Questions — Edexcel Paper 1 (133 questions)

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Edexcel Paper 1 2018 June Q1
  1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of
$$\frac { 1 - \cos 4 \theta } { 2 \theta \sin 3 \theta }$$ (3)
Edexcel Paper 1 2018 June Q2
  1. A curve \(C\) has equation
$$y = x ^ { 2 } - 2 x - 24 \sqrt { x } , \quad x > 0$$
  1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Verify that \(C\) has a stationary point when \(x = 4\)
  3. Determine the nature of this stationary point, giving a reason for your answer.
Edexcel Paper 1 2018 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-06_332_348_246_861} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
The angle \(A O B\) is \(\theta\) radians.
The area of the sector \(A O B\) is \(11 \mathrm {~cm} ^ { 2 }\) Given that the perimeter of the sector is 4 times the length of the arc \(A B\), find the exact value of \(r\).
Edexcel Paper 1 2018 June Q4
  1. The curve with equation \(y = 2 \ln ( 8 - x )\) meets the line \(y = x\) at a single point, \(x = \alpha\).
    1. Show that \(3 < \alpha < 4\)
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-08_666_1061_445_502} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the graph of \(y = 2 \ln ( 8 - x )\) and the graph of \(y = x\).
    A student uses the iteration formula $$x _ { n + 1 } = 2 \ln \left( 8 - x _ { n } \right) , \quad n \in \mathbb { N }$$ in an attempt to find an approximation for \(\alpha\).
    Using the graph and starting with \(x _ { 1 } = 4\)
  2. determine whether or not this iteration formula can be used to find an approximation for \(\alpha\), justifying your answer.
Edexcel Paper 1 2018 June Q5
  1. Given that
$$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { d y } { d \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(A\) is a rational constant to be found.
Edexcel Paper 1 2018 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-12_549_592_244_731} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Not to scale The circle \(C\) has centre \(A\) with coordinates (7,5).
The line \(l\), with equation \(y = 2 x + 1\), is the tangent to \(C\) at the point \(P\), as shown in Figure 3 .
  1. Show that an equation of the line \(P A\) is \(2 y + x = 17\)
  2. Find an equation for \(C\). The line with equation \(y = 2 x + k , \quad k \neq 1\) is also a tangent to \(C\).
  3. Find the value of the constant \(k\).
Edexcel Paper 1 2018 June Q7
  1. Given that \(k \in \mathbb { Z } ^ { + }\)
    1. show that \(\int _ { k } ^ { 3 k } \frac { 2 } { ( 3 x - k ) } \mathrm { d } x\) is independent of \(k\),
    2. show that \(\int _ { k } ^ { 2 k } \frac { 2 } { ( 2 x - k ) ^ { 2 } } \mathrm {~d} x\) is inversely proportional to \(k\).
Edexcel Paper 1 2018 June Q8
  1. The depth of water, \(D\) metres, in a harbour on a particular day is modelled by the formula
$$D = 5 + 2 \sin ( 30 t ) ^ { \circ } \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight. A boat enters the harbour at 6:30 am and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour.
  1. Find the depth of the water in the harbour when the boat enters the harbour.
  2. Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 1 2018 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-22_537_748_242_662} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } - 2 x y + 3 y ^ { 2 } = 50\)
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x } { 3 y - x }\) The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km .
    The points \(P\) and \(Q\) represent points that are furthest west and furthest east of the origin \(O\), as shown in Figure 4. Using part (a),
  2. find the exact coordinates of the point \(P\).
  3. Explain briefly how to find the coordinates of the point that is furthest north of the origin \(O\). (You do not need to carry out this calculation).
Edexcel Paper 1 2018 June Q10
  1. The height above ground, \(H\) metres, of a passenger on a roller coaster can be modelled by the differential equation
$$\frac { \mathrm { d } H } { \mathrm {~d} t } = \frac { H \cos ( 0.25 t ) } { 40 }$$ where \(t\) is the time, in seconds, from the start of the ride. Given that the passenger is 5 m above the ground at the start of the ride,
  1. show that \(H = 5 \mathrm { e } ^ { 0.1 \sin ( 0.25 t ) }\)
  2. State the maximum height of the passenger above the ground. The passenger reaches the maximum height, for the second time, \(T\) seconds after the start of the ride.
