Questions - Edexcel - A-Level Maths

Edexcel CP1 2021 June Q2

7 marks Standard +0.3

(a) Use the Maclaurin series expansion for $\cos x$ to determine the series expansion of $\cos ^ { 2 } \left( \frac { x } { 3 } \right)$ in ascending powers of $x$, up to and including the term in $x ^ { 4 }$

Give each term in simplest form.
(b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ (c) Use the integration function on your calculator to evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ Give your answer to 5 decimal places.
(d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).

Edexcel CP1 2021 June Q4

9 marks Standard +0.8

4 \sqrt { 3 } & - 4 \end{array} \right)$$

Determine
1. the value of $k$,
2. the smallest value of $\theta$ A square $S$ has vertices at the points with coordinates ( 0,0 ), ( $a , - a$ ), ( $2 a , 0$ ) and ( $a , a$ ) where $a$ is a constant. The square $S$ is transformed to the square $S ^ { \prime }$ by the transformation represented by $\mathbf { M }$.
Determine, in terms of $a$, the area of $S ^ { \prime }$
1. (a) Use the Maclaurin series expansion for $\cos x$ to determine the series expansion of $\cos ^ { 2 } \left( \frac { x } { 3 } \right)$ in ascending powers of $x$, up to and including the term in $x ^ { 4 }$
Give each term in simplest form.
Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
Use the integration function on your calculator to evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ Give your answer to 5 decimal places.
Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).
1. The cubic equation
$$a x ^ { 3 } + b x ^ { 2 } - 19 x - b = 0$$ where $a$ and $b$ are constants, has roots $\alpha , \beta$ and $\gamma$\\ The cubic equation $$w ^ { 3 } - 9 w ^ { 2 } - 97 w + c = 0$$ where $c$ is a constant, has roots $( 4 \alpha - 1 ) , ( 4 \beta - 1 )$ and $( 4 \gamma - 1 )$\\ Without solving either cubic equation, determine the value of $a$, the value of $b$ and the value of $c$.
1. 1. $\mathbf { A }$ is a 2 by 2 matrix and $\mathbf { B }$ is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate
$\mathbf { A B }$
$\mathbf { A } + \mathbf { B }$\\ (ii) Given that $$\left( \begin{array} { r r r } - 5 & 3 & 1
a & 0 & 0
b & a & b \end{array} \right) \left( \begin{array} { r r r } 0 & 5 & 0
2 & 12 & - 1
- 1 & - 11 & 3 \end{array} \right) = \lambda \mathbf { I }$$ where $a$, $b$ and $\lambda$ are constants,
determine
- the value of $\lambda$
- the value of $a$
- the value of $b$
- Hence deduce the inverse of the matrix $\left( \begin{array} { r r r } - 5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{array} \right)$\\ (iii) Given that
$$\mathbf { M } = \left( \begin{array} { c c c } 1 & 1 & 1
0 & \sin \theta & \cos \theta
0 & \cos 2 \theta & \sin 2 \theta \end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$ determine the values of $\theta$ for which the matrix $\mathbf { M }$ is singular.

Edexcel CP1 2021 June Q5

7 marks Moderate -0.3

1. Evaluate the improper integral

$$\int _ { 1 } ^ { \infty } 2 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x$$ (ii) The air temperature, $\theta ^ { \circ } \mathrm { C }$, on a particular day in London is modelled by the equation $$\theta = 8 - 5 \sin \left( \frac { \pi } { 12 } t \right) - \cos \left( \frac { \pi } { 6 } t \right) \quad 0 \leqslant t \leqslant 24$$ where $t$ is the number of hours after midnight.

Use calculus to show that the mean air temperature on this day is $8 ^ { \circ } \mathrm { C }$, according to the model. Given that the actual mean air temperature recorded on this day was higher than $8 ^ { \circ } \mathrm { C }$,
explain how the model could be refined.

Edexcel CP1 2021 June Q6

12 marks Challenging +1.2

A tourist decides to do a bungee jump from a bridge over a river.

