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Edexcel P1 2024 June Q4
6 marks Standard +0.3
  1. The curve \(C _ { 1 }\) has equation
$$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)
  1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
  2. Hence find the coordinates of \(P\)
Edexcel P1 2024 June Q5
7 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-10_529_1403_255_267} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\) Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)
  1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
  2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
  3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.
Edexcel P1 2024 June Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-14_899_901_251_584} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows
  • the line \(l\) with equation \(y - 5 x = 75\)
  • the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)
The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .
  1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
  2. Use inequalities to define the region \(R\).
Edexcel P1 2024 June Q7
8 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and \(k\) is a constant.
  1. Find, in simplest form,
    1. \(\mathrm { f } ^ { \prime } ( x )\)
    2. \(\mathrm { f } ^ { \prime \prime } ( x )\) The curve with equation \(y = \mathrm { f } ^ { \prime } ( x )\) intersects the curve with equation \(y = \mathrm { f } ^ { \prime \prime } ( x )\) at the points \(A\) and \(B\). Given that the \(x\) coordinate of \(A\) is 5
  2. find the value of \(k\).
  3. Hence find the coordinates of \(B\).
Edexcel P1 2024 June Q8
7 marks Standard +0.3
  1. The curve \(C _ { 1 }\) has equation
$$y = x \left( 4 - x ^ { 2 } \right)$$
  1. Sketch the graph of \(C _ { 1 }\) showing the coordinates of any points of intersection with the coordinate axes. The curve \(C _ { 2 }\) has equation \(y = \frac { A } { x }\) where \(A\) is a constant.
  2. Show that the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
  3. Hence find the range of possible values of \(A\) for which \(C _ { 1 }\) meets \(C _ { 2 }\) at 4 distinct points.
Edexcel P1 2024 June Q9
9 marks Moderate -0.8
  1. Given that
  • the point \(A\) has coordinates \(( 4,2 )\)
  • the point \(B\) has coordinates \(( 15,7 )\)
  • the line \(l _ { 1 }\) passes through \(A\) and \(B\)
    1. find an equation for \(l _ { 1 }\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found.
The line \(l _ { 2 }\) passes through \(A\) and is parallel to the \(x\)-axis.
The point \(C\) lies on \(l _ { 2 }\) so that the length of \(B C\) is \(5 \sqrt { 5 }\)
  • Find both possible pairs of coordinates of the point \(C\).
  • Hence find the minimum possible area of triangle \(A B C\).
    •
    Edexcel P1 2024 June Q10
    10 marks Moderate -0.3
    1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)
    Given that
    • \(\mathrm { f } ^ { \prime } ( x ) = 6 x - \frac { ( 2 x - 1 ) ( 3 x + 2 ) } { 2 \sqrt { x } }\)
    • the point \(P ( 4,12 )\) lies on \(C\)
      1. find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found,
      2. find \(\mathrm { f } ( x )\), giving each term in simplest form.
    Edexcel P1 2024 June Q11
    6 marks Moderate -0.8
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-30_686_707_205_680} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 12 \sin x$$ where \(x\) is measured in radians.
    The point \(P\) shown in Figure 4 is a maximum point on \(C _ { 1 }\)
    1. Find the coordinates of \(P\). The curve \(C _ { 2 }\) has equation $$y = 12 \sin x + k$$ where \(k\) is a constant.
      Given that the maximum value of \(y\) on \(C _ { 2 }\) is 3
    2. find the coordinates of the minimum point on \(C _ { 2 }\) which has the smallest positive \(x\) coordinate. The curve \(C _ { 3 }\) has equation $$y = 12 \sin ( x + B )$$ where \(B\) is a positive constant.
      Given that \(\left( \frac { \pi } { 4 } , A \right)\), where \(A\) is a constant, is the minimum point on \(C _ { 3 }\) which has the smallest positive \(x\) coordinate,
    3. find
      1. the value of \(A\),
      2. the smallest possible value of \(B\).
    Edexcel P1 2019 October Q1
    5 marks Moderate -0.8
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-02_488_376_287_790} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is 1.25 radians. Given that the area of the sector \(A O B\) is \(15 \mathrm {~cm} ^ { 2 }\)
    1. find the exact value of \(r\),
    2. find the exact length of the perimeter of the sector. Write your answer in simplest form.