  3. Find the value of \(T\).
Edexcel Paper 1 2018 June Q11
  1. (a) Use binomial expansions to show that \(\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }\)
A student substitutes \(x = \frac { 1 } { 2 }\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt { 6 }\)
(b) Give a reason why the student should not use \(x = \frac { 1 } { 2 }\)
(c) Substitute \(x = \frac { 1 } { 11 }\) into $$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 6 }\). Give your answer as a fraction in its simplest form.
Edexcel Paper 1 2018 June Q12
  1. The value, \(\pounds V\), of a vintage car \(t\) years after it was first valued on 1 st January 2001, is modelled by the equation
$$V = A p ^ { t } \quad \text { where } A \text { and } p \text { are constants }$$ Given that the value of the car was \(\pounds 32000\) on 1st January 2005 and \(\pounds 50000\) on 1st January 2012
    1. find \(p\) to 4 decimal places,
    2. show that \(A\) is approximately 24800
  2. With reference to the model, interpret
    1. the value of the constant \(A\),
    2. the value of the constant \(p\). Using the model,
  3. find the year during which the value of the car first exceeds \(\pounds 100000\)
Edexcel Paper 1 2018 June Q13
  1. Show that
$$\int _ { 0 } ^ { 2 } 2 x \sqrt { x + 2 } \mathrm {~d} x = \frac { 32 } { 15 } ( 2 + \sqrt { 2 } )$$
Edexcel Paper 1 2018 June Q14
  1. A curve \(C\) has parametric equations
$$x = 3 + 2 \sin t , \quad y = 4 + 2 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$
  1. Show that all points on \(C\) satisfy \(y = 6 - ( x - 3 ) ^ { 2 }\)
    1. Sketch the curve \(C\).
    2. Explain briefly why \(C\) does not include all points of \(y = 6 - ( x - 3 ) ^ { 2 } , \quad x \in \mathbb { R }\) The line with equation \(x + y = k\), where \(k\) is a constant, intersects \(C\) at two distinct points.
  3. State the range of values of \(k\), writing your answer in set notation.
Edexcel Paper 1 2019 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-04_670_1447_212_333} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a plot of part of the curve with equation \(y = \cos x\) where \(x\) is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.
  1. Use Diagram 1 to show why the equation $$\cos x - 2 x - \frac { 1 } { 2 } = 0$$ has only one real root, giving a reason for your answer. Given that the root of the equation is \(\alpha\), and that \(\alpha\) is small,
  2. use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.
    \includegraphics[max width=\textwidth, alt={}]{91a2f26a-add2-4b58-997d-2ae229548217-05_664_1452_246_333}
    \section*{Diagram 1}
Edexcel Paper 1 2019 June Q3
3. $$y = \frac { 5 x ^ { 2 } + 10 x } { ( x + 1 ) ^ { 2 } } \quad x \neq - 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A } { ( x + 1 ) ^ { n } }\) where \(A\) and \(n\) are constants to be found.
  2. Hence deduce the range of values for \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\)
Edexcel Paper 1 2019 June Q4
  1. (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of
$$\frac { 1 } { \sqrt { 4 - x } }$$ giving each coefficient in its simplest form. The expansion can be used to find an approximation to \(\sqrt { 2 }\)
Possible values of \(x\) that could be substituted into this expansion are:
  • \(x = - 14\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 18 } } = \frac { \sqrt { 2 } } { 6 }\)
  • \(x = 2\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 2 } } = \frac { \sqrt { 2 } } { 2 }\)
  • \(x = - \frac { 1 } { 2 }\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { \frac { 9 } { 2 } } } = \frac { \sqrt { 2 } } { 3 }\)
    (b) Without evaluating your expansion,
    1. state, giving a reason, which of the three values of \(x\) should not be used
    2. state, giving a reason, which of the three values of \(x\) would lead to the most accurate approximation to \(\sqrt { 2 }\)
Edexcel Paper 1 2019 June Q5
5. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$
  1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are integers to be found.
