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 January Q8
8. The line \(l _ { 1 }\) has equation $$2 x - 5 y + 7 = 0$$
  1. Find the gradient of \(l _ { 1 }\) Given that
    • the point \(A\) has coordinates \(( 6 , - 2 )\)
    • the line \(l _ { 2 }\) passes through \(A\) and is perpendicular to \(l _ { 1 }\)
    • find the equation of \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(M\).
  2. Using algebra and showing all your working, find the coordinates of \(M\).
    (Solutions relying on calculator technology are not acceptable.) Given that the diagonals of a square \(A B C D\) meet at \(M\),
  3. find the coordinates of the point \(C\).
Edexcel P1 2022 January Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c320b71-8793-461a-a078-e4f64c144a3a-28_784_1324_260_312} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows part of the curve with equation $$y = A \cos ( x - 30 ) ^ { \circ }$$ where \(A\) is a constant. The point \(P\) is a minimum point on the curve and has coordinates \(( 30 , - 3 )\) as shown in Figure 4.
  1. Write down the value of \(A\). The point \(Q\) is shown in Figure 4 and is a maximum point.
  2. Find the coordinates of \(Q\).
Edexcel P1 2022 January Q10
10. The curve \(C\) has equation $$y = \frac { 1 } { x ^ { 2 } } - 9$$
  1. Sketch the graph of \(C\). On your sketch
    • show the coordinates of any points of intersection with the coordinate axes
    • state clearly the equations of any asymptotes
    The curve \(D\) has equation \(y = k x ^ { 2 }\) where \(k\) is a constant. Given that \(C\) meets \(D\) at 4 distinct points,
  2. find the range of possible values for \(k\).
Edexcel P1 2023 January Q1
  1. A curve \(C\) has equation
$$y = 2 + 10 x ^ { \frac { 1 } { 2 } } - 2 x ^ { \frac { 3 } { 2 } } \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving your answer in simplest form.
  2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(x = 2\) giving your answer in simplest form.
    (Solutions relying on calculator technology are not acceptable.)
Edexcel P1 2023 January Q2
  1. The points \(P , Q\) and \(R\) have coordinates (-3, 7), (9, 11) and (12, 2) respectively.
    1. Prove that angle \(P Q R = 90 ^ { \circ }\)
    Given that the point \(S\) is such that \(P Q R S\) forms a rectangle,
  2. find the coordinates of \(S\).
Edexcel P1 2023 January Q3
  1. Find
$$\int \frac { 4 x ^ { 5 } + 3 } { 2 x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
Edexcel P1 2023 January Q4
  1. Given that the equation
    \(k x ^ { 2 } + 6 k x + 5 = 0 \quad\) where \(k\) is a non zero constant has no real roots, find the range of possible values for \(k\).
Edexcel P1 2023 January Q5
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
  2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$
Edexcel P1 2023 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb21001f-fe68-4776-992d-ede1aae233d7-12_438_816_246_621} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram NOT accurately drawn Figure 1 shows the plan view for the design of a stage.
The design consists of a sector \(O B C\) of a circle, with centre \(O\), joined to two congruent triangles \(O A B\) and \(O D C\). Given that
  • angle \(B O C = 2.4\) radians
  • area of sector \(B O C = 40 \mathrm {~m} ^ { 2 }\)
  • \(A O D\) is a straight line of length 12.5 m
    1. find the radius of the sector, giving your answer, in m , to 2 decimal places,
    2. find the size of angle \(A O B\), in radians, to 2 decimal places.
Hence find
  • the total area of the stage, giving your answer, in \(\mathrm { m } ^ { 2 }\), to one decimal place,
  • the total perimeter of the stage, giving your answer, in m , to one decimal place.
    •
    Edexcel P1 2023 January Q7
    1. (a) On Diagram 1, sketch a graph of the curve \(C\) with equation
    $$y = \frac { 6 } { x } \quad x \neq 0$$ The curve \(C\) is transformed onto the curve with equation \(y = \frac { 6 } { x - 2 } \quad x \neq 2\)
    (b) Fully describe this transformation. The curve with equation $$y = \frac { 6 } { x - 2 } \quad x \neq 2$$ and the line with equation $$y = k x + 7 \quad \text { where } k \text { is a constant }$$ intersect at exactly two points, \(P\) and \(Q\).
