Edexcel Paper 1 (Paper 1) 2023 June

1. Find

$$\int \frac { x ^ { \frac { 1 } { 2 } } ( 2 x - 5 ) } { 3 } \mathrm {~d} x$$ writing each term in simplest form.

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
2. Hence
   1. find the value of \(a\)
   2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$

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

• the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
• the point \(B\) has position vector \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\)
  where \(a\) is a positive integer.
  1. Show that \(| \overrightarrow { O A } | = \sqrt { 38 }\)
  2. Find the smallest value of \(a\) for which

$$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$

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

The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\)
Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 1 } { 2 } \cos x\)
• the curve has a stationary point with \(x\) coordinate \(\alpha\)
• \(\alpha\) is small
  1. use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.

The point \(P ( 0,3 )\) lies on \(C\)

Find the equation of the tangent to the curve at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

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

The table shows corresponding values of \(x\) and \(y\) for this curve, where \(a\) and \(b\) are constants. The trapezium rule is used, with all the \(y\) values in the table, to find an approximate area under the curve between \(x = 3\) and \(x = 4\) Given that this area is 17.59

1. show that \(a + 2 b = 51\) Given also that the sum of all the \(y\) values in the table is 97.2
2. find the value of \(a\) and the value of \(b\)

6. $$a = \log _ { 2 } x \quad b = \log _ { 2 } ( x + 8 )$$ Express in terms of \(a\) and/or \(b\)

1. \(\log _ { 2 } \sqrt { x }\)
2. \(\log _ { 2 } \left( x ^ { 2 } + 8 x \right)\)
3. \(\log _ { 2 } \left( 8 + \frac { 64 } { x } \right)\) Give your answer in simplest form.

1. The function f is defined by

$$f ( x ) = 3 + \sqrt { x - 2 } \quad x \in \mathbb { R } \quad x > 2$$

1. State the range of f
2. Find f-1 The function \(g\) is defined by $$g ( x ) = \frac { 15 } { x - 3 } \quad x \in \mathbb { R } \quad x \neq 3$$
3. Find \(g f ( 6 )\)
4. Find the exact value of the constant \(a\) for which $$\mathrm { f } \left( a ^ { 2 } + 2 \right) = \mathrm { g } ( a )$$

8. \begin{figure}[h] \end{figure} Figure 1 shows the plan view of a stage.
The plan view shows two congruent triangles \(A B O\) and \(G F O\) joined to a sector \(O C D E O\) of a circle, centre \(O\), where

• angle \(C O E = 2.3\) radians
• arc length \(C D E = 27.6 \mathrm {~m}\)
• \(A O G\) is a straight line of length 15 m
  1. Show that \(O C = 12 \mathrm {~m}\).
  2. Show that the size of angle \(A O B\) is 0.421 radians correct to 3 decimal places.

Given that the total length of the front of the stage, \(B C D E F\), is 35 m ,

find the total area of the stage, giving your answer to the nearest square metre.

1. The first three terms of a geometric sequence are

$$3 k + 4 \quad 12 - 3 k \quad k + 16$$ where \(k\) is a constant.

1. Show that \(k\) satisfies the equation $$3 k ^ { 2 } - 62 k + 40 = 0$$ Given that the sequence converges,
2. 1. find the value of \(k\), giving a reason for your answer,
   2. find the value of \(S _ { \infty }\)

1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 6 k x - 2 k y + 7 = 0$$ where \(k\) is a constant.

1. Find in terms of \(k\),
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\) The line with equation \(y = 2 x - 1\) intersects \(C\) at 2 distinct points.
2. Find the range of possible values of \(k\).

11. \begin{figure}[h] \end{figure} The value, \(V\) pounds, of a mobile phone, \(t\) months after it was bought, is modelled by $$V = a b ^ { t }$$ where \(a\) and \(b\) are constants.
Figure 2 shows the linear relationship between \(\log _ { 10 } V\) and \(t\).
The line passes through the points \(( 0,3 )\) and \(( 10,2.79 )\) as shown.
Using these points,

1. find the initial value of the phone,
2. find a complete equation for \(V\) in terms of \(t\), giving the exact value of \(a\) and giving the value of \(b\) to 3 significant figures. Exactly 2 years after it was bought, the value of the phone was \(\pounds 320\)
3. Use this information to evaluate the reliability of the model.

12. $$y = \sin x$$ where \(x\) is measured in radians.
Use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cos x$$ You may

• use without proof the formula for \(\sin ( A \pm B )\)
• assume that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)

1. On a roller coaster ride, passengers travel in carriages around a track.

On the ride, carriages complete multiple circuits of the track such that

• the maximum vertical height of a carriage above the ground is 60 m
• a carriage starts a circuit at a vertical height of 2 m above the ground
• the ground is horizontal

The vertical height, \(H \mathrm {~m}\), of a carriage above the ground, \(t\) seconds after the carriage starts the first circuit, is modelled by the equation $$H = a - b ( t - 20 ) ^ { 2 }$$ where \(a\) and \(b\) are positive constants.

1. Find a complete equation for the model.
2. Use the model to determine the height of the carriage above the ground when \(t = 40\) In an alternative model, the vertical height, \(H \mathrm {~m}\), of a carriage above the ground, \(t\) seconds after the carriage starts the first circuit, is given by $$H = 29 \cos ( 9 t + \alpha ) ^ { \circ } + \beta \quad 0 \leqslant \alpha < 360 ^ { \circ }$$ where \(\alpha\) and \(\beta\) are constants.
3. Find a complete equation for the alternative model. Given that the carriage moves continuously for 2 minutes,
4. give a reason why the alternative model would be more appropriate.

1. Prove, using algebra, that

$$( n + 1 ) ^ { 3 } - n ^ { 3 }$$ is odd for all \(n \in \mathbb { N }\)

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

$$\mathrm { f } ( x ) = \frac { 7 x \mathrm { e } ^ { x } } { \sqrt { \mathrm { e } ^ { 3 x } - 2 } } \quad x > \ln \sqrt [ 3 ] { 2 }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 7 \mathrm { e } ^ { x } \left( \mathrm { e } ^ { 3 x } ( 2 - x ) + A x + B \right) } { 2 \left( \mathrm { e } ^ { 3 x } - 2 \right) ^ { \frac { 3 } { 2 } } }$$ where \(A\) and \(B\) are constants to be found.
2. Hence show that the \(x\) coordinates of the turning points of the curve are solutions of the equation $$x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }$$ The equation \(x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) has two positive roots \(\alpha\) and \(\beta\) where \(\beta > \alpha\)
   A student uses the iteration formula $$x _ { n + 1 } = \frac { 2 \mathrm { e } ^ { 3 x _ { n } } - 4 } { \mathrm { e } ^ { 3 x _ { n } } + 4 }$$ in an attempt to find approximations for \(\alpha\) and \(\beta\)
   Diagram 1 shows a plot of part of the curve with equation \(y = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) and part of the line with equation \(y = x\) Using Diagram 1 on page 42
3. draw a staircase diagram to show that the iteration formula starting with \(x _ { 1 } = 1\) can be used to find an approximation for \(\beta\) Use the iteration formula with \(x _ { 1 } = 1\), to find, to 3 decimal places,
4. 1. the value of \(x _ { 2 }\)
   2. the value of \(\beta\) Using a suitable interval and a suitable function that should be stated
5. show that \(\alpha = 0.432\) to 3 decimal places. Only use the copy of Diagram 1 if you need to redraw your answer to part (c). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_736_812_372_143} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_738_815_370_1114} \captionsetup{labelformat=empty} \caption{copy of Diagram 1}
   \end{figure}