Questions FP1 (1385 questions)

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Edexcel FP1 2024 June Q3
  1. Use L'Hospital's rule to show that
$$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { \sin x } - \frac { 1 } { x } \right) = 0$$ (6)
Edexcel FP1 2024 June Q4
4. $$\left[ \begin{array} { l } \text { The Taylor series expansion of } \mathrm { f } ( x ) \text { about } x = a \text { is given by }
\mathrm { f } ( x ) = \mathrm { f } ( a ) + ( x - a ) \mathrm { f } ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } \mathrm { f } ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } \mathrm { f } ^ { ( r ) } ( a ) + \ldots \end{array} \right]$$ The curve with equation \(y = \mathrm { f } ( x )\) satisfies the differential equation $$\cos x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \sin x = 0$$ Given that \(\left( \frac { \pi } { 4 } , 1 \right)\) is a stationary point of the curve,
  1. determine the nature of this stationary point, giving a reason for your answer.
  2. Show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sqrt { 2 } - 2\) at this stationary point.
  3. Hence determine a series solution for \(y\), in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\), giving each coefficient in simplest form.
Edexcel FP1 2024 June Q5
5. $$y = \mathrm { e } ^ { 3 x } \sin x$$
  1. Use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = 28 \mathrm { e } ^ { 3 x } \sin x + 96 \mathrm { e } ^ { 3 x } \cos x$$
  2. Hence express \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) in the form $$\operatorname { Re } ^ { 3 \mathrm { x } } \sin ( \mathrm { x } + \alpha )$$ where \(R\) and \(\alpha\) are constants to be determined, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Edexcel FP1 2024 June Q6
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
Given that
  • the eccentricity of \(H\) is the reciprocal of the eccentricity of \(E\)
  • the coordinates of the foci of \(H\) are the same as the coordinates of the foci of \(E\) determine
    1. the value of \(a\)
    2. the value of \(b\)
Edexcel FP1 2024 June Q7
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
  1. Use the substitution \(t = \tan \left( \frac { \theta } { 2 } \right)\) to show that $$\int \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \int \frac { a } { ( t + b ) ^ { 2 } + c } d t$$ where \(a\), \(b\) and \(c\) are constants to be determined.
  2. Hence show that $$\int _ { \frac { \pi } { 2 } } ^ { \frac { 2 \pi } { 3 } } \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \ln \left( \frac { 2 \sqrt { 3 } } { 3 } \right)$$
Edexcel FP1 2024 June Q8
  1. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.
The point \(A \left( a t ^ { 2 } , 2 a t \right)\), where \(t \neq 0\), lies on \(P\).
  1. Use calculus to show that an equation of the tangent to \(P\) at \(A\) is $$y t = x + a t ^ { 2 }$$ The point \(B \left( 2 k ^ { 2 } , 4 k \right)\) and the point \(C \left( 2 k ^ { 2 } , - 4 k \right)\), where \(k\) is a constant, lie on \(P\).
    The tangent to \(P\) at \(B\) and the tangent to \(P\) at \(C\) intersect at the point \(D\).
    Given that the area of the triangle \(B C D\) is 432
  2. determine the coordinates of \(B\) and the coordinates of \(C\).
Edexcel FP1 2024 June Q9
    1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2
      - 3
      1 \end{array} \right) + \lambda \left( \begin{array} { r } 3
      4
      - 1 \end{array} \right)\)
The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } 13
5
8 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
5 \end{array} \right)\)
where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
  1. Determine the coordinates of \(P\). Given that the plane \(\Pi\) contains both \(l _ { 1 }\) and \(l _ { 2 }\)
  2. determine a Cartesian equation for \(\Pi\).
