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Edexcel P1 2020 October Q7
11 marks Standard +0.3
7. The curve \(C\) has equation $$y = \frac { 1 } { 2 - x }$$
  1. Sketch the graph of \(C\). On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line \(l\) has equation \(y = 4 x + k\), where \(k\) is a constant. Given that \(l\) meets \(C\) at two distinct points,
  2. show that $$k ^ { 2 } + 16 k + 48 > 0$$
  3. Hence find the range of possible values for \(k\).
Edexcel P1 2020 October Q8
11 marks Moderate -0.3
8. The curve \(C\) has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line \(l _ { 1 }\) is the tangent to \(C\) at the point where \(x = 6\)
  2. 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 line \(l _ { 2 }\) is the tangent to \(C\) at the point where \(x = \alpha\) Given that \(l _ { 1 }\) and \(l _ { 2 }\) are parallel and distinct,
  3. find the value of \(\alpha\)
Edexcel P1 2020 October Q9
6 marks Moderate -0.3
9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 9,10 )\). Given that $$f ^ { \prime } ( x ) = 27 x ^ { 2 } - \frac { 21 x ^ { 3 } - 5 x } { 2 \sqrt { x } } \quad x > 0$$ find \(\mathrm { f } ( x )\), fully simplifying each term.
Edexcel P1 2021 October Q1
5 marks Easy -1.2
  1. Find
$$\int 12 x ^ { 3 } + \frac { 1 } { 6 \sqrt { x } } - \frac { 3 } { 2 x ^ { 4 } } \mathrm {~d} x$$ giving each term in simplest form.
Edexcel P1 2021 October Q2
5 marks Moderate -0.8
2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).
Edexcel P1 2021 October Q3
6 marks Moderate -0.3
3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only. Given that
  • \(\quad l\) has a gradient of 3
  • \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
  • \(\quad C\) and \(l\) intersect on the negative \(x\)-axis
    use inequalities to define the region \(R\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-06_643_652_575_648} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only.
Given that
  • \(l\) has a gradient of 3
  • \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
  • \(C\) and \(l\) intersect on the negative \(x\)-axis
    use inequalities to define the region \(R\).
Edexcel P1 2021 October Q4
5 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-08_721_855_214_550} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \cos 2 x ^ { \circ } \quad 0 \leqslant x \leqslant k$$ The point \(Q\) and the point \(R ( k , 0 )\) lie on the curve and are shown in Figure 2.
  1. State
    1. the coordinates of \(Q\),
    2. the value of \(k\).
  2. Given that there are exactly two solutions to the equation $$\cos 2 x ^ { \circ } = p \quad \text { in the region } 0 \leqslant x \leqslant k$$ find the range of possible values for \(p\).
Edexcel P1 2021 October Q5
8 marks Moderate -0.8
5. The line \(l _ { 1 }\) has equation \(3 y - 2 x = 30\) The line \(l _ { 2 }\) passes through the point \(A ( 24,0 )\) and is perpendicular to \(l _ { 1 }\) Lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)
  1. Find, using algebra and showing your working, the coordinates of \(P\). Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(B\),
  2. find the area of triangle \(B P A\).
Edexcel P1 2021 October Q6
10 marks Moderate -0.8
6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$
  1. Sketch a graph of \(C\). Show on your graph the coordinates of the points where \(C\) cuts or meets the coordinate axes.
  2. Write \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\), where \(a , b , c\) and \(d\) are constants to be found.
  3. Hence, find the equation of the tangent to \(C\) at the point where \(x = \frac { 1 } { 3 }\)
Edexcel P1 2021 October Q7
10 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-18_428_894_210_525} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the design for a sign at a bird sanctuary.
The design consists of a kite \(O A B C\) joined to a sector \(O C X A\) of a circle centre \(O\).
In the design
  • \(O A = O C = 0.6 \mathrm {~m}\)
  • \(A B = C B = 1.4 \mathrm {~m}\)
  • Angle \(O A B =\) Angle \(O C B = 2\) radians
  • Angle \(A O C = \theta\) radians, as shown in Figure 3
Making your method clear,
  1. show that \(\theta = 1.64\) radians to 3 significant figures,
  2. find the perimeter of the sign, in metres to 2 significant figures,
  3. find the area of the sign, in \(\mathrm { m } ^ { 2 }\) to 2 significant figures.
