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Edexcel P1 2021 June Q8
9 marks Moderate -0.8
8. The curve \(C _ { 1 }\) has equation $$y = 3 x ^ { 2 } + 6 x + 9$$
  1. Write \(3 x ^ { 2 } + 6 x + 9\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The point \(P\) is the minimum point of \(C _ { 1 }\)
  2. Deduce the coordinates of \(P\). A different curve \(C _ { 2 }\) has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where \(A\), \(B\), \(C\) and \(D\) are constants. Given that \(C _ { 2 }\)
    • passes through \(P\)
    • intersects the \(x\)-axis at \(- 4 , - 2\) and 3
    • find, making your method clear, the values of \(A , B , C\) and \(D\). \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-27_2644_1840_118_111}
    \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-29_2646_1838_121_116}
Edexcel P1 2021 June Q9
7 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-30_707_1034_251_456} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation $$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ The line \(l\), shown in Figure 4, is an asymptote to \(y = \tan x\)
  1. State an equation for \(l\). A copy of Figure 4, labelled Diagram 1, is shown on the next page.
    1. On Diagram 1, sketch the curve with equation $$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ stating the equation of the horizontal asymptote of this curve.
    2. Hence, giving a reason, state the number of solutions of the equation
  3. State the number of solutions of the equation \(\tan x = \frac { 1 } { x } + 1\) in the region
    1. \(0 \leqslant x \leqslant 40 \pi\)
    2. \(- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi\) $$\begin{aligned} & \qquad \tan x = \frac { 1 } { x } + 1 \\ & \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi \end{aligned}$$" \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}
Edexcel P1 2022 June Q1
4 marks Easy -1.3
  1. Find
$$\int \left( 10 x ^ { 5 } + 6 x ^ { 3 } - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in simplest form.
Edexcel P1 2022 June Q2
5 marks Easy -1.2
2. In the triangle \(A B C\),
  • \(A B = 21 \mathrm {~cm}\)
  • \(B C = 13 \mathrm {~cm}\)
  • angle \(B A C = 25 ^ { \circ }\)
  • angle \(A C B = x ^ { \circ }\)
    1. Use the sine rule to find the value of \(\sin x ^ { \circ }\), giving your answer to 4 decimal places.
Given also that \(A B\) is the longest side of the triangle,
  • find the value of \(x\), giving your answer to 2 decimal places.
    •
    Edexcel P1 2022 June Q3
    5 marks Moderate -0.8
    3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    1. Show that \(\frac { \sqrt { 180 } - \sqrt { 80 } } { \sqrt { 5 } }\) is an integer and find its value.
    2. Simplify $$\frac { 4 \sqrt { 5 } - 5 } { 7 - 3 \sqrt { 5 } }$$ giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational numbers.
    Edexcel P1 2022 June Q4
    6 marks Moderate -0.8
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-08_604_1207_251_370} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\) The curve has a minimum at \(P ( - 1,0 )\) and a maximum at \(Q \left( \frac { 3 } { 2 } , 2 \right)\) The line with equation \(y = 1\) is the only asymptote to the curve. On separate diagrams sketch the curves with equation
    1. \(y = \mathrm { f } ( x ) - 2\)
    2. \(y = \mathrm { f } ( - x )\) On each sketch you must clearly state
      • the coordinates of the maximum and minimum points
      • the equation of the asymptote
    Edexcel P1 2022 June Q5
    9 marks Moderate -0.8
    1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\)
    Given that
    • \(\mathrm { f } ( x )\) is a quadratic expression
    • the maximum turning point on \(C\) has coordinates \(( - 2,12 )\)
    • \(C\) cuts the negative \(x\)-axis at - 5
      1. find \(\mathrm { f } ( x )\)
    The line \(l _ { 1 }\) has equation \(y = \frac { 4 } { 5 } x\) Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(( - 5,0 )\)
  • find an equation for \(l _ { 2 }\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
    (3) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve \(C\) and the lines \(l _ { 1 }\) and \(l _ { 2 }\)
  • Define the region \(R\), shown shaded in Figure 2, using inequalities.