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
    1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
    2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$
Edexcel Paper 1 2019 June Q6
  1. (a) Solve, for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), the equation
$$5 \sin 2 \theta = 9 \tan \theta$$ giving your answers, where necessary, to one decimal place.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
(b) Deduce the smallest positive solution to the equation $$5 \sin \left( 2 x - 50 ^ { \circ } \right) = 9 \tan \left( x - 25 ^ { \circ } \right)$$
Edexcel Paper 1 2019 June Q7
  1. In a simple model, the value, \(\pounds V\), of a car depends on its age, \(t\), in years.
The following information is available for \(\operatorname { car } A\)
  • its value when new is \(\pounds 20000\)
  • its value after one year is \(\pounds 16000\)
    1. Use an exponential model to form, for car \(A\), a possible equation linking \(V\) with \(t\).
The value of car \(A\) is monitored over a 10-year period.
Its value after 10 years is \(\pounds 2000\)
  • Evaluate the reliability of your model in light of this information. The following information is available for car \(B\)
    • it has the same value, when new, as car \(A\)
    • its value depreciates more slowly than that of \(\operatorname { car } A\)
    • Explain how you would adapt the equation found in (a) so that it could be used to model the value of car \(B\).
    •
    Edexcel Paper 1 2019 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-22_812_958_244_555} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = x ( x + 2 ) ( x - 4 )\).
    The region \(R _ { 1 }\) shown shaded in Figure 2 is bounded by the curve and the negative \(x\)-axis.
    1. Show that the exact area of \(R _ { 1 }\) is \(\frac { 20 } { 3 }\) The region \(R _ { 2 }\) also shown shaded in Figure 2 is bounded by the curve, the positive \(x\)-axis and the line with equation \(x = b\), where \(b\) is a positive constant and \(0 < b < 4\) Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
    2. verify that \(b\) satisfies the equation $$( b + 2 ) ^ { 2 } \left( 3 b ^ { 2 } - 20 b + 20 \right) = 0$$ The roots of the equation \(3 b ^ { 2 } - 20 b + 20 = 0\) are 1.225 and 5.442 to 3 decimal places. The value of \(b\) is therefore 1.225 to 3 decimal places.
    3. Explain, with the aid of a diagram, the significance of the root 5.442
    Edexcel Paper 1 2019 June Q9
    1. Given that \(a > b > 0\) and that \(a\) and \(b\) satisfy the equation
    $$\log a - \log b = \log ( a - b )$$
    1. show that $$a = \frac { b ^ { 2 } } { b - 1 }$$ (3)
    2. Write down the full restriction on the value of \(b\), explaining the reason for this restriction.
    Edexcel Paper 1 2019 June Q10
    1. (i) Prove that for all \(n \in \mathbb { N } , n ^ { 2 } + 2\) is not divisible by 4
      (ii) "Given \(x \in \mathbb { R }\), the value of \(| 3 x - 28 |\) is greater than or equal to the value of ( \(x - 9\) )." State, giving a reason, if the above statement is always true, sometimes true or never true.
      (2)
    Edexcel Paper 1 2019 June Q11
    1. A competitor is running a 20 kilometre race.
    She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be \(5 \%\) greater than the time that she took to complete the previous kilometre. Using the model,
    1. show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
    2. show that her estimated time, in minutes, to run the \(r\) th kilometre, for \(5 \leqslant r \leqslant 20\), is $$6 \times 1.05 ^ { r - 4 }$$
    3. estimate the total time, in minutes and seconds, that she will take to complete the race.
    Edexcel Paper 1 2019 June Q12
    12. $$\mathrm { f } ( x ) = 10 \mathrm { e } ^ { - 0.25 x } \sin x , \quad x \geqslant 0$$
    1. Show that the \(x\) coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\) satisfy the equation \(\tan x = 4\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-34_687_1029_495_518} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
    2. Sketch the graph of \(H\) against \(t\) where $$\mathrm { H } ( t ) = \left| 10 \mathrm { e } ^ { - 0.25 t } \sin t \right| \quad t \geqslant 0$$ showing the long-term behaviour of this curve. The function \(\mathrm { H } ( t )\) is used to model the height, in metres, of a ball above the ground \(t\) seconds after it has been kicked. Using this model, find
    3. the maximum height of the ball above the ground between the first and second bounce.
    4. Explain why this model should not be used to predict the time of each bounce.