Edexcel P1 2021 October Q8
10 marks Easy -1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-22_657_659_214_646} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation $$y = 4 + 12 x - 3 x ^ { 2 }$$ The point \(M\) is the maximum turning point on \(C\).
    1. Write \(4 + 12 x - 3 x ^ { 2 }\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are constants to be found.
    2. Hence, or otherwise, state the coordinates of \(M\). The line \(l _ { 1 }\) passes through \(O\) and \(M\), as shown in Figure 4.
      A line \(l _ { 2 }\) touches \(C\) and is parallel to \(l _ { 1 }\)
  2. Find an equation for \(l _ { 2 }\)
Edexcel P1 2021 October Q9
9 marks Moderate -0.3
9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-26_595_716_420_662} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \sqrt { x } \quad x > 0$$ The point \(P ( 9,3 )\) lies on the curve and is shown in Figure 5.
On the next page there is a copy of Figure 5 called Diagram 1.
  1. On Diagram 1, sketch and clearly label the graphs of $$y = \mathrm { f } ( 2 x ) \text { and } y = \mathrm { f } ( x ) + 3$$ Show on each graph the coordinates of the point to which \(P\) is transformed. The graph of \(y = \mathrm { f } ( 2 x )\) meets the graph of \(y = \mathrm { f } ( x ) + 3\) at the point \(Q\).
  2. Show that the \(x\) coordinate of \(Q\) is the solution of $$\sqrt { x } = 3 ( \sqrt { 2 } + 1 )$$
  3. Hence find, in simplest form, the coordinates of \(Q\).
    \includegraphics[max width=\textwidth, alt={}]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-27_599_720_274_660}
    \section*{Diagram 1} Turn over for a copy of Diagram 1 if you need to redraw your graphs. Only use this copy if you need to redraw your graphs. \includegraphics[max width=\textwidth, alt={}, center]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-29_600_718_1991_660} Copy of Diagram 1
Edexcel P1 2021 October Q10
7 marks Standard +0.3
10. A curve has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that
  • \(\mathrm { f } ^ { \prime } ( x ) = a x - 12 x ^ { \frac { 1 } { 3 } }\), where \(a\) is a constant
  • \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 27\)
  • the curve passes through the point \(( 1 , - 8 )\)
    1. find the value of \(a\).
    2. Hence find \(\mathrm { f } ( x )\).
Edexcel P1 2022 October Q1
6 marks Moderate -0.8
  1. The curve \(C\) has equation
$$y = \frac { x ^ { 3 } } { 4 } - x ^ { 2 } + \frac { 17 } { x } \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(R \left( 2 , \frac { 13 } { 2 } \right)\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at the point \(R\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
Edexcel P1 2022 October Q2
7 marks Moderate -0.8
  1. Given that
$$( x - 5 ) ( 2 x + 1 ) ( x + 3 ) \equiv a x ^ { 3 } + b x ^ { 2 } - 32 x - 15$$ where \(a\) and \(b\) are constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Hence find $$\int \frac { ( x - 5 ) ( 2 x + 1 ) ( x + 3 ) } { 5 \sqrt { x } } \mathrm {~d} x$$ writing each term in simplest form.
Edexcel P1 2022 October Q3
5 marks Easy -1.3
  1. The share price of a company is monitored.
Exactly 3 years after monitoring began, the share price was \(\pounds 1.05\) Exactly 5 years after monitoring began, the share price was \(\pounds 1.65\) The share price, \(\pounds V\), of the company is modelled by the equation $$V = p t + q$$ where \(t\) is the number of years after monitoring began and \(p\) and \(q\) are constants.
  1. Find the value of \(p\) and the value of \(q\). Exactly \(T\) years after monitoring began, the share price was \(\pounds 2.50\)
  2. Find the value of \(T\), according to the model, giving your answer to one decimal place.
Edexcel P1 2022 October Q4
6 marks Standard +0.3
  1. In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.
$$f ( x ) = x ^ { 2 } ( 2 x + 1 ) - 15 x$$
  1. Solve $$\mathrm { f } ( x ) = 0$$
  2. Hence solve $$y ^ { \frac { 4 } { 3 } } \left( 2 y ^ { \frac { 2 } { 3 } } + 1 \right) - 15 y ^ { \frac { 2 } { 3 } } = 0 \quad y > 0$$ giving your answer in simplified surd form.
