Questions P1 (1401 questions)

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Edexcel P1 2022 January Q4
7 marks Standard +0.3
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
7 marks Standard +0.3
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
11 marks Moderate -0.3
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 \(C\)
  • Find \(\mathrm { f } ( x )\), simplifying your answer.
Edexcel P1 2022 January Q7
11 marks Standard +0.3
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}
Edexcel P1 2022 January Q8
9 marks Standard +0.3
8. The line \(l _ { 1 }\) has equation $$2 x - 5 y + 7 = 0$$
  1. Find the gradient of \(l _ { 1 }\) Given that
    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
4 marks Easy -1.3
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
9 marks Challenging +1.2
10. The curve \(C\) has equation $$y = \frac { 1 } { x ^ { 2 } } - 9$$
  1. Sketch the graph of \(C\). On your sketch
    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
5 marks Easy -1.2
  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
5 marks Moderate -0.8
  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
5 marks Moderate -0.8
  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
4 marks Moderate -0.8
  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
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. 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
10 marks Standard +0.3
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
    10 marks Moderate -0.3
    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 marks Moderate -0.3
    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 \(C\)
      The region \(R\) is shown shaded in Figure 2.
    2. Use inequalities to define \(R\).
    Edexcel P1 2023 January Q9
    4 marks Moderate -0.3
    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 \(- \pi < x < \pi\)
    Edexcel P1 2023 January Q10
    10 marks Standard +0.3
    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
    8 marks Standard +0.3
    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
    5 marks Moderate -0.3
    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
    5 marks Moderate -0.8
    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
    7 marks Moderate -0.3
    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
    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. 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
    10 marks Standard +0.3
    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 marks Easy -1.3
    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
    9 marks Standard +0.8
    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 \(k\).