Questions — Edexcel FP1 (269 questions)

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Edexcel FP1 2022 June Q2
  1. During 2029, the number of hours of daylight per day in London, H, is modelled by the equation
$$H = 0.3 \sin \left( \frac { x } { 60 } \right) - 4 \cos \left( \frac { x } { 60 } \right) + 11.5 \quad 0 \leqslant x < 365$$ where \(x\) is the number of days after 1st January 2029 and the angle is in radians.
  1. Show that, according to the model, the number of hours of daylight in London on the 31st January 2029 will be 8.13 to 3 significant figures.
  2. Use the substitution \(t = \tan \left( \frac { x } { 120 } \right)\) to show that \(H\) can be written as $$H = \frac { a t ^ { 2 } + b t + c } { 1 + t ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be determined.
  3. Hence determine, according to the model, the date of the first day of 2029 when there will be at least 12 hours of daylight in London.
Edexcel FP1 2022 June Q3
  1. With respect to a fixed origin \(O\), the points \(A\) and \(B\) have coordinates \(( 2,2 , - 1 )\) and ( \(4,2 p , 1\) ) respectively, where \(p\) is a constant.
For each of the following, determine the possible values of \(p\) for which,
  1. \(O B\) makes an angle of \(45 ^ { \circ }\) with the positive \(x\)-axis
  2. \(\overrightarrow { O A } \times \overrightarrow { O B }\) is parallel to \(\left( \begin{array} { r } 4
    - p
    2 \end{array} \right)\)
  3. the area of triangle \(O A B\) is \(3 \sqrt { 2 }\)
Edexcel FP1 2022 June Q4
  1. The velocity \(v \mathrm {~ms} ^ { - 1 }\), of a raindrop, \(t\) seconds after it falls from a cloud, is modelled by the differential equation
$$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.1 v ^ { 2 } + 10 \quad t \geqslant 0$$ Initially the raindrop is at rest.
  1. Use two iterations of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the velocity of the raindrop 1 second after it falls from the cloud. Given that the initial acceleration of the raindrop is found to be smaller than is suggested by the current model,
  2. refine the model by changing the value of one constant.
Edexcel FP1 2022 June Q5
  1. The rectangular hyperbola \(H\) has equation \(x y = 36\)
    1. Use calculus to show that the equation of the tangent to \(H\) at the point \(P \left( 6 t , \frac { 6 } { t } \right)\) is
    $$y t ^ { 2 } + x = 12 t$$ The point \(Q \left( 12 t , \frac { 3 } { t } \right)\) also lies on \(H\).
  2. Find the equation of the tangent to \(H\) at the point \(Q\). The tangent at \(P\) and the tangent at \(Q\) meet at the point \(R\).
  3. Show that as \(t\) varies the locus of \(R\) is also a rectangular hyperbola.
Edexcel FP1 2022 June Q6
  1. The points \(P , Q\) and \(R\) have position vectors \(\left( \begin{array} { r } 1
    - 2
    4 \end{array} \right) , \left( \begin{array} { r } 3
    1
    - 5 \end{array} \right)\) and \(\left( \begin{array} { l } 2
    0
    3 \end{array} \right)\) respectively.
    1. Determine a vector equation of the plane that passes through the points \(P , Q\) and \(R\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), where \(\lambda\) and \(\mu\) are scalar parameters.
    2. Determine the coordinates of the point of intersection of the plane with the \(x\)-axis.
Edexcel FP1 2022 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7de3f581-eff1-4671-87a9-55ca1bb97890-20_591_962_312_548} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. The equation $$\left| x ^ { 2 } - 8 \right| = m x + c$$ has exactly 3 roots, as shown in Figure 1.
  1. Show that $$m ^ { 2 } - 4 c + 32 = 0$$ Given that \(c = 3 m\)
  2. determine the value of \(m\) and the value of \(c\)
  3. Hence solve $$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$
Edexcel FP1 2022 June Q8
  1. The Taylor series expansion of \(f ( x )\) about \(x = a\) is given by
$$f ( x ) = f ( a ) + ( x - a ) f ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } f ^ { ( r ) } ( a ) + \ldots$$
  1. (a) Use differentiation to determine the Taylor series expansion of \(\ln x\), in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 2 }\)
    (b) Hence prove that $$\lim _ { x \rightarrow 1 } \left( \frac { \ln x } { x - 1 } \right) = 1$$
  2. Use L'Hospital's rule to determine $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { ( x + 3 ) \tan ( 6 x ) \operatorname { cosec } ( 2 x ) } \right)$$ (Solutions relying entirely on calculator technology are not acceptable.)
Edexcel FP1 2022 June Q9
  1. A particle \(P\) moves along a straight line.
At time \(t\) minutes, the displacement, \(x\) metres, of \(P\) from a fixed point \(O\) on the line is modelled by the differential equation $$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x + 16 t ^ { 2 } x = 4 t ^ { 3 } \sin 2 t$$
  1. Show that the transformation \(x =\) ty transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 16 y = 4 \sin 2 t$$
  2. Hence find a general solution for the displacement of \(P\) from \(O\) at time \(t\) minutes.
Edexcel FP1 2024 June Q1
  1. (a) Given that
$$y = \ln \left( 3 + x ^ { 2 } \right)$$ complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 significant figures.
\(\boldsymbol { x }\)22.533.544.55
\(\boldsymbol { y }\)1.9462.2252.7252.9443.1463.332
In part (b) you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} (b) Use Simpson's rule with all the values of \(y\) in the completed table to estimate, to 3 significant figures, the value of $$\int _ { 2 } ^ { 5 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$$ (c) Using your answer to part (b) and making your method clear, estimate the value of $$\int _ { 2 } ^ { 5 } \ln \sqrt { \left( 3 + x ^ { 2 } \right) } \mathrm { d } x$$
Edexcel FP1 2024 June Q2
  1. Use algebra to determine the values of \(x\) for which
$$\left| x ^ { 2 } - 2 x \right| \leqslant x$$
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 $$4 s y + r x = 4$$