    (ii) Determine a Cartesian equation for each of the two lines that
    • pass through \(( 0,0,0 )\)
    • make an angle of \(60 ^ { \circ }\) with the \(x\)-axis
    • make an angle of \(45 ^ { \circ }\) with the \(y\)-axis
Edexcel FP1 2024 June Q10
  1. The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t ( t + 1 ) \frac { \mathrm { d } x } { \mathrm {~d} t } + 2 ( t + 1 ) x = 8 t ^ { 3 } \mathrm { e } ^ { t }$$ where \(P\) has displacement \(x\) metres from the origin \(O\) at time \(t\) minutes, \(t > 0\)
  1. Show that the transformation \(x = t u\) transforms the differential equation (I) into the differential equation $$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} u } { \mathrm {~d} t } = 8 \mathrm { e } ^ { t }$$ Given that \(P\) is at \(O\) when \(t = \ln 3\) and when \(t = \ln 5\)
  2. determine the particular solution of the differential equation (I)
Edexcel FP1 Specimen Q1
  1. Use Simpson's Rule with 6 intervals to estimate
$$\int _ { 1 } ^ { 4 } \sqrt { 1 + x ^ { 3 } } d x$$
Edexcel FP1 Specimen Q2
  1. Given \(k\) is a constant and that
$$y = x ^ { 3 } \mathrm { e } ^ { k x }$$ use Leibnitz theorem to show that $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = k ^ { n - 3 } \mathrm { e } ^ { k x } \left( k ^ { 3 } x ^ { 3 } + 3 n k ^ { 2 } x ^ { 2 } + 3 n ( n - 1 ) k x + n ( n - 1 ) ( n - 2 ) \right)$$
Edexcel FP1 Specimen Q3
  1. A vibrating spring, fixed at one end, has an external force acting on it such that the centre of the spring moves in a straight line. At time \(t\) seconds, \(t \geqslant 0\), the displacement of the centre \(C\) of the spring from a fixed point \(O\) is \(x\) micrometres.
The displacement of \(C\) from \(O\) is modelled by the differential equation $$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 2 + t ^ { 2 } \right) x = t ^ { 4 }$$
  1. Show that the transformation \(x = t v\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} t ^ { 2 } } + v = t$$
  2. Hence find the general equation for the displacement of \(C\) from \(O\) at time \(t\) seconds.
    1. State what happens to the displacement of \(C\) from \(O\) as \(t\) becomes large.
    2. Comment on the model with reference to this long term behaviour.
Edexcel FP1 Specimen Q4
4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$
  1. Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a x \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$$ where \(a\) and \(b\) are integers to be found.
  2. Hence find a series solution, in ascending powers of \(x\), as far as the term in \(x ^ { 5 }\), of the differential equation (I) where \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
Edexcel FP1 Specimen Q5
  1. The normal to the parabola \(y ^ { 2 } = 4 a x\) at the point \(P \left( a p ^ { 2 } , 2 a p \right)\) passes through the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\).
The line \(O P\) is perpendicular to the line \(O Q\), where \(O\) is the origin.
Prove that \(p ^ { 2 } = 2\)
Edexcel FP1 Specimen Q6
  1. A tetrahedron has vertices \(A ( 1,2,1 ) , B ( 0,1,0 ) , C ( 2,1,3 )\) and \(D ( 10,5,5 )\).
Find
  1. a Cartesian equation of the plane \(A B C\).
  2. the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) has equation \(2 x - 3 y + 3 = 0\)
    The point \(E\) lies on the line \(A C\) and the point \(F\) lies on the line \(A D\).
    Given that \(\Pi\) contains the point \(B\), the point \(E\) and the point \(F\),
  3. find the value of \(k\) such that \(\overrightarrow { A E } = k \overrightarrow { A C }\). Given that \(\overrightarrow { A F } = \frac { 1 } { 9 } \overrightarrow { A D }\)
  4. show that the volume of the tetrahedron \(A B C D\) is 45 times the volume of the tetrahedron \(A B E F\).
Edexcel FP1 Specimen Q7
  1. \(P\) and \(Q\) are two distinct points on the ellipse described by the equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)
The line \(l\) passes through the point \(P\) and the point \(Q\).
The tangent to the ellipse at \(P\) and the tangent to the ellipse at \(Q\) intersect at the point \(( r , s )\).
