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Edexcel P1 2023 October Q9
7 marks Standard +0.3
9. Diagram NOT accurately drawn \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-24_581_1491_340_296} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the plan view of the area being used for a ball-throwing competition.
Competitors must stand within the circle \(C\) and throw a ball as far as possible into the target area, \(P Q R S\), shown shaded in Figure 3. Given that
  • circle \(C\) has centre \(O\)
  • \(P\) and \(S\) are points on \(C\)
  • \(O P Q R S O\) is a sector of a circle with centre \(O\)
  • the length of arc \(P S\) is 0.72 m
  • the size of angle \(P O S\) is 0.6 radians
    1. show that \(O P = 1.2 \mathrm {~m}\)
Given also that
$$5 x ^ { 2 } + 12 x - 1500 = 0$$
  • Hence calculate the total perimeter of the target area, \(P Q R S\), giving your answer to the nearest metre.
    •
    Edexcel P1 2023 October Q10
    6 marks Moderate -0.8
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-28_538_652_255_708} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 3 \cos \left( \frac { x } { n } \right) ^ { \circ } \quad x \geqslant 0$$ where \(n\) is a constant.
    The curve \(C _ { 1 }\) cuts the positive \(x\)-axis for the first time at point \(P ( 270,0 )\), as shown in Figure 4.
      1. State the value of \(n\)
      2. State the period of \(C _ { 1 }\) The point \(Q\), shown in Figure 4, is a minimum point of \(C _ { 1 }\)
    2. State the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 2 \sin x ^ { \circ } + k\), where \(k\) is a constant.
      The point \(R \left( a , \frac { 12 } { 5 } \right)\) and the point \(S \left( - a , - \frac { 3 } { 5 } \right)\), both lie on \(C _ { 2 }\) Given that \(a\) is a constant less than 90
    3. find the value of \(k\).
    Edexcel P1 2023 October Q11
    10 marks Easy -1.2
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-30_595_869_255_568} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$
    1. Write \(2 x ^ { 2 } - 12 x + 14\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. Given that \(C\) has a minimum at the point \(P\)
    2. state the coordinates of \(P\) The line \(l\) intersects \(C\) at \(( - 1,28 )\) and at \(P\) as shown in Figure 5.
    3. Find the equation of \(l\) giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. The finite region \(R\), shown shaded in Figure 5, is bounded by the \(x\)-axis, \(l\), the \(y\)-axis, and \(C\).
    4. Use inequalities to define the region \(R\).
    CAIE P1 2021 March Q10
    8 marks Challenging +1.2
    1. For the case where angle \(B A C = \frac { 1 } { 6 } \pi\) radians, find \(k\) correct to 4 significant figures.
    2. For the general case in which angle \(B A C = \theta\) radians, where \(0 < \theta < \frac { 1 } { 2 } \pi\), it is given that \(\frac { \theta } { \sin \theta } > 1\). Find the set of possible values of \(k\).
    CAIE P1 2022 March Q6
    8 marks Standard +0.3
    1. Find, by calculation, the coordinates of \(A\) and \(B\).
    2. Find an equation of the circle which has its centre at \(C\) and for which the line with equation \(y = 3 x - 20\) is a tangent to the circle.
    CAIE P1 2022 March Q10
    8 marks Standard +0.3
    1. Find the perimeter of the shaded region.
    2. Find the area of the shaded region.
    CAIE P1 2024 March Q10
    12 marks Standard +0.3
    1. Find the equation of the tangent to the circle at the point \(( - 6,9 )\).
    2. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
    3. Find the value of \(\theta\) correct to 4 significant figures.
    4. Find the perimeter and area of the segment shaded in the diagram.
    CAIE P1 2020 November Q10
    10 marks Moderate -0.3
    [diagram]
    The diagram shows a sector \(CAB\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D\), \(E\), and \(F\), where \(COD\) is a straight line and angle \(ACD\) is \(\theta\) radians.
    1. Find \(C D\) in terms of \(r\) and \(\sin \theta\).
      It is now given that \(r = 4\) and \(\theta = \frac { 1 } { 6 } \pi\).
    2. Find the perimeter of sector \(C A B\) in terms of \(\pi\).
    3. Find the area of the shaded region in terms of \(\pi\) and \(\sqrt { 3 }\).
    CAIE P1 2021 November Q6
    7 marks Standard +0.3
    1. Find the perimeter of the plate, giving your answer in terms of \(\pi\).
    2. Find the area of the plate, giving your answer in terms of \(\pi\) and \(\sqrt { 3 }\).
    CAIE P1 2022 November Q10
    8 marks Standard +0.3
    1. Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
    2. Find the difference in area of the two triangles \(A O B\) and \(A P B\), giving your answer correct to 2 decimal places.
