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CAIE FP1 2018 November Q5
Standard +0.3
5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.
  1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
    The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) as eigenvectors.
  2. Find the corresponding eigenvalues.
    The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) respectively.
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).
CAIE FP1 2018 November Q6
Standard +0.8
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) intersects the \(x\)-axis twice.
  3. Justifying your answer, find the number of stationary points on \(C\).
  4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.
CAIE FP1 2018 November Q7
Challenging +1.2
7
  1. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
  2. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.
CAIE FP1 2018 November Q8
5 marks Standard +0.3
8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5 \\ 1 \\ 0 \end{array} \right) + s \left( \begin{array} { r } - 4 \\ 1 \\ 3 \end{array} \right) + t \left( \begin{array} { l } 0 \\ 1 \\ 2 \end{array} \right)$$
  1. Find a cartesian equation of \(\Pi _ { 1 }\).
    The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
  3. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]
CAIE FP1 2018 November Q9
Challenging +1.2
9 The curve \(C\) has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to 1 decimal place.
    Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
  2. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
  3. Find the maximum distance of \(C\) from the initial line.
  4. Sketch \(C\).
CAIE FP1 2018 November Q10
Challenging +1.3
10
  1. Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$ given that \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\).
  2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = 6\).
CAIE FP1 2018 November Q11 EITHER
Standard +0.8
  1. By considering \(( 2 r + 1 ) ^ { 2 } - ( 2 r - 1 ) ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
  2. By considering \(( 2 r + 1 ) ^ { 4 } - ( 2 r - 1 ) ^ { 4 }\), use the method of differences and the result given in part (i) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ The sums \(S\) and \(T\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \ldots + ( 2 N ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } , \\ & T = 1 ^ { 3 } + 3 ^ { 3 } + 5 ^ { 3 } + 7 ^ { 3 } + \ldots + ( 2 N - 1 ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } . \end{aligned}$$
  3. Use the result given in part (ii) to show that \(S = ( 2 N + 1 ) ^ { 2 } ( N + 1 ) ^ { 2 }\).
  4. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.
  5. Deduce the value of \(\frac { S } { T }\) as \(N \rightarrow \infty\).
CAIE FP1 2018 November Q11 OR
Challenging +1.8
The curve \(C\) has equation $$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$
  1. Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
    For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\).
  2. Show that \(A _ { 1 } = 0\).
  3. Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
    Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\).
  4. Show that for \(n \geqslant 2\), $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
  5. Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
OCR MEI FP1 Q9
Standard +0.3
9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
  2. Explain how your answer to part (i) relates to this information.
  3. By investigating the invariant points of the reflection, find the equation of the mirror line.
  4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
  5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
  6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS
    4755
    \textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}
    Tuesday 7 JUNE 2005Afternoon1 hour 30 minutes
    Additional materials:
    Answer booklet
    Graph paper
    MEI Examination Formulae and Tables (MF2)
    TIME 1 hour 30 minutes
OCR MEI FP1 Q10
Standard +0.3
10
  1. You are given that $$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$ Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
  2. Hence find the sum of the infinite series $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$ RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS} Further Concepts For Advanced Mathematics (FP1)
    Wednesday 18 JANUARY 2006 Afternoon ..... 1 hour 30 minutes
    Additional materials:
    8 page answer booklet
    Graph paper
    MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes
Edexcel FP1 2023 June Q1
Moderate -0.8
  1. Use Simpson's rule with 4 intervals to find an estimate for $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sin ^ { 2 } x } \mathrm {~d} x$$ Give your answer to 3 significant figures. Given that \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \mathrm { sin } ^ { 2 } x } \mathrm {~d} x = 3.855\) to 4 significant figures,
  2. comment on the accuracy of your answer to part (a).
Edexcel FP1 2023 June Q2
Standard +0.8
  1. The vertical height, \(h \mathrm {~m}\), above horizontal ground, of a passenger on a fairground ride, \(t\) seconds after the ride starts, where \(t \leqslant 5\), is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} h } { \mathrm {~d} t } + 2 h = t ^ { 3 }$$
  1. Given that \(t = \mathrm { e } ^ { x }\), show that
    1. \(t \frac { \mathrm {~d} h } { \mathrm {~d} t } = \frac { \mathrm { d } h } { \mathrm {~d} x }\)
    2. \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } h } { \mathrm {~d} x }\)
  2. Hence show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} h } { \mathrm {~d} x } + 2 h = \mathrm { e } ^ { 3 x }$$
  3. Hence show that $$h = A t + B t ^ { 2 } + \frac { 1 } { 2 } t ^ { 3 }$$ where \(A\) and \(B\) are constants. Given that when \(t = 1 , h = 2.5\) and when \(t = 2 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = - 1\)
  4. determine the height of the passenger above the ground 5 seconds after the start of the ride.