One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist.
The tourist jumps off the bridge.
At time $t$ seconds after the tourist reaches their lowest point, their vertical displacement is $x$ metres above a fixed point 30 metres vertically above the river. When $t = 0$

$x = - 20$
the velocity of the tourist is $0 \mathrm {~ms} ^ { - 1 }$
the acceleration of the tourist is $13.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation $$5 k \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 17 x = 0 \quad t \geqslant 0$$ where $k$ is a positive constant.

Determine the value of $k$
Determine the particular solution to the differential equation.
Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point.
Give a limitation of the model.

Edexcel CP1 2021 June Q7

8 marks Standard +0.8

The plane $\Pi$ has equation

$$\mathbf { r } = \left( \begin{array} { l } 3 \\ 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters.

Show that vector $2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$ is perpendicular to $\Pi$.
Hence find a Cartesian equation of $\Pi$. The line $l$ has equation $$\mathbf { r } = \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 6 \\ - 3 \end{array} \right)$$ where $t$ is a scalar parameter.
The point $A$ lies on $l$.
Given that the shortest distance between $A$ and $\Pi$ is $2 \sqrt { 29 }$
determine the possible coordinates of $A$.

Edexcel CP1 2021 June Q8

9 marks Standard +0.3

Two different colours of paint are being mixed together in a container.

The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint.
The colour of the paint mixture is now altered by

adding red paint to the container at a rate of 2 litres per second
adding blue paint to the container at a rate of 1 litre per second
pumping fully mixed paint from the container at a rate of 3 litres per second.

Let $r$ litres be the amount of red paint in the container at time $t$ seconds after the colour of the paint mixture starts to be altered.

Show that the amount of red paint in the container can be modelled by the differential equation $$\frac { \mathrm { dr } } { \mathrm { dt } } = 2 - \frac { \mathrm { r } } { \alpha }$$ where $\alpha$ is a positive constant to be determined.
By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.
Use this information to evaluate the model, giving a reason for your answer.

Edexcel CP1 2021 June Q9

11 marks Challenging +1.8

(a) Use a hyperbolic substitution and calculus to show that

$$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } \left[ x \sqrt { x ^ { 2 } - 1 } + \operatorname { arcosh } x \right] + k$$ where $k$ is an arbitrary constant. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a62583d5-0109-4bde-ac4e-c331f373d021-32_730_803_616_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = \frac { 4 } { 15 } x \operatorname { arcosh } x \quad x \geqslant 1$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve $C$, the $x$-axis and the line with equation $x = 3$
(b) Using algebraic integration and the result from part (a), show that the area of $R$ is given by $$\frac { 1 } { 15 } [ 17 \ln ( 3 + 2 \sqrt { 2 } ) - 6 \sqrt { 2 } ]$$

Edexcel CP1 2022 June Q1

6 marks Moderate -0.3

1. $$\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } + 52 \quad \text { where } a \text { is a real constant }$$ Given that $2 - 3 \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

write down the other complex root.
Hence
1. solve completely $\mathrm { f } ( \mathrm { z } ) = 0$
2. determine the value of $a$
Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.

Edexcel CP1 2022 June Q2

4 marks Standard +0.3

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

Determine the values of $x$ for which $$64 \cosh ^ { 4 } x - 64 \cosh ^ { 2 } x - 9 = 0$$ Give your answers in the form $q \ln 2$ where $q$ is rational and in simplest form.

Edexcel CP1 2022 June Q3

6 marks Standard +0.3

(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

7 marks Challenging +1.2

(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

6 marks Standard +0.3

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

7 marks Standard +0.8

(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

7 marks Standard +0.3

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

12 marks Challenging +1.2

(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

6 marks Standard +0.8

(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

14 marks Challenging +1.2

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

5 marks Moderate -0.3

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

6 marks Standard +0.3

(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

10 marks Standard +0.3

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

5 marks Standard +0.3

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

12 marks Challenging +1.8

5 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 $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where $\mu$ is a scalar parameter. The plane $\Pi _ { 2 }$ contains the line $l _ { 1 }$ and the line $l _ { 2 }$
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,
determine the value of $a$ and the value of $b$.

Edexcel CP1 2023 June Q6

12 marks Challenging +1.2

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

12 marks Standard +0.8

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

Challenging +1.2

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.)

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