    3. Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.
    CAIE P1 2022 November Q10
    10 marks Standard +0.3
    1. Find the coordinates of \(A\).
    2. Find the volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in the form \(\frac { \pi } { a } ( b \sqrt { c } - d )\), where \(a , b , c\) and \(d\) are integers.
    3. Find an exact expression for the perimeter of the shaded region.
    CAIE P1 2017 June Q4
    7 marks Moderate -0.8
    1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
    2. In the case where \(r = 5\) and \(\theta = \frac { 1 } { 6 } \pi\), find the area of the shaded region.
    CAIE P1 2018 June Q5
    6 marks Moderate -0.5
    1. Express each of the vectors \(\overrightarrow { D A }\) and \(\overrightarrow { C A }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
    2. Use a scalar product to find angle \(C A D\).
    CAIE P1 2017 March Q4
    6 marks Standard +0.3
    1. Show that angle \(C B D = \frac { 9 } { 14 } \pi\) radians.
    2. Find the perimeter of the shaded region.
    CAIE P1 2005 November Q5
    7 marks Standard +0.3
    1. Express \(h\) in terms of \(r\) and hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 12 \pi r ^ { 2 } - 2 \pi r ^ { 3 }$$
    2. Given that \(r\) varies, find the stationary value of \(V\).
    CAIE P1 2015 November Q10
    9 marks Standard +0.3
    1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
    2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).
    CAIE P1 2016 November Q4
    6 marks Easy -1.2
    1. Find the equation of the line \(C D\), giving your answer in the form \(y = m x + c\).
    2. Find the distance \(A D\).
    CAIE P1 Specimen Q10
    9 marks Standard +0.3
    1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
    2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).
    CAIE P1 2023 June Q1
    3 marks Moderate -0.8
    The equation of a curve is such that \(\frac{dy}{dx} = \frac{4}{(x-3)^2}\) for \(x > 3\). The curve passes through the point \((4, 5)\). Find the equation of the curve. [3]
    CAIE P1 2023 June Q2
    4 marks Standard +0.3
    The coefficient of \(x^4\) in the expansion of \((x + a)^6\) is \(p\) and the coefficient of \(x^2\) in the expansion of \((ax + 3)^4\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\). [4]
    CAIE P1 2023 June Q3
    3 marks Moderate -0.8
    1. Express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\). [2]
    2. Hence or otherwise find the set of values of \(p\) for which the equation \(4x^2 - 24x + p = 0\) has no real roots. [1]
    CAIE P1 2023 June Q4
    3 marks Standard +0.3
    Solve the equation \(8x^6 + 215x^3 - 27 = 0\). [3]
    CAIE P1 2023 June Q5
    4 marks Moderate -0.3
    \includegraphics{figure_5} The diagram shows the curve with equation \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) for \(x > 0\). The curve meets the \(x\)-axis at the points \((0, 0)\) and \((4, 0)\). Find the area of the shaded region. [4]
    CAIE P1 2023 June Q6
    6 marks Standard +0.3
    \includegraphics{figure_6} The diagram shows a sector \(OAB\) of a circle with centre \(O\). Angle \(AOB = \theta\) radians and \(OP = AP = x\).
    1. Show that the arc length \(AB\) is \(2x\theta \cos \theta\). [2]
    2. Find the area of the shaded region \(APB\) in terms of \(x\) and \(\theta\). [4]
    CAIE P1 2023 June Q7
    11 marks Standard +0.3
      1. By first expanding \((\cos \theta + \sin \theta)^2\), find the three solutions of the equation $$(\cos \theta + \sin \theta)^2 = 1$$ for \(0 \leqslant \theta \leqslant \pi\). [3]
      2. Hence verify that the only solutions of the equation \(\cos \theta + \sin \theta = 1\) for \(0 < \theta < \pi\) are \(0\) and \(\frac{1}{2}\pi\). [2]
    2. Prove the identity \(\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} \equiv \frac{\cos \theta + \sin \theta - 1}{1 - 2\sin^2 \theta}\). [3]
    3. Using the results of (a)(ii) and (b), solve the equation $$\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} = 2(\cos \theta + \sin \theta - 1)$$ for \(0 \leqslant \theta \leqslant \pi\). [3]