Edexcel Paper 2 (Paper 2) 2021 October

1. In an arithmetic series

• the first term is 16
• the 21 st term is 24
  1. Find the common difference of the series.
  2. Hence find the sum of the first 500 terms of the series.

1. The functions f and g are defined by

$$\begin{aligned} & f ( x ) = 7 - 2 x ^ { 2 } \quad x \in \mathbb { R }
& \operatorname { g } ( x ) = \frac { 3 x } { 5 x - 1 } \quad x \in \mathbb { R } \quad x \neq \frac { 1 } { 5 } \end{aligned}$$

1. State the range of f
2. Find gf (1.8)
3. Find \(\mathrm { g } ^ { - 1 } ( x )\)

1. Using the laws of logarithms, solve the equation

$$\log _ { 3 } ( 12 y + 5 ) - \log _ { 3 } ( 1 - 3 y ) = 2$$

1. Given that \(\theta\) is small and measured in radians, use the small angle approximations to show that

$$4 \sin \frac { \theta } { 2 } + 3 \cos ^ { 2 } \theta \approx a + b \theta + c \theta ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.

1. The curve \(C\) has equation

$$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. 1. Verify that \(C\) has a stationary point at \(x = 1\)
   2. Show that this stationary point is a point of inflection, giving reasons for your answer.

6. \begin{figure}[h] \end{figure} The shape \(O A B C D E F O\) shown in Figure 1 is a design for a logo.
In the design

• \(O A B\) is a sector of a circle centre \(O\) and radius \(r\)
• sector \(O F E\) is congruent to sector \(O A B\)
• \(O D C\) is a sector of a circle centre \(O\) and radius \(2 r\)
• \(A O F\) is a straight line

Given that the size of angle \(C O D\) is \(\theta\) radians,

1. write down, in terms of \(\theta\), the size of angle \(A O B\)
2. Show that the area of the logo is $$\frac { 1 } { 2 } r ^ { 2 } ( 3 \theta + \pi )$$
3. Find the perimeter of the logo, giving your answer in simplest form in terms of \(r , \theta\) and \(\pi\).

1. In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\)
The line \(l\) is the tangent to \(C\) at \(P\)

1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
3. Use algebraic integration to find the exact area of \(R\).

1. The curve \(C\) has equation

$$p x ^ { 3 } + q x y + 3 y ^ { 2 } = 26$$ where \(p\) and \(q\) are constants.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a p x ^ { 2 } + b q y } { q x + c y }$$ where \(a\), \(b\) and \(c\) are integers to be found. Given that
   • the point \(P ( - 1 , - 4 )\) lies on \(C\)
   • the normal to \(C\) at \(P\) has equation \(19 x + 26 y + 123 = 0\)
   • find the value of \(p\) and the value of \(q\).

1. Show that

$$\sum _ { n = 2 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n } \cos ( 180 n ) ^ { \circ } = \frac { 9 } { 28 }$$

1. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula

$$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.

1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-26_581_888_749_625} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
   The student recorded the value of \(T\) for different values of \(l\).
   Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data.
   The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\)
   Using this information,
2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(a\).

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = k - | 2 x - 3 k |$$ stating
   • the coordinates of the maximum point
   • the coordinates of any points where the graph cuts the coordinate axes
   • Find, in terms of \(k\), the set of values of \(x\) for which
   $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 f \left( \frac { 1 } { 2 } x \right)$$

1. (a) Use the substitution \(u = 1 + \sqrt { x }\) to show that

$$\int _ { 0 } ^ { 16 } \frac { \mathrm { x } } { 1 + \sqrt { \mathrm { x } } } \mathrm {~d} x = \int _ { p } ^ { q } \frac { 2 ( u - 1 ) ^ { 3 } } { u } \mathrm {~d} u$$ where \(p\) and \(q\) are constants to be found.
(b) Hence show that $$\int _ { 0 } ^ { 16 } \frac { \mathrm { x } } { 1 + \sqrt { \mathrm { x } } } \mathrm {~d} x = A - B \ln 5$$ where \(A\) and \(B\) are constants to be found.

1. The curve \(C\) has parametric equations

$$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\)
2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\)

14. \begin{figure}[h] \end{figure} Water flows at a constant rate into a large tank.
The tank is a cuboid, with all sides of negligible thickness.
The base of the tank measures 8 m by 3 m and the height of the tank is 5 m .
There is a tap at a point \(T\) at the bottom of the tank, as shown in Figure 5.
At time \(t\) minutes after the tap has been opened

• the depth of water in the tank is \(h\) metres
• water is flowing into the tank at a constant rate of \(0.48 \mathrm {~m} ^ { 3 }\) per minute
• water is modelled as leaving the tank through the tap at a rate of \(0.1 h \mathrm {~m} ^ { 3 }\) per minute
  1. Show that, according to the model,

$$1200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 24 - 5 h$$ Given that when the tap was opened, the depth of water in the tank was 2 m ,

show that, according to the model, $$h = A + B \mathrm { e } ^ { - k t }$$ where \(A , B\) and \(k\) are constants to be found. Given that the tap remains open,

determine, according to the model, whether the tank will ever become full, giving a reason for your answer.

1. (a) Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. \begin{figure}[h] \end{figure} Figure 6 shows the cross-section of a water wheel.
The wheel is free to rotate about a fixed axis through the point \(C\).
The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
The water level is assumed to be horizontal and of constant height.
The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where \(t\) is the time in seconds after the wheel starts rotating.
Using the model, find
(b) (i) the maximum height of \(P\) above the water level,
(ii) the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
(c) find the value of \(T\) giving your answer to 3 significant figures.
(Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
(d) Explain how the equation of the model should be refined to take this into account.