Edexcel CP1 (Core Pure 1) 2023 June

1. 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.

1. (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.

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

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

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

1. Prove by induction that for \(n \in \mathbb { N }\)

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

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

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

1. 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 \(\Pi _ { 1 }\)
2. 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 \(l _ { 2 }\)
(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 \(b\).

1. 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.

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

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 \(b\).

1. 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.

1. 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 $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$

1. A colony of small mammals is being studied.

In the study, the mammals are divided into 3 categories (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.)