    •
    Edexcel P1 2022 June Q6
    7 marks Standard +0.3
    6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    1. Given that $$2 x y - 3 x ^ { 2 } = 50$$ and $$y - x ^ { 3 } + 6 x = 0$$ show that $$2 x ^ { 4 } - 15 x ^ { 2 } - 50 = 0$$
    2. Hence solve the simultaneous equations $$\begin{aligned} 2 x y - 3 x ^ { 2 } & = 50 \\ y - x ^ { 3 } + 6 x & = 0 \end{aligned}$$ Give your answers in fully simplified surd form. \includegraphics[max width=\textwidth, alt={}, center]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-14_2257_52_312_1982}
    Edexcel P1 2022 June Q7
    9 marks Moderate -0.3
    7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that
    • \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \sqrt { x } } + \frac { A } { x ^ { 2 } } + 3\), where \(A\) is a constant
    • \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 4\)
      1. find the value of \(A\).
    Given also that
    • \(\mathrm { f } ( x ) = 8 \sqrt { 3 }\), when \(x = 12\)
    • find \(\mathrm { f } ( x )\), giving each term in simplest form.
    Edexcel P1 2022 June Q8
    10 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-22_922_876_246_539} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the outline of the face of a ceiling fan viewed from below.
    The fan consists of three identical sections congruent to \(O A B C D O\), shown in Figure 3, where
    • \(O A B O\) is a sector of a circle with centre \(O\) and radius 9 cm
    • \(O B C D O\) is a sector of a circle with centre \(O\) and radius 84 cm
    • angle \(A O D = \frac { 2 \pi } { 3 }\) radians
    Given that the length of the arc \(A B\) is 15 cm ,
    1. show that the length of the arc \(C D\) is 35.9 cm to one decimal place. The face of the fan is modelled to be a flat surface.
      Find, according to the model,
    2. the perimeter of the face of the fan, giving your answer to the nearest cm,
    3. the surface area of the face of the fan. Give your answer to 3 significant figures and make your units clear.
    Edexcel P1 2022 June Q9
    8 marks Moderate -0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-26_428_1354_251_287} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows part of the graph of the curve with equation \(y = \sin x\) Given that \(\sin \alpha = p\), where \(0 < \alpha < 90 ^ { \circ }\)
    1. state, in terms of \(p\), the value of
      1. \(2 \sin \left( 180 ^ { \circ } - \alpha \right)\)
      2. \(\sin \left( \alpha - 180 ^ { \circ } \right)\)
      3. \(3 + \sin \left( 180 ^ { \circ } + \alpha \right)\) A copy of Figure 4, labelled Diagram 1, is shown on page 27. On Diagram 1,
    2. sketch the graph of \(y = \sin 2 x\)
    3. Hence find, in terms of \(\alpha\), the \(x\) coordinates of any points in the interval \(0 < x < 180 ^ { \circ }\) where $$\sin 2 x = p$$
      \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-27_433_1331_296_310}
      \section*{Diagram 1}
    Edexcel P1 2022 June Q10
    12 marks Standard +0.3
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-28_655_869_255_541} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where \(k\) is a constant.
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
    2. show that the \(x\) coordinate of the point \(B\) is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
    3. Hence find the \(x\) coordinate of \(B\), justifying your answer. Given that the \(y\) intercept of \(l\) is - 1
    4. find the value of \(k\).
      \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}
    Edexcel P1 2023 June Q1
    4 marks Moderate -0.8
    1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    Solve the inequality $$4 x ^ { 2 } - 3 x + 7 \geq 4 x + 9$$
    Edexcel P1 2023 June Q2
    6 marks Easy -1.3
    1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    A rectangular sports pitch has length \(x\) metres and width \(y\) metres, where \(x > y\) Given that the perimeter of the pitch is 350 m ,
    1. write down an equation linking \(x\) and \(y\) Given also that the area of the pitch is \(7350 \mathrm {~m} ^ { 2 }\)
    2. write down a second equation linking \(x\) and \(y\)
    3. hence find the value of \(x\) and the value of \(y\)
    Edexcel P1 2023 June Q3
    6 marks Moderate -0.8
    1. (a) Express \(3 x ^ { 2 } + 12 x + 13\) in the form
    $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
    (b) Hence sketch the curve with equation \(y = 3 x ^ { 2 } + 12 x + 13\) On your sketch show clearly
    • the coordinates of the \(y\) intercept
    • the coordinates of the turning point of the curve
    Edexcel P1 2023 June Q4
    7 marks Moderate -0.8
    1. In this question you must show all stages of your working.
      1. Write
      $$y = \frac { 5 x ^ { 2 } + \sqrt { x ^ { 3 } } } { \sqrt [ 3 ] { 8 x } }$$ in the form $$y = A x ^ { p } + B x ^ { q }$$ where \(A , B , p\) and \(q\) are constants to be found.
    2. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each coefficient in simplest form.
    Edexcel P1 2023 June Q5
    10 marks Moderate -0.3
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-10_488_784_310_667} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the plan for a garden.
