Edexcel Paper 2 (Paper 2) 2023 June

1. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 8 x + 5$$

1. Find \(f ^ { \prime \prime } ( x )\)
2. 1. Solve \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\)
   2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is concave.

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

$$\begin{aligned} u _ { 1 } & = 35
u _ { n + 1 } & = u _ { n } + 7 \cos \left( \frac { n \pi } { 2 } \right) - 5 ( - 1 ) ^ { n } \end{aligned}$$

1. 1. Show that \(u _ { 2 } = 40\)
   2. Find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\) Given that the sequence is periodic with order 4
2. 1. write down the value of \(u _ { 5 }\)
   2. find the value of \(\sum _ { r = 1 } ^ { 25 } u _ { r }\)

1. Given that

$$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$

1. show that $$3 x ^ { 2 } - 13 x - 30 = 0$$
2. 1. Write down the roots of the equation $$3 x ^ { 2 } - 13 x - 30 = 0$$
   2. Hence state which of the roots in part (b)(i) is not a solution of $$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$ giving a reason for your answer.

1. Coffee is poured into a cup.

The temperature of the coffee, \(H ^ { \circ } \mathrm { C } , t\) minutes after being poured into the cup is modelled by the equation $$H = A \mathrm { e } ^ { - B t } + 30$$ where \(A\) and \(B\) are constants.
Initially, the temperature of the coffee was \(85 ^ { \circ } \mathrm { C }\).

1. State the value of \(A\). Initially, the coffee was cooling at a rate of \(7.5 ^ { \circ } \mathrm { C }\) per minute.
2. Find a complete equation linking \(H\) and \(t\), giving the value of \(B\) to 3 decimal places.

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

The curve

• passes through the point \(P ( 3 , - 10 )\)
• has a turning point at \(P\)

Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } - 9 x ^ { 2 } + 5 x + k$$ where \(k\) is a constant,

1. show that \(k = 12\)
2. Hence find the coordinates of the point where \(C\) crosses the \(y\)-axis.

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

• \(A\) is the point with position vector \(12 \mathbf { i }\)
• \(B\) is the point with position vector \(16 \mathbf { j }\)
• \(C\) is the point with position vector \(( 50 \mathbf { i } + 136 \mathbf { j } )\)
• \(D\) is the point with position vector \(( 22 \mathbf { i } + 24 \mathbf { j } )\)
  1. Show that \(A D\) is parallel to \(B C\).

Points \(A , B , C\) and \(D\) are used to model the vertices of a running track in the shape of a quadrilateral. Runners complete one lap by running along all four sides of the track.
The lengths of the sides are measured in metres. Given that a particular runner takes exactly 5 minutes to complete 2 laps,

calculate the average speed of this runner, giving the answer in kilometres per hour.

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

Solutions relying on calculator technology are not acceptable.
A curve has equation $$x ^ { 3 } + 2 x y + 3 y ^ { 2 } = 47$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) The point \(P ( - 2,5 )\) lies on the curve.
2. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

1. (a) Express \(2 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
   Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.

The first three terms of an arithmetic sequence are $$\cos x \quad \cos x + \sin x \quad \cos x + 2 \sin x \quad x \neq n \pi$$ Given that \(S _ { 9 }\) represents the sum of the first 9 terms of this sequence as \(x\) varies,
(b) (i) find the exact maximum value of \(S _ { 9 }\)
(ii) deduce the smallest positive value of \(x\) at which this maximum value of \(S _ { 9 }\) occurs.

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

$$x = t ^ { 2 } + 6 t - 16 \quad y = 6 \ln ( t + 3 ) \quad t > - 3$$

1. Show that a Cartesian equation for \(C\) is $$y = A \ln ( x + B ) \quad x > - B$$ where \(A\) and \(B\) are integers to be found. The curve \(C\) cuts the \(y\)-axis at the point \(P\)
2. Show that the equation of the tangent to \(C\) at \(P\) can be written in the form $$a x + b y = c \ln 5$$ where \(a\), \(b\) and \(c\) are integers to be found.