Edexcel P1 2022 October Q5
9 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.
The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that
  • \(\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 12 } { \sqrt { \mathrm { x } } } + \frac { x } { 3 } - 4\)
  • the point \(P ( 9,8 )\) lies on \(C\)
    1. find, in simplest form, \(\mathrm { f } ( x )\)
The line \(l\) is the normal to \(C\) at \(P\)
  • Find the coordinates of the point at which \(l\) crosses the \(y\)-axis.
    •
    Edexcel P1 2022 October Q6
    6 marks Moderate -0.3
    1. (a) Given that \(k\) is a positive constant such that \(0 < k < 4\) sketch, on separate axes, the graphs of
      1. \(y = ( 2 x - k ) ( x + 4 ) ^ { 2 }\)
      2. \(y = \frac { k } { x ^ { 2 } }\) showing the coordinates of any points where the graphs cross or meet the coordinate axes, leaving coordinates in terms of \(k\), where appropriate.
        (b) State, with a reason, the number of roots of the equation
      $$( 2 x - k ) ( x + 4 ) ^ { 2 } = \frac { k } { x ^ { 2 } }$$
    Edexcel P1 2022 October Q7
    8 marks Moderate -0.8
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-16_732_1071_248_497} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
    The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.
    1. Using Figure 1, find the range of values of \(x\) for which $$\mathrm { f } ( x ) < 6$$
    2. State the largest solution of the equation $$f ( 2 x ) = 6$$
      1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
      2. Hence find the set of values of \(x\) for which $$f ( - x ) \geqslant 6 \text { and } x < 0$$
    Edexcel P1 2022 October Q8
    14 marks Moderate -0.3
    8. \section*{Diagram NOT to scale} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-20_461_1036_296_534} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the plan view of a design for a pond.
    The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).
    • \(B C = 8 \mathrm {~m}\)
    • \(O A = O B = 3 \mathrm {~m}\)
    • angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
    • angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
      1. Calculate (i) the exact area of the sector \(A O B X\),
        (ii) the exact perimeter of the sector \(A O B X\).
      2. Calculate the exact area of the triangle \(A O B\).
      3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
      4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.
    Edexcel P1 2022 October Q9
    14 marks Standard +0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-24_889_666_258_703} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$
    1. Write \(\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a , b\) and \(c\) are constants to be found. The point \(M\) is the minimum turning point of \(C\), as shown in Figure 3.
    2. Deduce the coordinates of \(M\) The line \(l\) is the normal to \(C\) at the point \(P\), as shown in Figure 3.
      Given that \(l\) has equation \(y = k - \frac { 1 } { 8 } x\), where \(k\) is a constant,
      1. find the coordinates of \(P\)
      2. find the value of \(k\) Question 9 continues on the next page \begin{figure}[h]
        \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605} \captionsetup{labelformat=empty} \caption{Figure 4}
        \end{figure} Figure 4 is a copy of Figure 3. The finite region \(R\), shown shaded in Figure 4, is bounded by \(l , C\) and the line through \(M\) parallel to the \(y\)-axis.
    4. Identify the inequalities that define \(R\).
    Edexcel P1 2023 October Q1
    5 marks Easy -1.3
    1. Given that
    $$y = 5 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } - 7 x \quad x > 0$$ find, in simplest form,
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
    Edexcel P1 2023 October Q2
    4 marks Easy -1.2
    1. Given that
    $$a = \frac { 1 } { 64 } x ^ { 2 } \quad b = \frac { 16 } { \sqrt { x } }$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are simplified constants.
    1. \(a ^ { \frac { 1 } { 2 } }\)
    2. \(\frac { 16 } { b ^ { 3 } }\)
    3. \(\left( \frac { a b } { 2 } \right) ^ { - \frac { 4 } { 3 } }\)
    Edexcel P1 2023 October Q3
    6 marks Moderate -0.3
    1. In this question you must show all stages of your working.
    Solutions relying on calculator technology are not acceptable.
    1. Write \(\frac { 8 - \sqrt { 15 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(a \sqrt { 3 } + b \sqrt { 5 }\) where \(a\) and \(b\) are integers to be found.
    2. Hence, or otherwise, solve $$( x + 5 \sqrt { 3 } ) \sqrt { 5 } = 40 - 2 x \sqrt { 3 }$$ giving your answer in simplest form.