Questions — Edexcel FP1 (310 questions)

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Edexcel FP1 2024 June Q8
8 marks Challenging +1.2
  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
10 marks Standard +0.8
    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
12 marks Challenging +1.3
  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
5 marks Moderate -0.8
  1. Use Simpson's Rule with 6 intervals to estimate
$$\int _ { 1 } ^ { 4 } \sqrt { 1 + x ^ { 3 } } d x$$
Edexcel FP1 Specimen Q2
4 marks Challenging +1.8
  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
14 marks Challenging +1.2
  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
9 marks Challenging +1.2
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
9 marks Challenging +1.2
  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
11 marks Challenging +1.2
  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
8 marks Challenging +1.8
  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
15 marks Challenging +1.2
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.
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}
Edexcel FP1 Q1
5 marks Moderate -0.3
$$\text{f}(x) = 2x^3 - 8x^2 + 7x - 3.$$ Given that \(x = 3\) is a solution of the equation f\((x) = 0\), solve f\((x) = 0\) completely. [5]
Edexcel FP1 Q2
7 marks Moderate -0.3
  1. Show, using the formulae for \(\sum r\) and \(\sum r^2\), that $$\sum_{r=1}^n (6r^2 + 4r - 1) = n(n + 2)(2n + 1).$$ [5]
  2. Hence, or otherwise, find the value of \(\sum_{r=1}^n (6r^2 + 4r - 1)\). [2]
Edexcel FP1 Q3
4 marks Moderate -0.8
The rectangular hyperbola, \(H\), has parametric equations \(x = 5t, y = \frac{5}{t}, t \neq 0\).
  1. Write the cartesian equation of \(H\) in the form \(xy = c^2\). [1]
  2. Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively. Find the coordinates of the mid-point of \(AB\). [3]
Edexcel FP1 Q4
5 marks Moderate -0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), $$\sum_{r=1}^n \frac{1}{r(r+1)} = \frac{n}{n+1}$$ [5]
Edexcel FP1 Q5
9 marks Moderate -0.3
$$\text{f}(x) = 3\sqrt{x} + \frac{18}{\sqrt{x}} - 20.$$
  1. Show that the equation f\((x) = 0\) has a root \(a\) in the interval \([1.1, 1.2]\). [2]
  2. Find \(f'(x)\). [3]
  3. Using \(x_0 = 1.1\) as a first approximation to \(a\), apply the Newton-Raphson procedure once to f\((x)\) to find a second approximation to \(a\), giving your answer to 3 significant figures. [4]
Edexcel FP1 Q6
5 marks Standard +0.3
A series of positive integers \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 6 \text{ and } u_{n+1} = 6u_n - 5, \text{ for } n \geq 1.$$ Prove by induction that \(u_n = 5 \times 6^{n-1} + 1\), for \(n \geq 1\). [5]
Edexcel FP1 Q7
6 marks Standard +0.3
Given that \(\mathbf{X} = \begin{pmatrix} 2 & a \\ -1 & -1 \end{pmatrix}\), where \(a\) is a constant, and \(a \neq 2\).
  1. find \(\mathbf{X}^{-1}\) in terms of \(a\). [3]
  2. Given that \(\mathbf{X} + \mathbf{X}^{-1} = \mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix, find the value of \(a\). [3]