1. \(\mathrm { f } ( x ) = \frac { 3 k x - 18 } { ( x + 4 ) ( x - 2 ) } \quad\) where \(k\) is a positive constant
   1. Express \(\mathrm { f } ( x )\) in partial fractions in terms of \(k\).
   2. Hence find the exact value of \(k\) for which
   $$\int _ { - 3 } ^ { 1 } f ( x ) d x = 21$$

11. \begin{figure}[h] \end{figure} A tank in the shape of a cuboid is being filled with water.
The base of the tank measures 20 m by 10 m and the height of the tank is 5 m , as shown in Figure 1. At time \(t\) minutes after water started flowing into the tank the height of the water was \(h \mathrm {~m}\) and the volume of water in the tank was \(V \mathrm {~m} ^ { 3 }\) In a model of this situation

• the sides of the tank have negligible thickness
• the rate of change of \(V\) is inversely proportional to the square root of \(h\)
  1. Show that

$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { \lambda } { \sqrt { h } }$$ where \(\lambda\) is a constant. Given that

• initially the height of the water in the tank was 1.44 m
• exactly 8 minutes after water started flowing into the tank the height of the water was 3.24 m
• use the model to find an equation linking \(h\) with \(t\), giving your answer in the form

$$h ^ { \frac { 3 } { 2 } } = A t + B$$ where \(A\) and \(B\) are constants to be found.

Hence find the time taken, from when water started flowing into the tank, for the tank to be completely full.

12. \begin{figure}[h] \end{figure} The number of subscribers to two different music streaming companies is being monitored. The number of subscribers, \(N _ { \mathrm { A } }\), in thousands, to company \(\mathbf { A }\) is modelled by the equation $$N _ { \mathrm { A } } = | t - 3 | + 4 \quad t \geqslant 0$$ where \(t\) is the time in years since monitoring began.
The number of subscribers, \(N _ { \mathrm { B } }\), in thousands, to company B is modelled by the equation $$N _ { \mathrm { B } } = 8 - | 2 t - 6 | \quad t \geqslant 0$$ where \(t\) is the time in years since monitoring began.
Figure 2 shows a sketch of the graph of \(N _ { \mathrm { A } }\) and the graph of \(N _ { \mathrm { B } }\) over a 5-year period.
Use the equations of the models to answer parts (a), (b), (c) and (d).

1. Find the initial difference between the number of subscribers to company \(\mathbf { A }\) and the number of subscribers to company B. When \(t = T\) company A reduced its subscription prices and the number of subscribers increased.
2. Suggest a value for \(T\), giving a reason for your answer.
3. Find the range of values of \(t\) for which \(N _ { \mathrm { A } } > N _ { \mathrm { B } }\) giving your answer in set notation.
4. State a limitation of the model used for company B.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of
   $$( 3 + x ) ^ { - 2 }$$ writing each term in simplest form.
2. Using the answer to part (a) and using algebraic integration, estimate the value of $$\int _ { 0.2 } ^ { 0.4 } \frac { 6 x } { ( 3 + x ) ^ { 2 } } d x$$ giving your answer to 4 significant figures.
3. Find, using algebraic integration, the exact value of $$\int _ { 0.2 } ^ { 0.4 } \frac { 6 x } { ( 3 + x ) ^ { 2 } } d x$$ giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are constants to be found.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$2 \tan \theta \left( 8 \cos \theta + 23 \sin ^ { 2 } \theta \right) = 8 \sin 2 \theta \left( 1 + \tan ^ { 2 } \theta \right)$$ may be written as $$\sin 2 \theta \left( A \cos ^ { 2 } \theta + B \cos \theta + C \right) = 0$$ where \(A , B\) and \(C\) are constants to be found.
2. Hence, solve for \(360 ^ { \circ } \leqslant x \leqslant 540 ^ { \circ }\) $$2 \tan x \left( 8 \cos x + 23 \sin ^ { 2 } x \right) = 8 \sin 2 x \left( 1 + \tan ^ { 2 } x \right) \quad x \in \mathbb { R } \quad x \neq 450 ^ { \circ }$$

1. A student attempts to answer the following question:

Given that \(x\) is an obtuse angle, use algebra to prove by contradiction that $$\sin x - \cos x \geqslant 1$$ The student starts the proof with: Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\begin{aligned} & \Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1
& \Rightarrow \ldots \end{aligned}$$ The start of the student's proof is reprinted below.
Complete the proof. Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1$$