Edexcel FP1 2023 June Q3
Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0ac1e1e-16bf-4a06-9eaa-8dcf01177722-08_748_814_392_621} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | }\) and the line with equation \(y = 5 - 4 x\) Use algebra to determine the values of \(x\) for which $$\frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | } < 5 - 4 x$$
Edexcel FP1 2023 June Q4
Challenging +1.2
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$
  1. Determine the exact value of the eccentricity of \(E\) The points \(P ( 4 \cos \theta , 3 \sin \theta )\) and \(Q ( 4 \cos \theta , - 3 \sin \theta )\) lie on \(E\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l _ { 1 }\) is the normal to \(E\) at the point \(P\)
  2. Use calculus to show that \(l _ { 1 }\) has equation $$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$ The line \(l _ { 2 }\) passes through the origin and the point \(Q\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\)
  3. Determine, in simplest form, the coordinates of \(R\)
  4. Hence show that, as \(\theta\) varies, \(R\) lies on an ellipse which has the same eccentricity as ellipse \(E\)
Edexcel FP1 2023 June Q5
Challenging +1.2
  1. Show that the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) transforms the integral $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$ into the integral $$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$
  2. Hence determine $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$
Edexcel FP1 2023 June Q6
Challenging +1.2
6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
  2. Determine \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  3. Hence determine the first 3 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)\), giving each coefficient in simplest form.
  4. Use the Maclaurin series expansion for \(\ln ( 1 + x )\) to write down the first 4 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln ( 1 + k x )\), where \(k\) is a constant.
  5. Hence determine the value of \(k\) for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.
Edexcel FP1 2023 June Q7
Challenging +1.8
  1. With respect to a fixed origin \(O\) the point \(A\) has coordinates \(( 3,6,5 )\) and the line \(l\) has equation
$$( \mathbf { r } - ( 12 \mathbf { i } + 30 \mathbf { j } + 39 \mathbf { k } ) ) \times ( 7 \mathbf { i } + 13 \mathbf { j } + 24 \mathbf { k } ) = \mathbf { 0 }$$ The points \(B\) and \(C\) lie on \(l\) such that \(A B = A C = 15\) Given that \(A\) does not lie on \(l\) and that the \(x\) coordinate of \(B\) is negative,
  1. determine the coordinates of \(B\) and the coordinates of \(C\)
  2. Hence determine a Cartesian equation of the plane containing the points \(A , B\) and \(C\) The point \(D\) has coordinates \(( - 2,1 , \alpha )\), where \(\alpha\) is a constant.
    Given that the volume of the tetrahedron \(A B C D\) is 147
  3. determine the possible values of \(\alpha\) Given that \(\alpha > 0\)
  4. determine the shortest distance between the line \(l\) and the line passing through the points \(A\) and \(D\), giving your answer to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c0ac1e1e-16bf-4a06-9eaa-8dcf01177722-24_2267_50_312_1980}
WJEC Further Unit 6 2024 June Q1
Standard +0.8
  1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.
Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Find the velocity of \(A\) immediately after the collision.
  2. Calculate the coefficient of restitution between \(A\) and \(B\).
  3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
  4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q2
Challenging +1.2
An object, of mass 1.8 kg , is falling vertically downwards under gravity. During the motion, it experiences a variable resistance of \(0 \cdot 2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the object at time \(t\) seconds.
  1. Show that \(v\) satisfies the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 9 g - v ^ { 2 } } { 9 }$$ At time \(t = 0\), the object passes a point \(A\) with a speed of \(\sqrt { g } \mathrm {~ms} ^ { - 1 }\). The object then hits the ground with a speed of \(8 \mathrm {~ms} ^ { - 1 }\).
  2. Calculate the time taken for the object to hit the ground.
    …………………………………………………………………………………………………………………………………………………... \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_67_1614_639_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_76_1614_717_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_79_1614_801_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_72_1609_895_267} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_75_1614_979_264}
  3. Given that the distance of the object from \(A\) at time \(t\) is \(x\) metres, form another differential equation to find an expression for \(x\) in terms of \(v\). Hence, find the height of \(A\) above the ground.
