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 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)
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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.
Edexcel P1 2021 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-08_625_835_264_557} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation \(y = 4 \cos x ^ { \circ }\) The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 1.
  1. State
    1. the coordinates of \(P\),
    2. the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 4 \cos x ^ { \circ } + k\), where \(k\) is a constant.
      Curve \(C _ { 2 }\) has a minimum \(y\) value of - 1
      The point \(R\) is the maximum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
  2. State the coordinates of \(R\).
Edexcel P1 2021 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-10_583_866_260_539} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The points \(P\) and \(Q\), as shown in Figure 2, have coordinates ( \(- 2,13\) ) and ( \(4 , - 5\) ) respectively. The straight line \(l\) passes through \(P\) and \(Q\).
  1. Find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found. The quadratic curve \(C\) passes through \(P\) and has a minimum point at \(Q\).
  2. Find an equation for \(C\). The region \(R\), shown shaded in Figure 2, lies in the second quadrant and is bounded by \(C\) and \(l\) only.
  3. Use inequalities to define region \(R\). \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P1 2021 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-14_470_940_246_500} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\). Given that
  • \(O A = O C = 6 \mathrm {~m}\)
  • \(A D = 14 \mathrm {~m}\)
  • angle \(A D C = 0.43\) radians
  • angle \(A O D\) is an obtuse angle
  • \(O C D\) is a straight line
    find
    1. the size of angle \(A O D\), in radians, to 3 decimal places,
    2. the length of arc \(A B C\), in metres, to one decimal place,
    3. the total area of the viewing platform, in \(\mathrm { m } ^ { 2 }\), to one decimal place.
Edexcel P1 2021 January Q6
6. (a) Sketch the curve with equation $$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$ (b) On a separate diagram, sketch the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ stating the coordinates of the point of intersection with the \(x\)-axis and, in terms of \(k\), the equation of the horizontal asymptote.
(c) Find the range of possible values of \(k\) for which the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ does not touch or intersect the line with equation \(y = 3 x + 4\) \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}
Edexcel P1 2021 January Q7
7. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = 2 x - 3 \sqrt { x } - 5 \quad x > 0$$
  1. Solve the equation $$f ( x ) = 9$$
  2. Solve the equation $$\mathrm { f } ^ { \prime \prime } ( x ) = 6$$
Edexcel P1 2021 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-26_718_1076_260_434} \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 $$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$
  1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) > 0\)
  2. Expand f(x) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found. The line \(l\), also shown in Figure 4, passes through the \(y\) intercept of \(C\) and is parallel to the \(x\)-axis. The line \(l\) cuts \(C\) again at points \(P\) and \(Q\), also shown in Figure 4 .
  3. Using algebra and showing your working, find the length of line \(P Q\). Write your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be found.
    (Solutions relying entirely on calculator technology are not acceptable.)
Edexcel P1 2021 January Q9
9. (i) Find $$\int \frac { ( 3 x + 2 ) ^ { 2 } } { 4 \sqrt { x } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.
(ii) A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given
  • \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } + a x + b\) where \(a\) and \(b\) are constants
  • the \(y\) intercept of \(C\) is - 8
  • the point \(P ( 3 , - 2 )\) lies on \(C\)
  • the gradient of \(C\) at \(P\) is 2
    find, in simplest form, \(\mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-31_2255_50_314_34}
Edexcel P1 2022 January Q1
  1. Find
$$\int \left( \frac { 8 x ^ { 3 } } { 5 } - \frac { 2 } { 3 x ^ { 4 } } - 1 \right) d x$$ giving each term in simplest form.
Edexcel P1 2022 January Q2
2. $$f ( x ) = 11 - 4 x - 2 x ^ { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.
  2. Sketch the graph of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), showing clearly the coordinates of the point where the curve crosses the \(y\)-axis.
  3. Write down the equation of the line of symmetry of \(C\).
Edexcel P1 2022 January Q3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. $$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$ Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 2 } + b x \sqrt { 2 } + c\) where \(a , b\) and \(c\) are integers to be found.
  2. Solve the equation $$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are simplified fractions to be found.
Edexcel P1 2022 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c320b71-8793-461a-a078-e4f64c144a3a-10_689_917_264_507} \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.} Figure 1 shows a line \(l\) with equation \(x + y = 6\) and a curve \(C\) with equation \(y = 6 x - 2 x ^ { 2 } + 1\) The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.
  1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 1, is bounded by \(C , l\) and the \(x\)-axis.
  2. Use inequalities to define the region \(R\).
    VIIV SIHI NI IIIIM IONOOVIIIV SIHI NI JIIIM I ON OOVIAV SIHI NI III HM ION OC
Edexcel P1 2022 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c320b71-8793-461a-a078-e4f64c144a3a-12_401_677_219_635} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a plan view of a semicircular garden \(A B C D E O A\) The semicircle has
  • centre \(O\)
  • diameter \(A O E\)
  • radius 3 m
The straight line \(B D\) is parallel to \(A E\) and angle \(B O A\) is 0.7 radians.
  1. Show that, to 4 significant figures, angle \(B O D\) is 1.742 radians. The flowerbed \(R\), shown shaded in Figure 2, is bounded by \(B D\) and the arc \(B C D\).
  2. Find the area of the flowerbed, giving your answer in square metres to one decimal place.
  3. Find the perimeter of the flowerbed, giving your answer in metres to one decimal place.
Edexcel P1 2022 January Q6
6. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\) Given that
  • \(\mathrm { f } ^ { \prime } ( x ) = \frac { ( x + 3 ) ^ { 2 } } { x \sqrt { x } }\)
  • the point \(P ( 4,20 )\) lies on \(C\)
    1. (i) find the value of the gradient at \(P\)
      (ii) Hence find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
    2. Find \(\mathrm { f } ( x )\), simplifying your answer.
Edexcel P1 2022 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c320b71-8793-461a-a078-e4f64c144a3a-20_618_841_267_555} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 4 ) ( x - 2 ) ( 2 x - 9 )$$ Given that the curve with equation \(y = \mathrm { f } ( x ) - p\) passes through the point with coordinates \(( 0,50 )\)
  1. find the value of the constant \(p\). Given that the curve with equation \(y = \mathrm { f } ( x + q )\) passes through the origin,
  2. write down the possible values of the constant \(q\).
  3. Find \(\mathrm { f } ^ { \prime } ( x )\).
  4. Hence find the range of values of \(x\) for which the gradient of the curve with equation \(y = \mathrm { f } ( x )\) is less than - 18
    \includegraphics[max width=\textwidth, alt={}, center]{6c320b71-8793-461a-a078-e4f64c144a3a-23_68_37_2617_1914}