    Given that the \(x\) coordinate of point \(P\) is - 4
    (c) find the value of \(k\),
    (d) find, using algebra, the coordinates of point \(Q\).
    (Solutions relying entirely on calculator technology are not acceptable.)
    \includegraphics[max width=\textwidth, alt={}]{bb21001f-fe68-4776-992d-ede1aae233d7-17_710_743_248_662}
    \section*{Diagram 1} Only use this copy of Diagram 1 if you need to redraw your graph.
    \includegraphics[max width=\textwidth, alt={}, center]{bb21001f-fe68-4776-992d-ede1aae233d7-19_709_739_1802_664} Copy of Diagram 1
    (Total for Question 7 is 10 marks)
    Edexcel P1 2023 January Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb21001f-fe68-4776-992d-ede1aae233d7-20_728_885_248_584} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the straight line \(l\) and the curve \(C\).
    Given that \(l\) cuts the \(y\)-axis at - 12 and cuts the \(x\)-axis at 4 , as shown in Figure 2,
    1. find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. Given that \(C\)
      • has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression
      • has a minimum point at \(( 7 , - 18 )\)
      • cuts the \(x\)-axis at 4 and at \(k\), where \(k\) is a constant
      • deduce the value of \(k\),
      • find \(\mathrm { f } ( x )\).
      The region \(R\) is shown shaded in Figure 2.
    2. Use inequalities to define \(R\).
    Edexcel P1 2023 January Q9
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb21001f-fe68-4776-992d-ede1aae233d7-24_675_835_251_616} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of
    • the curve with equation \(y = \tan x\)
    • the straight line l with equation \(y = \pi x\)
      in the interval \(- \pi < x < \pi\)
      1. State the period of \(\tan x\)
      2. Write down the number of roots of the equation
        1. \(\tan x = ( \pi + 2 ) x\) in the interval \(- \pi < x < \pi\)
        2. \(\tan x = \pi x\) in the interval \(- 2 \pi < x < 2 \pi\)
        3. \(\tan x = \pi x\) in the interval \(- 100 \pi < x < 100 \pi\)
    Edexcel P1 2023 January Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb21001f-fe68-4776-992d-ede1aae233d7-26_902_896_248_587} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$$
    1. Use the given information to state the values of \(x\) for which $$f ( x ) > 0$$
    2. Expand \(( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )\), writing your answer as a polynomial in simplest form. The straight line \(l\) is the tangent to \(C\) at the point where \(C\) cuts the \(y\)-axis.
      Given that \(l\) cuts \(C\) at the point \(P\), as shown in Figure 4,
    3. find, using algebra, the \(x\) coordinate of \(P\)
      (Solutions based on calculator technology are not acceptable.)
    Edexcel P1 2023 January Q11
    1. A curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\)
    Given that
    • \(\mathrm { f } ^ { \prime \prime } ( x ) = 4 x + \frac { 1 } { \sqrt { x } }\)
    • the point \(P\) has \(x\) coordinate 4 and lies on \(C\)
    • the tangent to \(C\) at \(P\) has equation \(y = 3 x + 4\)
      1. find an equation of the normal to \(C\) at \(P\)
      2. find \(\mathrm { f } ( x )\), writing your answer in simplest form.
    Edexcel P1 2024 January Q1
    1. Find
    $$\int ( 2 x - 5 ) ( 3 x + 2 ) ( 2 x + 5 ) \mathrm { d } x$$ writing your answer in simplest form.
    Edexcel P1 2024 January Q2
    1. The triangle \(A B C\) is such that
    • \(A B = 15 \mathrm {~cm}\)
    • \(A C = 25 \mathrm {~cm}\)
    • angle \(B A C = \theta ^ { \circ }\)
    • area triangle \(A B C = 100 \mathrm {~cm} ^ { 2 }\)
      1. Find the value of \(\sin \theta ^ { \circ }\)
    Given that \(\theta > 90\)
  • find the length of \(B C\), in cm , to 3 significant figures.