    In the plan
    • \(O A\) and \(C D\) are perpendicular to \(O D\)
    • \(A B\) is an arc of the circle with centre \(O\) and radius 4 metres
    • \(\quad B C\) is parallel to \(O D\)
    • \(O D\) is 6 metres, \(O A\) is 4 metres and \(C D\) is 1.5 metres
      1. Show that angle \(A O B\) is 1.186 radians to 4 significant figures.
      2. Find the perimeter of the garden, giving your answer in metres to 3 significant figures.
      3. Find the area of the garden, giving your answer in square metres to 3 significant figures.
    Edexcel P1 2023 June Q6
    7 marks Moderate -0.8
    1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
      1. Expand and simplify
      $$\left( r - \frac { 1 } { r } \right) ^ { 2 }$$
    2. Express \(\frac { 1 } { 3 + 2 \sqrt { 2 } }\) in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are integers.
    3. Use the results of parts (a) and (b), or otherwise, to show that $$\sqrt { 3 + 2 \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 + 2 \sqrt { 2 } } } = 2$$
    Edexcel P1 2023 June Q7
    6 marks Moderate -0.3
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-18_737_951_301_587} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The region \(R _ { 1 }\), shown shaded in Figure 2, is defined by the inequalities $$0 \leqslant y \leqslant 2 \quad y \leqslant 10 - 2 x \quad y \leqslant k x$$ where \(k\) is a constant.
    The line \(x = a\), where \(a\) is a constant, passes through the intersection of the lines \(y = 2\) and \(y = k x\) Given that the area of \(R _ { 1 }\) is \(\frac { 27 } { 4 }\) square units,
    1. find
      1. the value of \(a\)
      2. the value of \(k\)
    2. Define the region \(R _ { 2 }\), also shown shaded in Figure 2, using inequalities.
    Edexcel P1 2023 June Q8
    10 marks Moderate -0.3
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Find the equation of the tangent to the curve with equation $$y = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$ at the point \(P ( 4,12 )\) Give your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The curve with equation \(y = \mathrm { f } ( x )\) also passes through the point \(P ( 4,12 )\) Given that $$f ^ { \prime } ( x ) = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$
    2. find \(\mathrm { f } ( x )\) giving the coefficients in simplest form.
    Edexcel P1 2023 June Q9
    9 marks Moderate -0.3
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_536_933_255_568} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows part of the graph of the trigonometric function with equation \(y = \mathrm { f } ( x )\)
    1. Write down an expression for \(\mathrm { f } ( x )\) On a separate diagram,
    2. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)\) Show clearly the coordinates of all the points where the curve intersects the coordinate axes.
      (ii) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 shows part of the graph of the trigonometric function with equation \(y = \mathrm { g } ( x )\)
    3. Write down an expression for \(\mathrm { g } ( x )\) On a separate diagram,
    4. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { g } ( x ) - 2\) Show clearly the coordinates of the \(y\) intercept.
    Edexcel P1 2023 June Q10
    10 marks Standard +0.3
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-28_903_1010_219_539} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,
    • has gradient \(\frac { 1 } { 2 }\)
    • intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
      1. (i) Write down an equation for \(l _ { 1 }\) (ii) Find, in terms of \(b\), the coordinates of \(P\)
    Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and intersects \(C\) on the positive \(x\)-axis,
  • find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)
  • show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
  • Hence, or otherwise, find the value of \(b\)
    •
    Edexcel P1 2024 June Q1
    3 marks Easy -1.2
    1. Find
    $$\int \left( 10 x ^ { 4 } - \frac { 3 } { 2 x ^ { 3 } } - 7 \right) \mathrm { d } x$$ giving each term in simplest form.
    Edexcel P1 2024 June Q2
    6 marks Easy -1.2
      1. Given that \(m = 2 ^ { n }\), express each of the following in simplest form in terms of \(m\).
        1. \(2 ^ { n + 3 }\)
      2. \(16 ^ { 3 n }\) (ii) In this question you must show all stages of your working.
      Solutions relying on calculator technology are not acceptable. Solve the equation $$x \sqrt { 3 } - 3 = x + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are integers.
    Edexcel P1 2024 June Q3
    6 marks Moderate -0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-06_688_771_251_648} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
    The curve passes through the points \(( - 1,0 )\) and \(( 0,2 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). On separate diagrams, sketch the curve with equation
    1. \(y = \mathrm { f } ( \mathrm { x } + 3 )\)
    2. \(y = \mathrm { f } ( - 3 x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.