Edexcel Paper 2 (Paper 2) 2022 June

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 3 - 2 x |\)
Solve $$| 3 - 2 x | = 7 + x$$

1. (a) Sketch the curve with equation

$$y = 4 ^ { x }$$ stating any points of intersection with the coordinate axes.
(b) Solve $$4 ^ { x } = 100$$ giving your answer to 2 decimal places.

1. A sequence of terms \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$\begin{aligned} a _ { 1 } & = 3
a _ { n + 1 } & = 8 - a _ { n } \end{aligned}$$

1. 1. Show that this sequence is periodic.
   2. State the order of this periodic sequence.
2. Find the value of $$\sum _ { n = 1 } ^ { 85 } a _ { n }$$

1. Given that

$$y = 2 x ^ { 2 }$$ use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x$$

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } 2 x\) The values of \(y\) are given to 2 decimal places as appropriate.

1. Using the trapezium rule with all the values of \(y\) in the table, find an estimate for $$\int _ { 3 } ^ { 9 } \log _ { 3 } 2 x \mathrm {~d} x$$ Using your answer to part (a) and making your method clear, estimate
2. 1. \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( 2 x ) ^ { 10 } \mathrm {~d} x\)
   2. \(\int _ { 3 } ^ { 9 } \log _ { 3 } 18 x \mathrm {~d} x\)

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 8 \sin \left( \frac { 1 } { 2 } x \right) - 3 x + 9 \quad x > 0$$ and \(x\) is measured in radians.
The point \(P\), shown in Figure 2, is a local maximum point on the curve.
Using calculus and the sketch in Figure 2,

1. find the \(x\) coordinate of \(P\), giving your answer to 3 significant figures. The curve crosses the \(x\)-axis at \(x = \alpha\), as shown in Figure 2 .
   Given that, to 3 decimal places, \(f ( 4 ) = 4.274\) and \(f ( 5 ) = - 1.212\)
2. explain why \(\alpha\) must lie in the interval \([ 4,5 ]\)
3. Taking \(x _ { 0 } = 5\) as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show your method and give your answer to 3 significant figures.

1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\sqrt { 4 - 9 x }$$ writing each term in simplest form. A student uses this expansion with \(x = \frac { 1 } { 9 }\) to find an approximation for \(\sqrt { 3 }\)
Using the answer to part (a) and without doing any calculations,
(b) state whether this approximation will be an overestimate or an underestimate of \(\sqrt { 3 }\) giving a brief reason for your answer.

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

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of a curve with equation $$y = \frac { ( x - 2 ) ( x - 4 ) } { 4 \sqrt { x } } \quad x > 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.
Find the exact area of \(R\), writing your answer in the form \(a \sqrt { 2 } + b\), where \(a\) and \(b\) are constants to be found.

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 4 shows a sketch of a Ferris wheel.
The height above the ground, \(H \mathrm {~m}\), of a passenger on the Ferris wheel, \(t\) seconds after the wheel starts turning, is modelled by the equation $$H = \left| A \sin ( b t + \alpha ) ^ { \circ } \right|$$ where \(A\), \(b\) and \(\alpha\) are constants.
Figure 5 shows a sketch of the graph of \(H\) against \(t\), for one revolution of the wheel.
Given that

• the maximum height of the passenger above the ground is 50 m
• the passenger is 1 m above the ground when the wheel starts turning
• the wheel takes 720 seconds to complete one revolution
  1. find a complete equation for the model, giving the exact value of \(A\), the exact value of \(b\) and the value of \(\alpha\) to 3 significant figures.
  2. Explain why an equation of the form

$$H = \left| A \sin ( b t + \alpha ) ^ { \circ } \right| + d$$ where \(d\) is a positive constant, would be a more appropriate model.

