Edexcel Paper 2 (Paper 2) 2020 October

1 The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { 1 + x } }\)
The values of \(y\) are given to 4 significant figures.

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for $$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { x } { 1 + x } } \mathrm {~d} x$$ giving your answer to 3 significant figures.
2. Using your answer to part (a), deduce an estimate for \(\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x\) Given that $$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x = 4.535 \text { to } 4 \text { significant figures }$$
3. comment on the accuracy of your answer to part (b).

1. Relative to a fixed origin, points \(P , Q\) and \(R\) have position vectors \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) respectively.

Given that

• \(\quad P , Q\) and \(R\) lie on a straight line
• \(Q\) lies one third of the way from \(P\) to \(R\)
  show that

$$\mathbf { q } = \frac { 1 } { 3 } ( \mathbf { r } + 2 \mathbf { p } )$$

1. (a) Given that

$$2 \log ( 4 - x ) = \log ( x + 8 )$$ show that $$x ^ { 2 } - 9 x + 8 = 0$$ (b) (i) Write down the roots of the equation $$x ^ { 2 } - 9 x + 8 = 0$$ (ii) State which of the roots in (b)(i) is not a solution of $$2 \log ( 4 - x ) = \log ( x + 8 )$$ giving a reason for your answer.

1. In the binomial expansion of
   \(( a + 2 x ) ^ { 7 } \quad\) where \(a\) is a constant
   the coefficient of \(x ^ { 4 }\) is 15120
   Find the value of \(a\).

1. The curve with equation \(y = 3 \times 2 ^ { x }\) meets the curve with equation \(y = 15 - 2 ^ { x + 1 }\) at the point \(P\). Find, using algebra, the exact \(x\) coordinate of \(P\).

1. (a) Given that

$$\frac { x ^ { 2 } + 8 x - 3 } { x + 2 } \equiv A x + B + \frac { C } { x + 2 } \quad x \in \mathbb { R } \quad x \neq - 2$$ find the values of the constants \(A , B\) and \(C\)
(b) Hence, using algebraic integration, find the exact value of $$\int _ { 0 } ^ { 6 } \frac { x ^ { 2 } + 8 x - 3 } { x + 2 } d x$$ giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers to be found.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 4 x ^ { 2 } + x } { 2 \sqrt { x } } - 4 \ln x \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 x ^ { 2 } + x - 16 \sqrt { x } } { 4 x \sqrt { x } }$$ The point \(P\), shown in Figure 1, is the minimum turning point on \(C\).
2. Show that the \(x\) coordinate of \(P\) is a solution of $$x = \left( \frac { 4 } { 3 } - \frac { \sqrt { x } } { 12 } \right) ^ { \frac { 2 } { 3 } }$$
3. Use the iteration formula $$x _ { n + 1 } = \left( \frac { 4 } { 3 } - \frac { \sqrt { x _ { n } } } { 12 } \right) ^ { \frac { 2 } { 3 } } \quad \text { with } x _ { 1 } = 2$$ to find (i) the value of \(x _ { 2 }\) to 5 decimal places,
   (ii) the \(x\) coordinate of \(P\) to 5 decimal places.

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\)

Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + a x - 23\) where \(a\) is a constant
• the \(y\) intercept of \(C\) is - 12
• ( \(x + 4\) ) is a factor of \(\mathrm { f } ( x )\)
  find, in simplest form, \(\mathrm { f } ( x )\)

1. A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) seconds after heating began, is modelled by the equation $$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$ where \(A\) and \(B\) are positive constants.
Given that

• the initial temperature of the ethanol was \(18 ^ { \circ } \mathrm { C }\)
• after 10 seconds the temperature of the ethanol was \(44 ^ { \circ } \mathrm { C }\)
  1. find a complete equation for the model, giving the values of \(A\) and \(B\) to 3 significant figures.

Ethanol has a boiling point of approximately \(78 ^ { \circ } \mathrm { C }\)

Use this information to evaluate the model.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$
2. Hence solve, for \(- 90 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), the equation $$1 - \cos 3 x = \sin ^ { 2 } x$$

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point \(P\), shown in Figure 2.

1. Find the coordinates of \(P\).
2. Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line \(l\) has equation \(y = a x\), where \(a\) is a constant.
   Given that \(l\) intersects \(y = 2 | x + 4 | - 5\) at least once,
3. find the range of possible values of \(a\), writing your answer in set notation.

12. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = 6 \sin t \quad y = 5 \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.

1. 1. Show that the area of \(R\) is given by \(\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \sin t \cos ^ { 2 } t \mathrm {~d} t\)
   2. Hence show, by algebraic integration, that the area of \(R\) is exactly 20 \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_451_570_1416_742} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that
      • \(x\) and \(y\) are in metres
2. the vertical wall of the dam is 4.2 metres high
3. there is a horizontal walkway of width \(M N\) along the top of the dam
4. calculate the width of the walkway.

1. The function \(g\) is defined by

$$g ( x ) = \frac { 3 \ln ( x ) - 7 } { \ln ( x ) - 2 } \quad x > 0 \quad x \neq k$$ where \(k\) is a constant.

1. Deduce the value of \(k\).
2. Prove that $$\mathrm { g } ^ { \prime } ( x ) > 0$$ for all values of \(x\) in the domain of g .
3. Find the range of values of \(a\) for which $$g ( a ) > 0$$

1. A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2 x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
2. find the two possible values of \(r\), giving your answers as fully simplified surds.

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} A geometric series has common ratio \(r\) and first term \(a\).
Given \(r \neq 1\) and \(a \neq 0\)

1. prove that $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ Given also that \(S _ { 10 }\) is four times \(S _ { 5 }\)
2. find the exact value of \(r\).

1. Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3