    Edexcel P1 2019 October Q2
    5 marks Easy -1.3
    2. A tree was planted in the ground. Exactly 2 years after it was planted, the height of the tree was 1.85 m . Exactly 7 years after it was planted, the height of the tree was 3.45 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted in the ground, can be modelled by the equation $$H = a t + b$$ where \(a\) and \(b\) are constants,
    1. find the value of \(a\) and the value of \(b\).
    2. State, according to the model, the height of the tree when it was planted.
    Edexcel P1 2019 October Q3
    10 marks Standard +0.3
    3. 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]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_583_588_395_680} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = x ^ { 2 } - 5 x + 13\) The point \(M\) is the minimum point of \(C\). The straight line \(l\) passes through the origin \(O\) and intersects \(C\) at the points \(M\) and \(N\) as shown. Find, showing your working,
    1. the coordinates of \(M\),
    2. the coordinates of \(N\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_531_561_1793_680} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} Figure 3 shows the curve \(C\) and the line \(l\). The finite region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
    3. Use inequalities to define the region \(R\).
    Edexcel P1 2019 October Q4
    5 marks Standard +0.3
    4. A parallelogram \(A B C D\) has area \(40 \mathrm {~cm} ^ { 2 }\) Given that \(A B\) has length \(10 \mathrm {~cm} , B C\) has length 6 cm and angle \(D A B\) is obtuse, find
    1. the size of angle \(D A B\), in degrees, to 2 decimal places,
    2. the length of diagonal \(B D\), in cm , to one decimal place.
    Edexcel P1 2019 October Q5
    7 marks Moderate -0.8
    5. A curve has equation $$y = \frac { x ^ { 3 } } { 6 } + 4 \sqrt { x } - 15 \quad x \geqslant 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in simplest form. The point \(P \left( 4 , \frac { 11 } { 3 } \right)\) lies on the curve.
    2. Find the equation of the normal to the curve at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
      VIIIV SIHI NI III M I I N OCVIIV SIHI NI IM IMM ION OCVI4V SIHI NI JIIYM IONOO
    Edexcel P1 2019 October Q6
    8 marks Standard +0.8
    6. The curve \(C\) has equation \(y = \frac { 4 } { x } + k\), where \(k\) is a positive constant.
    1. Sketch a graph of \(C\), stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the \(x\)-axis. The line with equation \(y = 10 - 2 x\) is a tangent to \(C\).
    2. Find the possible values for \(k\). \(\_\_\_\_\) -
    Edexcel P1 2019 October Q7
    6 marks Moderate -0.8
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-16_648_822_296_561} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows part of the curve with equation \(y = 2 x ^ { 2 } + 5\) The point \(P ( 2,13 )\) lies on the curve.
    1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(2 + h\) also lies on the curve.
    2. Find, in terms of \(h\), the gradient of the line \(P Q\). Give your answer in simplest form.
    3. Explain briefly the relationship between the answer to (b) and the answer to (a).
    Edexcel P1 2019 October Q8
    5 marks Standard +0.3
    8. Solve, using algebra, the equation $$x - 6 x ^ { \frac { 1 } { 2 } } + 4 = 0$$ Fully simplify your answers, writing them in the form \(a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers to be found.
    (5)
    Edexcel P1 2019 October Q9
    4 marks Moderate -0.8
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-20_671_856_303_548} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \sin \left( \frac { x } { 12 } \right)\), where \(x\) is measured in radians. The point \(M\) shown in Figure 5 is a minimum point on \(C\).
    1. State the period of \(C\).
    2. State the coordinates of \(M\). The smallest positive solution of the equation \(\sin \left( \frac { x } { 12 } \right) = k\), where \(k\) is a constant, is \(\alpha\). Find, in terms of \(\alpha\),
      1. the negative solution of the equation \(\sin \left( \frac { x } { 12 } \right) = k\) that is closest to zero,
      2. the smallest positive solution of the equation \(\cos \left( \frac { x } { 12 } \right) = k\).