Show that an equation of the line \(l\) is $$4 s y + r x = 4$$
Edexcel FP1 Specimen Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a52911da-4b69-4d86-975e-d10e3a481e1d-16_407_1100_201_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of the function \(\mathrm { h } ( x )\) with equation $$h ( x ) = 45 + 15 \sin x + 21 \sin \left( \frac { x } { 2 } \right) + 25 \cos \left( \frac { x } { 2 } \right) \quad x \in [ 0,40 ]$$
  1. Show that $$\frac { \mathrm { d } h } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 6 t - 17 \right) \left( 9 t ^ { 2 } + 4 t - 3 \right) } { 2 \left( 1 + t ^ { 2 } \right) ^ { 2 } }$$ where \(t = \tan \left( \frac { x } { 4 } \right)\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a52911da-4b69-4d86-975e-d10e3a481e1d-16_581_1403_1263_331} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Source: \({ } ^ { 1 }\) Data taken on 29th December 2016 from \href{http://www.ukho.gov.uk/easytide/EasyTide}{http://www.ukho.gov.uk/easytide/EasyTide} Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from 08:00 on the 3rd January 2017 to the end of the 4th January \(2017 { } ^ { 1 }\). The graph of \(k \mathrm {~h} ( x )\), where \(k\) is a constant and \(x\) is the number of hours after 08:00 on 3rd of January, can be used to model the predicted tide heights, in metres, for this period of time.
    1. Suggest a value of \(k\) that could be used for the graph of \(k \mathrm {~h} ( x )\) to form a suitable model.
    2. Why may such a model be suitable to predict the times when the tide heights are at their peaks, but not to predict the heights of these peaks?
  3. Use Figure 2 and the result of part (a) to estimate, to the nearest minute, the time of the highest tide height on the 4th January 2017.
CAIE FP1 2017 November Q8
  1. Find the value of \(I _ { 2 }\).
  2. Show that, for \(n > 2\), $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
  3. The curve \(C\) has equation \(y = \sec ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2019 November Q5
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
AQA FP1 2013 January Q7
  1. Show that there is a linear relationship between \(Y\) and \(X\).
  2. The graph of \(Y\) against \(X\) is shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593} Find the value of \(n\) and the value of \(a\).
AQA FP1 2015 June Q6
  1. Sketch the curve \(C _ { 1 }\), stating the values of its intercepts with the coordinate axes.
  2. The curve \(C _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } k
    0 \end{array} \right]\), where \(k < 0\), to give a curve \(C _ { 2 }\). Given that \(C _ { 2 }\) passes through the origin \(( 0,0 )\), find the equations of the asymptotes of \(C _ { 2 }\).
    [0pt] [3 marks]
OCR FP1 2011 January Q7
  1. Write down the matrix, \(\mathbf { A }\), that represents a shear with \(x\)-axis invariant in which the image of the point \(( 1,1 )\) is \(( 4,1 )\).
  2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0
    0 & \sqrt { 3 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
  3. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 2 & 6
    0 & 2 \end{array} \right)\).
    (a) Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
    (b) Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\). 8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).
  4. Show that \(p = \frac { 5 } { 6 }\).
  5. Find the value of \(q\). 9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1
    3 & a & 1
    4 & 2 & 1 \end{array} \right)\).
  6. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
  7. Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
  8. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k
    & 3 x + 6 y + z = 0
    & 4 x + 2 y + z = k \end{aligned}$$ where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
  9. Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
  10. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
  11. Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).
OCR FP1 2016 June Q8
  1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
  2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), giving your answer as a single fraction.
  3. Find \(\sum _ { r = n } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), giving your answer as a single fraction.
Edexcel FP1 Q1
  1. Given that \(z = 22 + 4 \mathrm { i }\) and \(\frac { z } { w } = 6 - 8 \mathrm { i }\), find
    1. \(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
    2. the argument of \(z\), in radians to 2 decimal places.
    3. (a) Prove that \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 ) = \frac { 1 } { 6 } n ( n - 1 ) ( 2 n + 5 )\).
    4. Deduce that \(n ( n - 1 ) ( 2 n + 5 )\) is divisible by 6 for all \(n > 1\).
      [0pt] [P4 January 2002 Qn 3]
    $$\mathrm { f } ( x ) = x ^ { 3 } + x - 3$$ The equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between 1 and 2 .
  2. By considering \(\mathrm { f } ^ { \prime } ( x )\), show that \(\alpha\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\).
  3. Taking 1.2 as your first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 significant figures.
  4. Prove that your answer to part (b) gives the value of \(\alpha\) correct to 3 significant figures.
Edexcel FP1 Q4
4. Given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 2 } + b z + c = 0 , \text { where } b \text { and } c \text { are real constants, }$$
  1. write down the other root of the equation,
  2. find the value of \(b\) and the value of \(c\).
Edexcel FP1 Q5
5. Prove that $$\sum _ { r = 1 } ^ { n } 6 \left( r ^ { 2 } - 1 \right) \equiv ( n - 1 ) n ( 2 n + 5 )$$