Questions P1 (1374 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel P1 2022 June Q8
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
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
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
  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
  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
  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
  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
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
  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
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
  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
\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. \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
    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
      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
    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.
    Edexcel P1 2024 June Q4
    1. The curve \(C _ { 1 }\) has equation
    $$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
    Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)
    1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
    2. Hence find the coordinates of \(P\)
    Edexcel P1 2024 June Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-10_529_1403_255_267} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the plan view of a garden.
    The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\)
    Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)
    1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
    2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
    3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.
    Edexcel P1 2024 June Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-14_899_901_251_584} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} \section*{In this question you must show all stages of your working.
    Solutions relying on calculator technology are not acceptable.} Figure 3 shows
    • the line \(l\) with equation \(y - 5 x = 75\)
    • the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)
    The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .
    1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
    2. Use inequalities to define the region \(R\).
    Edexcel P1 2024 June Q7
    1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
    $$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and \(k\) is a constant.
    1. Find, in simplest form,
      1. \(\mathrm { f } ^ { \prime } ( x )\)
      2. \(\mathrm { f } ^ { \prime \prime } ( x )\) The curve with equation \(y = \mathrm { f } ^ { \prime } ( x )\) intersects the curve with equation \(y = \mathrm { f } ^ { \prime \prime } ( x )\) at the points \(A\) and \(B\). Given that the \(x\) coordinate of \(A\) is 5
    2. find the value of \(k\).
    3. Hence find the coordinates of \(B\).
    Edexcel P1 2024 June Q8
    1. The curve \(C _ { 1 }\) has equation
    $$y = x \left( 4 - x ^ { 2 } \right)$$
    1. Sketch the graph of \(C _ { 1 }\) showing the coordinates of any points of intersection with the coordinate axes. The curve \(C _ { 2 }\) has equation \(y = \frac { A } { x }\) where \(A\) is a constant.
    2. Show that the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
    3. Hence find the range of possible values of \(A\) for which \(C _ { 1 }\) meets \(C _ { 2 }\) at 4 distinct points.
    Edexcel P1 2024 June Q9
    1. Given that
    • the point \(A\) has coordinates \(( 4,2 )\)
    • the point \(B\) has coordinates \(( 15,7 )\)
    • the line \(l _ { 1 }\) passes through \(A\) and \(B\)
      1. find an equation for \(l _ { 1 }\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found.
    The line \(l _ { 2 }\) passes through \(A\) and is parallel to the \(x\)-axis.
    The point \(C\) lies on \(l _ { 2 }\) so that the length of \(B C\) is \(5 \sqrt { 5 }\)
  • Find both possible pairs of coordinates of the point \(C\).
  • Hence find the minimum possible area of triangle \(A B C\).
    •
    Edexcel P1 2024 June Q10
    1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)
    Given that
    • \(\mathrm { f } ^ { \prime } ( x ) = 6 x - \frac { ( 2 x - 1 ) ( 3 x + 2 ) } { 2 \sqrt { x } }\)
    • the point \(P ( 4,12 )\) lies on \(C\)
      1. find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found,
      2. find \(\mathrm { f } ( x )\), giving each term in simplest form.
    Edexcel P1 2024 June Q11
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-30_686_707_205_680} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 12 \sin x$$ where \(x\) is measured in radians.
    The point \(P\) shown in Figure 4 is a maximum point on \(C _ { 1 }\)
    1. Find the coordinates of \(P\). The curve \(C _ { 2 }\) has equation $$y = 12 \sin x + k$$ where \(k\) is a constant.
      Given that the maximum value of \(y\) on \(C _ { 2 }\) is 3
    2. find the coordinates of the minimum point on \(C _ { 2 }\) which has the smallest positive \(x\) coordinate. The curve \(C _ { 3 }\) has equation $$y = 12 \sin ( x + B )$$ where \(B\) is a positive constant.
      Given that \(\left( \frac { \pi } { 4 } , A \right)\), where \(A\) is a constant, is the minimum point on \(C _ { 3 }\) which has the smallest positive \(x\) coordinate,
    3. find
      1. the value of \(A\),
      2. the smallest possible value of \(B\).
    Edexcel P1 2019 October Q1
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-02_488_376_287_790} \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 1.25 radians. Given that the area of the sector \(A O B\) is \(15 \mathrm {~cm} ^ { 2 }\)
    1. find the exact value of \(r\),
    2. find the exact length of the perimeter of the sector. Write your answer in simplest form.