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CAIE FP1 2015 November Q11 EITHER
Challenging +1.8
The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\). Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\). Find the position vector of the point \(Q\). Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).
CAIE FP1 2015 November Q11 OR
Standard +0.8
The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\). Sketch \(C\). Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\). Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) ,$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). {www.cie.org.uk} after the live examination series.
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CAIE FP1 2018 November Q1
Moderate -0.3
1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2 \\ 9 \\ 0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0 \\ - 8 \\ 3 \end{array} \right) .$$
  1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
  2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
CAIE FP1 2018 November Q2
Standard +0.8
2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.
  1. Show that $$2 p \alpha + q = 0$$
  2. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$
CAIE FP1 2018 November Q3
Challenging +1.2
3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
  1. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
  2. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
CAIE FP1 2018 November Q4
Challenging +1.2
4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).
  1. Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant \(a\) is to be found.
  2. Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).
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
    • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
    • Answer all the questions.
    • You are permitted to use a graphical calculator in this paper.
    • The number of marks is given in brackets [ ] at the end of each question or part question.
    • You are advised that an answer may receive no marks unless you show sufficient defail of the working to indicate that a correct method is being used.
    • Final answers should be given to a degree of accuracy appropriate to the context.
    • The total number of marks for this paper is 72.
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
    • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
    • Answer all the questions.
    • You are permitted to use a graphical calculator in this paper.
    • Final answers should be given to a degree of accuracy appropriate to the context.
    • The number of marks is given in brackets [ ] at the end of each question or part question.
    • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
    • The total number of marks for this paper is 72.
Edexcel FP1 2023 June Q1
Moderate -0.8
  1. (a) 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,
(b) 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. (a) 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 $$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$ (b) 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}
CAIE FP1 2003 November Q1
6 marks Challenging +1.2
\includegraphics{figure_1} The curve \(C\) has polar equation $$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$ where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]
CAIE FP1 2003 November Q2
6 marks Challenging +1.2
Given that $$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$ find \(S_N = \sum_{n=N+1}^{2N} u_n\) in terms of \(N\). [3] Find a number \(M\) such that \(S_N < 10^{-20}\) for all \(N > M\). [3]