Questions — Edexcel P1 (172 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 2019 January Q1
  1. Find
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { 2 x ^ { 3 } } + 5 \right) d x$$ simplifying your answer.
Edexcel P1 2019 January Q2
  1. Given
$$\frac { 3 ^ { x } } { 3 ^ { 4 y } } = 27 \sqrt { 3 }$$ find \(y\) as a simplified function of \(x\).
Edexcel P1 2019 January Q3
  1. The line \(l _ { 1 }\) has equation \(3 x + 5 y - 7 = 0\)
    1. Find the gradient of \(l _ { 1 }\)
    The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 6 , - 2 )\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel P1 2019 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-08_857_857_251_548} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a line \(l _ { 1 }\) with equation \(2 y = x\) and a curve \(C\) with equation \(y = 2 x - \frac { 1 } { 8 } x ^ { 2 }\) The region \(R\), shown unshaded in Figure 1, is bounded by the line \(l _ { 1 }\), the curve \(C\) and a line \(l _ { 2 }\) Given that \(l _ { 2 }\) is parallel to the \(y\)-axis and passes through the intercept of \(C\) with the positive \(x\)-axis, identify the inequalities that define \(R\).
Edexcel P1 2019 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-10_677_1036_260_456} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a plot of part of the curve with equation \(y = \cos 2 x\) with \(x\) being measured in radians. The point \(P\), shown on Figure 2, is a minimum point on the curve.
  1. State the coordinates of \(P\). A copy of Figure 2, called Diagram 1, is shown at the top of the next page.
  2. Sketch, on Diagram 1, the curve with equation \(y = \sin x\)
  3. Hence, or otherwise, deduce the number of solutions of the equation
    1. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 20 \pi\)
    2. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 21 \pi\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-11_693_1050_301_447} \captionsetup{labelformat=empty} \caption{
      Diagram 1}\}
      \end{figure} \textbackslash section*\{Diagram 1
Edexcel P1 2019 January Q6
  1. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Given $$\mathrm { f } ( x ) = 2 x ^ { \frac { 5 } { 2 } } - 40 x + 8 \quad x > 0$$
  1. solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\)
  2. solve the equation \(\mathrm { f } ^ { \prime \prime } ( x ) = 5\)
Edexcel P1 2019 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-14_327_595_251_676} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Not to scale Figure 3 shows the design for a structure used to support a roof. The structure consists of four wooden beams, \(A B , B D , B C\) and \(A D\). Given \(A B = 6.5 \mathrm {~m} , B C = B D = 4.7 \mathrm {~m}\) and angle \(B A C = 35 ^ { \circ }\)
  1. find, to one decimal place, the size of angle \(A C B\),
  2. find, to the nearest metre, the total length of wood required to make this structure.
Edexcel P1 2019 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-16_647_970_306_488} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is shown in Figure 4. The curve \(C\)
  • has a single turning point, a maximum at ( 4,9 )
  • crosses the coordinate axes at only two places, \(( - 3,0 )\) and \(( 0,6 )\)
  • has a single asymptote with equation \(y = 4\)
    as shown in Figure 4.
    1. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( - x )\).
    2. State the coordinates of the turning point on the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)\).
Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,
  • state the possible values for \(k\). The curve \(C\) is transformed to a new curve that passes through the origin.
    1. Given that the new curve has equation \(y = \mathrm { f } ( x ) - a\), state the value of the constant \(a\).
    2. Write down an equation for another single transformation of \(C\) that also passes through the origin.
    •
    Edexcel P1 2019 January Q9
    1. The equation
    $$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
    Find the range of possible values of \(c\).
    Edexcel P1 2019 January Q10
    1. A sector \(A O B\), of a circle centre \(O\), has radius \(r \mathrm {~cm}\) and angle \(\theta\) radians.
    Given that the area of the sector is \(6 \mathrm {~cm} ^ { 2 }\) and that the perimeter of the sector is 10 cm ,
    1. show that $$3 \theta ^ { 2 } - 13 \theta + 12 = 0$$
    2. Hence find possible values of \(r\) and \(\theta\).
      □
      \includegraphics[max width=\textwidth, alt={}, center]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-21_131_19_2627_1882}
    Edexcel P1 2019 January Q11
    11. (a) On Diagram 1 sketch the graphs of
    1. \(y = x ( 3 - x )\)
    2. \(y = x ( x - 2 ) ( 5 - x )\)
      showing clearly the coordinates of the points where the curves cross the coordinate axes.
      (b) Show that the \(x\) coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 13 \right) = 0\) The point \(P\) lies on both curves. Given that \(P\) lies in the first quadrant,
      (c) find, using algebra and showing your working, the exact coordinates of \(P\).
      \includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
      \section*{Diagram 1}
    Edexcel P1 2019 January Q12
    12. The curve with equation \(y = \mathrm { f } ( x ) , x > 0\), passes through the point \(P ( 4 , - 2 )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x \sqrt { x } - 10 x ^ { - \frac { 1 } { 2 } }$$
    1. find the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
    2. Find \(\mathrm { f } ( x )\).
    Edexcel P1 2020 January Q1
    1. Find, in simplest form,
    $$\int \left( \frac { 8 x ^ { 3 } } { 3 } - \frac { 1 } { 2 \sqrt { x } } - 5 \right) \mathrm { d } x$$
    Edexcel P1 2020 January Q2
    2. Given \(y = 3 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.
    1. \(3 ^ { 3 x }\)
    2. \(\frac { 1 } { 3 ^ { x - 2 } }\)
    3. \(\frac { 81 } { 9 ^ { 2 - 3 x } }\)
    Edexcel P1 2020 January Q3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{28839dd5-b9c1-4cbd-981e-8f79c43ba086-06_652_654_269_646} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the curve with equation \(y = x ^ { 2 } + 3 x - 2\) The point \(P ( 3,16 )\) lies on the curve.