    Edexcel P1 2019 October Q10
    10 marks Moderate -0.8
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-22_592_665_251_676} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$
    1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\) The curve crosses the \(y\)-axis at the point \(P\), as shown.
    2. Expand \(\mathrm { f } ( x )\) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
    3. Hence, or otherwise, find
      1. the coordinates of \(P\),
      2. the gradient of the curve at \(P\). The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
      1. Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
      2. Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).
    Edexcel P1 2019 October Q11
    10 marks Moderate -0.8
    1. A curve has equation \(y = \mathrm { f } ( x )\).
    The point \(P \left( 4 , \frac { 32 } { 3 } \right)\) lies on the curve.
    Given that
    • \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3\)
    • \(\quad \mathrm { f } ^ { \prime } ( x ) = 5\) at \(P\) find
      1. 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 constants to be found,
      2. \(\mathrm { f } ( x )\).
    Edexcel P1 2020 October Q1
    3 marks Easy -1.2
    1. Given that
    $$\left( 3 p q ^ { 2 } \right) ^ { 4 } \times 2 p \sqrt { q ^ { 8 } } \equiv a p ^ { b } q ^ { c }$$ find the values of the constants \(a , b\) and \(c\).
    Edexcel P1 2020 October Q2
    7 marks Moderate -0.3
    2. $$f ( x ) = 3 + 12 x - 2 x ^ { 2 }$$
    1. Express \(\mathrm { f } ( x )\) in the form
      2. \(\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }\)
    2. Express \(\mathrm { f } ( x )\) in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 } \\ & \text { where } a , b \text { and } c \text { are integers to be found. } \\ & \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses } \\ & \text { te } y \text {-axis at the point } R \text {. } \\ & \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and } \\ & n \text { are integers to be found. } \end{aligned}$$ \(\_\_\_\_\) "
    Edexcel P1 2020 October Q3
    10 marks Moderate -0.8
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-08_885_1388_260_287} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the design for a badge.
    The design consists of two congruent triangles, \(A O C\) and \(B O C\), joined to a sector \(A O B\) of a circle centre \(O\).
    • Angle \(A O B = \alpha\)
    • \(A O = O B = 3 \mathrm {~cm}\)
    • \(O C = 5 \mathrm {~cm}\)
    Given that the area of sector \(A O B\) is \(7.2 \mathrm {~cm} ^ { 2 }\)
    1. show that \(\alpha = 1.6\) radians.
    2. Hence find
      1. the area of the badge, giving your answer in \(\mathrm { cm } ^ { 2 }\) to 2 significant figures,
      2. the perimeter of the badge, giving your answer in cm to one decimal place.
        VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    Edexcel P1 2020 October Q4
    7 marks Moderate -0.3
    4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4 \\ x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.
    Edexcel P1 2020 October Q5
    9 marks Moderate -0.3
    5. (i) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_572_1025_212_463} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
    The curve passes through the points \(( - 5,0 )\) and \(( 0 , - 3 )\) and touches the \(x\)-axis at the point \(( 2,0 )\). On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( x + 2 )\)
    2. \(y = \mathrm { f } ( - x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
      (ii) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} Figure 3 shows a sketch of the curve with equation $$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$ where \(k\) is a constant.
      The curve meets the \(y\)-axis at the point \(( 0 , \sqrt { 3 } )\) and passes through the points \(( p , 0 )\) and ( \(q , 0\) ). Find
    3. the value of \(k\),
    4. the exact value of \(p\) and the exact value of \(q\).
    Edexcel P1 2020 October Q6
    11 marks Moderate -0.3
    6. The point \(A\) has coordinates \(( - 4,11 )\) and the point \(B\) has coordinates \(( 8,2 )\).
    1. Find the gradient of the line \(A B\), giving your answer as a fully simplified fraction. The point \(M\) is the midpoint of \(A B\). The line \(l\) passes through \(M\) and is perpendicular to \(A B\).
    2. Find an equation for \(l\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found. The point \(C\) lies on \(l\) such that the area of triangle \(A B C\) is 37.5 square units.
    3. Find the two possible pairs of coordinates of point \(C\).
      VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO