$, the line $x = \frac { 1 } { 8 }$, the $x$-axis and the line $x = 8$ is rotated through $\theta$ radians about the $x$-axis to form a solid of revolution. Given that the volume of the solid formed is $\frac { 461 } { 2 }$ units cubed, use algebraic integration to find the angle $\theta$ through which the region is rotated.

Edexcel CP AS 2019 June Q10

The population of chimpanzees in a particular country consists of juveniles and adults. Juvenile chimpanzees do not reproduce.

In a study, the numbers of juvenile and adult chimpanzees were estimated at the start of each year. A model for the population satisfies the matrix system $$\binom { J _ { n + 1 } } { A _ { n + 1 } } = \left( \begin{array} { c c } a & 0.15
0.08 & 0.82 \end{array} \right) \binom { J _ { n } } { A _ { n } } \quad n = 0,1,2 , \ldots$$ where $a$ is a constant, and $J _ { n }$ and $A _ { n }$ are the respective numbers of juvenile and adult chimpanzees $n$ years after the start of the study.

Interpret the meaning of the constant $a$ in the context of the model. At the start of the study, the total number of chimpanzees in the country was estimated to be 64000 According to the model, after one year the number of juvenile chimpanzees is 15360 and the number of adult chimpanzees is 43008
1. Find, in terms of $a$ $$\left( \begin{array} { c c } a & 0.15
  0.08 & 0.82 \end{array} \right) ^ { - 1 }$$
2. Hence, or otherwise, find the value of $a$.
3. Calculate the change in the number of juvenile chimpanzees in the first year of the study, according to this model. Given that the number of juvenile chimpanzees is known to be in decline in the country,
comment on the short-term suitability of this model. A study of the population revealed that adult chimpanzees stop reproducing at the age of 40 years.
Refine the matrix system 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 CP AS 2020 June Q1

A system of three equations is defined by

$$\begin{aligned} k x + 3 y - z & = 3
3 x - y + z & = - k
- 16 x - k y - k z & = k \end{aligned}$$ where $k$ is a positive constant.
Given that there is no unique solution to all three equations,

show that $k = 2$ Using $k = 2$
determine whether the three equations are consistent, justifying your answer.
Interpret the answer to part (b) geometrically.

Edexcel CP AS 2020 June Q2

Given that

$$\begin{aligned} z _ { 1 } & = 2 + 3
\left| z _ { 1 } z _ { 2 } \right| & = 39 \sqrt { 2 }
\arg \left( z _ { 1 } z _ { 2 } \right) & = \frac { \pi } { 4 } \end{aligned}$$ where $z _ { 1 }$ and $z _ { 2 }$ are complex numbers,

write $z _ { 1 }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$ Give the exact value of $r$ and give the value of $\theta$ in radians to 4 significant figures.
Find $z _ { 2 }$ giving your answer in the form $a + \mathrm { i } b$ where $a$ and $b$ are integers.

Edexcel CP AS 2020 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{09bd14c0-c368-4ae1-bee0-cc8bf82abecc-06_582_588_255_758} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a circle with radius $r$ and centre at the origin.
The region $R$, shown shaded in Figure 1, is bounded by the $x$-axis and the part of the circle for which $y > 0$
The region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis to create a sphere with volume $V$
Use integration to show that $V = \frac { 4 } { 3 } \pi r ^ { 3 }$

Edexcel CP AS 2020 June Q4

All units in this question are in metres.

A lawn is modelled as a plane that contains the points $L ( - 2 , - 3 , - 1 ) , M ( 6 , - 2,0 )$ and $N ( 2,0,0 )$, relative to a fixed origin $O$.

Determine a vector equation of the plane that models the lawn, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }$
1. Show that, according to the model, the lawn is perpendicular to the vector $\left( \begin{array} { c } 1
  2
  - 10 \end{array} \right)$
2. Hence determine a Cartesian equation of the plane that models the lawn. There are two posts set in the lawn.
  There is a washing line between the two posts.
  The washing line is modelled as a straight line through points at the top of each post with coordinates $P ( - 10,8,2 )$ and $Q ( 6,4,3 )$.
Determine a vector equation of the line that models the washing line.
State a limitation of one of the models. The point $R ( 2,5,2.75 )$ lies on the washing line.
Determine, according to the model, the shortest distance from the point $R$ to the lawn, giving your answer to the nearest cm. Given that the shortest distance from the point $R$ to the lawn is actually 1.5 m ,
use your answer to part (e) to evaluate the model, explaining your reasoning.

Edexcel CP AS 2020 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{09bd14c0-c368-4ae1-bee0-cc8bf82abecc-12_351_655_246_705} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A block has length $( r + 2 ) \mathrm { cm }$, width $( r + 1 ) \mathrm { cm }$ and height $r \mathrm {~cm}$, as shown in Figure 2.
In a set of $n$ such blocks, the first block has a height of 1 cm , the second block has a height of 2 cm , the third block has a height of 3 cm and so on.

Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that the total volume, $V$, of all $n$ blocks in the set is given by $$V = \frac { n } { 4 } ( n + 1 ) ( n + 2 ) ( n + 3 ) \quad n \geqslant 1$$ Given that the total volume of all $n$ blocks is $$\left( n ^ { 4 } + 6 n ^ { 3 } - 11710 \right) \mathrm { cm } ^ { 3 }$$
determine how many blocks make up the set.

Edexcel CP AS 2020 June Q6

$$\mathbf { A } = \left( \begin{array} { c c } 2 & a
a - 4 & b \end{array} \right)$$ where $a$ and $b$ are non-zero constants.
Given that the matrix $\mathbf { A }$ is self-inverse,

determine the value of $b$ and the possible values for $a$. The matrix $\mathbf { A }$ represents a linear transformation $M$.
Using the smaller value of $a$ from part (a),
show that the invariant points of the linear transformation $M$ form a line, stating the equation of this line.
(ii) $$\mathbf { P } = \left( \begin{array} { c c } p & 2 p
- 1 & 3 p \end{array} \right)$$ where $p$ is a positive constant.
The matrix $\mathbf { P }$ represents a linear transformation $U$.
The triangle $T$ has vertices at the points with coordinates ( 1,2 ), ( 3,2 ) and ( 2,5 ). The area of the image of $T$ under the linear transformation $U$ is 15
Determine the value of $p$. The transformation $V$ consists of a stretch scale factor 3 parallel to the $x$-axis with the $y$-axis invariant followed by a stretch scale factor - 2 parallel to the $y$-axis with the $x$-axis invariant. The transformation $V$ is represented by the matrix $\mathbf { Q }$.
Write down the matrix $\mathbf { Q }$. Given that $U$ followed by $V$ is the transformation $W$, which is represented by the matrix $\mathbf { R }$, (c) find the matrix $\mathbf { R }$.

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