    1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(3 + h\) also lies on the curve.
    2. Find, in terms of \(h\), the gradient of the line \(P Q\). Write your answer in simplest form.
    3. Explain briefly the relationship between the answer to (b) and the answer to (a).
    Edexcel P1 2020 January Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{28839dd5-b9c1-4cbd-981e-8f79c43ba086-08_622_894_258_683} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the plan view of a house \(A B C D\) and a lawn \(A P C D A\).
    \(A B C D\) is a rectangle with \(A B = 16 \mathrm {~m}\).
    \(A P C O A\) is a sector of a circle centre \(O\) with radius 12 m . The point \(O\) lies on the line \(D C\), as shown in Figure 2.
    1. Show that the size of angle \(A O D\) is 1.231 radians to 3 decimal places. The lawn \(A P C D A\) is shown shaded in Figure 2.
    2. Find the area of the lawn, in \(\mathrm { m } ^ { 2 }\), to one decimal place.
    3. Find the perimeter of the lawn, in metres, to one decimal place.
    Edexcel P1 2020 January Q5
    5. (a) Find, using algebra, all solutions of $$20 x ^ { 3 } - 50 x ^ { 2 } - 30 x = 0$$ (b) Hence find all real solutions of $$20 ( y + 3 ) ^ { \frac { 3 } { 2 } } - 50 ( y + 3 ) - 30 ( y + 3 ) ^ { \frac { 1 } { 2 } } = 0$$
    Edexcel P1 2020 January Q6
    6. The line \(l _ { 1 }\) has equation \(3 x - 4 y + 20 = 0\) The line \(l _ { 2 }\) cuts the \(x\)-axis at \(R ( 8,0 )\) and is parallel to \(l _ { 1 }\)
    1. Find the equation of \(l _ { 2 }\), writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. The line \(l _ { 1 }\) cuts the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\).
      Given that \(P Q R S\) is a parallelogram, find
    2. the area of \(P Q R S\),
    3. the coordinates of \(S\).
    Edexcel P1 2020 January Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{28839dd5-b9c1-4cbd-981e-8f79c43ba086-18_599_723_274_614} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows part of the curve \(C _ { 1 }\) with equation \(y = 3 \sin x\), where \(x\) is measured in degrees. The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 3.
    1. State
      1. the coordinates of \(P\),
      2. the coordinates of \(Q\). A different curve \(C _ { 2 }\) has equation \(y = 3 \sin x + k\), where \(k\) is a constant.
        The curve \(C _ { 2 }\) has a maximum \(y\) value of 10
        The point \(R\) is the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
    2. State the coordinates of \(R\). Figure 3
    Edexcel P1 2020 January Q8
    8. The straight line \(l\) has equation \(y = k ( 2 x - 1 )\), where \(k\) is a constant. The curve \(C\) has equation \(y = x ^ { 2 } + 2 x + 11\)
    Find the set of values of \(k\) for which \(l\) does not cross or touch \(C\).
    (6)
    Edexcel P1 2020 January Q9
    9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A curve has equation $$y = \frac { 4 x ^ { 2 } + 9 } { 2 \sqrt { x } } \quad x > 0$$ Find the \(x\) coordinate of the point on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
    Edexcel P1 2020 January Q10
    10. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 4 x - 3 ) ( x - 5 ) ^ { 2 }$$
    1. Sketch \(C _ { 1 }\) showing the coordinates of any point where the curve touches or crosses the coordinate axes.
    2. Hence or otherwise
      1. find the values of \(x\) for which \(\mathrm { f } \left( \frac { 1 } { 4 } x \right) = 0\)
      2. find the value of the constant \(p\) such that the curve with equation \(y = \mathrm { f } ( x ) + p\) passes through the origin. A second curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \mathrm { f } ( x + 1 )\)
      1. Find, in simplest form, \(\mathrm { g } ( x )\). You may leave your answer in a factorised form.
      2. Hence, or otherwise, find the \(y\) intercept of curve \(C _ { 2 }\)
    Edexcel P1 2020 January Q11
    11. A curve has equation \(y = \mathrm { f } ( x )\), where $$f ^ { \prime \prime } ( x ) = \frac { 6 } { \sqrt { x ^ { 3 } } } + x \quad x > 0$$ The point \(P ( 4 , - 50 )\) lies on the curve.
    Given that \(\mathrm { f } ^ { \prime } ( x ) = - 4\) at \(P\),
    1. find the equation of the normal at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
      (3)
    2. find \(\mathrm { f } ( x )\).
      (8)
      VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM ION OO
    Edexcel P1 2021 January Q1
    1. A curve has equation
    $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - \frac { 3 } { 2 x } + 7 \quad x > 0$$
    1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) lies on the curve and has \(x\) coordinate \(\frac { 1 } { 2 }\)
    2. Find an equation of the normal to the curve at \(P\), writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
      VILU SIHI NI JIIIM ION OCVIUV SIHI NI III M M I ON OOVIAV SIHI NI JIIIM I ION OC
    Edexcel P1 2021 January Q2
    1. A tree was planted.
    Exactly 3 years after it was planted, the height of the tree was 2 m . Exactly 5 years after it was planted, the height of the tree was 2.4 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted, can be modelled by the equation $$H ^ { 3 } = p t ^ { 2 } + q$$ where \(p\) and \(q\) are constants,
    1. find, to 3 significant figures where necessary, the value of \(p\) and the value of \(q\). Exactly \(T\) years after the tree was planted, its height was 5 m .
    2. Find the value of \(T\) according to the model, giving your answer to one decimal place.