1. The function f is defined by

$$f ( x ) = \frac { 8 x + 5 } { 2 x + 3 } \quad x > - \frac { 3 } { 2 }$$

1. Find \(\mathrm { f } ^ { - 1 } \left( \frac { 3 } { 2 } \right)\)
2. Show that $$\mathrm { f } ( x ) = A + \frac { B } { 2 x + 3 }$$ where \(A\) and \(B\) are constants to be found. The function \(g\) is defined by $$g ( x ) = 16 - x ^ { 2 } \quad 0 \leqslant x \leqslant 4$$
3. State the range of \(\mathrm { g } ^ { - 1 }\)
4. Find the range of \(\mathrm { fg } ^ { - 1 }\)

1. Prove, using algebra, that

$$n \left( n ^ { 2 } + 5 \right)$$ is even for all \(n \in \mathbb { N }\).

1. The function f is defined by

$$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where \(k\) is a positive constant.

1. Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where \(\mathrm { g } ( x )\) is a function to be found. Given that the curve with equation \(y = \mathrm { f } ( x )\) has at least one stationary point, (b) find the range of possible values of \(k\).

1. Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(4 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(4 \mathbf { j } + 6 \mathbf { k }\)
• the point \(C\) has position vector \(- 16 \mathbf { i } + p \mathbf { j } + 10 \mathbf { k }\)
  where \(p\) is a constant.
  Given that \(A , B\) and \(C\) lie on a straight line,
  1. find the value of \(p\).

The line segment \(O B\) is extended to a point \(D\) so that \(\overrightarrow { C D }\) is parallel to \(\overrightarrow { O A }\) (b) Find \(| \overrightarrow { O D } |\), writing your answer as a fully simplified surd.

1. (a) Express \(\frac { 3 } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.

When chemical \(A\) and chemical \(B\) are mixed, oxygen is produced.
A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. The total volume of oxygen produced, \(V \mathrm {~m} ^ { 3 } , t\) hours after the chemicals were mixed, is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 3 V } { ( 2 t - 1 ) ( t + 1 ) } \quad V \geqslant 0 \quad t \geqslant k$$ where \(k\) is a constant.
Given that exactly 2 hours after the chemicals were mixed, a total volume of \(3 \mathrm {~m} ^ { 3 }\) of oxygen had been produced,
(b) solve the differential equation to show that $$V = \frac { 3 ( 2 t - 1 ) } { ( t + 1 ) }$$ The scientist noticed that

• there was a time delay between the chemicals being mixed and oxygen being produced
• there was a limit to the total volume of oxygen produced

Deduce from the model
(c) (i) the time delay giving your answer in minutes,
(ii) the limit giving your answer in \(\mathrm { m } ^ { 3 }\)

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

\section*{Solutions relying on calculator technology are not acceptable.} Given that the first three terms of a geometric series are $$12 \cos \theta \quad 5 + 2 \sin \theta \quad \text { and } \quad 6 \tan \theta$$

1. show that $$4 \sin ^ { 2 } \theta - 52 \sin \theta + 25 = 0$$ Given that \(\theta\) is an obtuse angle measured in radians,
2. solve the equation in part (a) to find the exact value of \(\theta\)
3. show that the sum to infinity of the series can be expressed in the form $$k ( 1 - \sqrt { 3 } )$$ where \(k\) is a constant to be found.

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \tan t + 1 \quad y = 2 \sec ^ { 2 } t + 3 \quad - \frac { \pi } { 4 } \leqslant t \leqslant \frac { \pi } { 3 }$$ The line \(l\) is the normal to \(C\) at the point \(P\) where \(t = \frac { \pi } { 4 }\)

1. Using parametric differentiation, show that an equation for \(l\) is $$y = - \frac { 1 } { 2 } x + \frac { 17 } { 2 }$$
2. Show that all points on \(C\) satisfy the equation $$y = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 } + 5$$ The straight line with equation $$y = - \frac { 1 } { 2 } x + k \quad \text { where } k \text { is a constant }$$ intersects \(C\) at two distinct points.
3. Find the range of possible values for \(k\).