    •
    Edexcel P1 2024 January Q3
    1. The curve \(C\) has equation
    $$y = \frac { 5 x ^ { 3 } - 8 } { 2 x ^ { 2 } } \quad x > 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) writing your answer in simplest form. The point \(P ( 2,4 )\) lies on \(C\).
    2. Find an equation for the tangent to \(C\) at \(P\) writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    Edexcel P1 2024 January Q4
    1. In this question you must show all stages of your working.
    Solutions relying on calculator technology are not acceptable.
    1. By substituting \(p = 2 ^ { x }\), show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
    2. Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$
    Edexcel P1 2024 January Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2043b938-ed3f-4b69-9ea9-b4ab62e2a8ce-10_891_850_295_609} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The straight line \(l _ { 1 }\), shown in Figure 1, passes through the points \(P ( - 2,9 )\) and \(Q ( 10,6 )\).
    1. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The straight line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)
      The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\) as shown in Figure 1.
    2. Find the coordinates of \(R\)
    3. Find the exact area of triangle \(O P Q\).
    Edexcel P1 2024 January Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2043b938-ed3f-4b69-9ea9-b4ab62e2a8ce-14_919_954_299_559} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = 5 \cos x$$ with \(x\) being measured in degrees.
    The point \(P\), shown in Figure 2, is a minimum point on \(C _ { 1 }\)
    1. State the coordinates of \(P\) The point \(Q\) lies on a different curve \(C _ { 2 }\)
      Given that point \(Q\)
      • is a maximum point on the curve
      • is the maximum point with the smallest \(x\) coordinate, \(x > 0\)
      • find the coordinates of \(Q\) when
        1. \(C _ { 2 }\) has equation \(y = 5 \cos x - 2\)
        2. \(C _ { 2 }\) has equation \(y = - 5 \cos x\)
    Edexcel P1 2024 January Q7
    1. (a) Sketch the graph of the curve \(C\) with equation
    $$y = \frac { 4 } { x - k }$$ where \(k\) is a positive constant.
    Show on your sketch
    • the coordinates of any points where \(C\) cuts the coordinate axes
    • the equation of the vertical asymptote to \(C\)
    Given that the straight line with equation \(y = 9 - x\) does not cross or touch \(C\)
    (b) find the range of values of \(k\).
    Edexcel P1 2024 January Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2043b938-ed3f-4b69-9ea9-b4ab62e2a8ce-18_680_933_294_589} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the plan view of a platform.
    The plan view of the platform consists of a sector \(D O C\) of a circle centre \(O\) joined to a sector \(A O B E A\) of a different circle, also with centre \(O\). Given that
    • angle \(A O B = 0.8\) radians
    • arc length \(C D = 9 \mathrm {~m}\)
    • \(D A : A O = 3 : 5\)
      1. show that \(A O = 7.03 \mathrm {~m}\) to 3 significant figures.
      2. Find the perimeter of the platform, in m , to 3 significant figures.
      3. Find the total area of the platform, giving your answer in \(\mathrm { m } ^ { 2 }\) to the nearest whole number.
    Edexcel P1 2024 January Q9
    1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).
    Given that
    • \(\mathrm { f } ( x )\) is a quadratic expression
    • \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
    • \(C _ { 1 }\) passes through the origin
      1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
      2. find an expression for \(\mathrm { f } ( x )\).
    The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\)
    Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\)
    Given that the \(x\) coordinate of the point \(P\) is negative,
  • using algebra and showing all stages of your working, find the coordinates of \(P\)
    •
    Edexcel P1 2024 January Q10
    1. In this question you must show all stages of your working.
    The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\)
    Given that
    • the point \(P ( 2,8 \sqrt { 2 } )\) lies on \(C\)
    • \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }\) where \(k\) is a constant
    • \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(P\)
      1. find the exact value of \(k\),
      2. find \(\mathrm { f } ( x )\), giving your answer in simplest form.
    Edexcel P1 2019 June Q1
    1. The curve \(C\) has equation \(y = \frac { 1 } { 8 } x ^ { 3 } - \frac { 24 } { \sqrt { x } } + 1\)
      1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in its simplest form.
        (3)
      The point \(P ( 4 , - 3 )\) lies on \(C\).
    2. Find the equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.