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q3
Standard +0.8
\(A C B\) is the diameter of a semi-circular lamina of radius \(2 a\) and centre \(C\). Another semi-circular lamina, having \(A C\) as its diameter, is added to form a uniform lamina, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-10_755_521_520_772}
    1. Show that the distance of the centre of mass of the lamina from \(A B\) is \(\frac { 28 } { 15 \pi } a\).
    2. Calculate the distance of the centre of mass of the lamina from a line drawn through \(A\) that is perpendicular to \(A B\).
      □
      □
    (b) Suppose that the lamina is suspended in equilibrium by means of two vertical wires attached at \(A\) and \(B\) so that \(A B\) is horizontal. Find the fraction of the lamina's weight that is supported by the wire attached at \(B\).
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q4
Standard +0.3
  1. The diagram below shows part of a game at a funfair that consists of a target moving along a straight horizontal line \(A B\). The centre of the target may be modelled as a particle moving with Simple Harmonic Motion about centre \(O\), where \(O\) is the midpoint of \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-14_245_1145_525_452}
When the target is at a distance of 84 cm from \(O\), its speed is \(52 \mathrm { cms } ^ { - 1 }\) and the magnitude of its acceleration is \(1344 \mathrm { cms } ^ { - 2 }\).
  1. Show that the period of the motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
    (b) Determine the maximum speed of the target.Examiner
  2. During a game, players fire a ball at the target. A timer is started when the target is at \(A\). Players must wait for the target to complete at least one full cycle before firing. Given that the target is hit when it is at a distance of 67 cm from \(O\), calculate the two earliest possible times taken to hit the target.
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q5
Standard +0.8
  1. The diagram below shows a uniform rod \(A B\) of weight \(W N\) and length \(2 l\), with its lower end \(A\) resting on a rough horizontal floor. A light cable is attached to the other end \(B\). The rod is in equilibrium when it is inclined at an angle of \(\theta\) to the floor, where \(0 ^ { \circ } < \theta \leqslant 45 ^ { \circ }\). The tension in the cable is \(T \mathrm {~N}\) acting at an angle of \(2 \theta\) to the rod, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-18_508_1105_559_479}
      1. Show that \(T = \frac { W } { 4 } \operatorname { cosec } \theta\).
      2. Hence determine the normal reaction of the floor on the rod at \(A\), giving your answer in terms of \(W\).
      (b) Given that the coefficient of friction between the floor and the rod is \(\frac { \sqrt { 3 } } { 3 }\), calculate the minimum possible value for \(\theta\).
    2. The region \(R\), shown in the diagram below, is bounded by the coordinate axes and the curve
    $$y = \frac { a } { b } \sqrt { b ^ { 2 } - x ^ { 2 } }$$ where \(a , b\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-21_451_1116_644_468} The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 3 } \pi a ^ { 2 } b\).
    1. Use integration to show that the distance of the centre of mass of \(S\) from the \(y\)-axis is \(\frac { 3 b } { 8 }\).
      The diagram below shows a small tree growing in a pot. The uniform solid \(S\) described on the previous page may be used to model the part of the tree above the pot. This part of the tree has height \(h \mathrm {~cm}\) and base radius \(\frac { h } { 4 } \mathrm {~cm}\). The pot, including its contents, may be modelled as a solid cylinder of height 50 cm and radius 25 cm . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-22_846_839_1596_612} You may assume that the density of the pot, including its contents, is equal to 20 times the density of the part of the tree above the pot.
    2. A gardener suggests that a tree is said to have outgrown its pot if the centre of mass, of both the tree and its pot, lies above the height of the pot. Determine the maximum value of \(h\) before the tree outgrows its pot.
    3. Identify one possible limitation of the model used that could affect your answer to part (b). \section*{END OF PAPER} Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}
Edexcel PURE 2024 October Q1
Moderate -0.8
The line \(l _ { 1 }\) passes through the point \(A ( - 5,20 )\) and the point \(B ( 3 , - 4 )\).
  1. Find an equation for \(l _ { 1 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\)
  2. Find an equation for \(l _ { 2 }\) giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
Edexcel PURE 2024 October Q2
Easy -1.2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
  2. Find the exact value of \(a\) such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form \(p \sqrt { 3 } + q\) where \(p\) and \(q\) are fully simplified rational constants.
Edexcel PURE 2024 October Q3
Standard +0.3
  1. In this question you must show all stages of your working.
$$f ( x ) = \frac { ( x + 5 ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$
  1. Find \(\int f ( x ) d x\)
    1. Show that when \(\mathrm { f } ^ { \prime } ( x ) = 0\) $$3 x ^ { 2 } + 10 x - 25 = 0$$
    2. Hence state the value of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) = 0$$