Questions — Edexcel FP1 (310 questions)

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Edexcel FP1 Q9
14 marks Standard +0.3
9. (a) A sequence of numbers is defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = 3 u _ { n } + 4 \text { for } n \geqslant 1 .$$ Prove by induction that $$u _ { n } = 3 ^ { n } + 2 \left( 3 ^ { n - 1 } - 1 \right) \text { for } n \in \mathbb { Z } ^ { + } \text {. }$$ (b) $$\mathbf { A } = \left( \begin{array} { l l } 4 & 0 \\ 9 & 1 \end{array} \right)$$
  1. Prove by induction that $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right) \text { for } n \in \mathbb { Z } ^ { + } .$$
  2. Determine whether the result \(\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right)\) is also valid for \(n = - 1\).
Edexcel FP1 Specimen Q1
6 marks Moderate -0.8
1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$
  1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
  2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel FP1 Specimen Q2
7 marks Moderate -0.5
2. The rectangle \(R\) has vertices at the points \(( 0,0 ) , ( 1,0 ) , ( 1,2 )\) and \(( 0,2 )\).
  1. Find the coordinates of the vertices of the image of \(R\) under the transformation given by the matrix \(\mathbf { A } = \left( \begin{array} { c c } a & 4 \\ - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  2. Find det \(\mathbf { A }\), giving your answer in terms of \(a\). Given that the area of the image of \(R\) is 18 ,
  3. find the value of \(a\).
Edexcel FP1 Specimen Q3
5 marks Standard +0.3
3. The matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)\)
  1. Find \(\mathbf { R } ^ { 2 }\).
  2. Describe the geometrical transformation represented by \(\mathbf { R } ^ { 2 }\).
  3. Describe the geometrical transformation represented by \(\mathbf { R }\).
Edexcel FP1 Specimen Q4
3 marks Moderate -0.8
4. $$f ( x ) = 2 ^ { x } - 6 x$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4,5]. Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\).
Edexcel FP1 Specimen Q5
9 marks Standard +0.3
5. (a) Show that \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 1 \right) = \frac { 1 } { 3 } ( n - 2 ) n ( n + 2 )\).
(b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 40 } \left( r ^ { 2 } - r - 1 \right)\).
Edexcel FP1 Specimen Q6
10 marks Moderate -0.8
6. Given that \(z = - 3 + 4 \mathrm { i }\),
  1. find the modulus of \(z\),
  2. the argument of \(z\) in radians to 2 decimal places. Given also that \(w = \frac { - 14 + 2 \mathrm { i } } { z }\),
  3. use algebra to find \(w\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex numbers \(z\) and \(w\) are represented by points \(A\) and \(B\) on an Argand diagram.
  4. Show the points \(A\) and \(B\) on an Argand diagram.
Edexcel FP1 Specimen Q7
12 marks Standard +0.8
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. The point \(\left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
  1. Find the value of \(a\).
  2. Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is $$y t = x + 4 t ^ { 2 } .$$ The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
Edexcel FP1 Specimen Q8
9 marks Moderate -0.3
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),
  1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\),
  3. find the value of \(p\).
Edexcel FP1 Specimen Q9
14 marks Standard +0.3
9. Use the method of mathematical induction to prove that, for \(n \in \mathbb { Z } ^ { + }\),
  1. \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & 0 \end{array} \right) ^ { n } = \left( \begin{array} { c c } n + 1 & n \\ - n & 1 - n \end{array} \right)\)
  2. \(\mathrm { f } ( n ) = 4 ^ { n } + 6 n - 1\) is divisible by 3 .
Edexcel FP1 2019 June Q1
5 marks Moderate -0.8
  1. Use Simpson's rule with 4 intervals to estimate
$$\int _ { 0.4 } ^ { 2 } e ^ { x ^ { 2 } } d x$$
Edexcel FP1 2019 June Q2
4 marks Challenging +1.8
  1. Given that \(k\) is a real non-zero constant and that
$$y = x ^ { 3 } \sin k x$$ use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = \left( k ^ { 2 } x ^ { 2 } + A \right) k ^ { 3 } x \cos k x + B \left( k ^ { 2 } x ^ { 2 } + C \right) k ^ { 2 } \sin k x$$ where \(A\), \(B\) and \(C\) are integers to be determined.
Edexcel FP1 2019 June Q3
9 marks Challenging +1.2
3. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x - y ^ { 2 }$$
  1. Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a y \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + c \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) ^ { 2 }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
  2. Hence find a series solution, in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\), of the differential equation (I), given that \(y = 1\) at \(x = 0\)
Edexcel FP1 2019 June Q4
8 marks Challenging +1.2
  1. The parabola \(C\) has equation
$$y ^ { 2 } = 16 x$$ The distinct points \(P \left( p ^ { 2 } , 4 p \right)\) and \(Q \left( q ^ { 2 } , 4 q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0\) The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) meet at the point \(R ( - 28,6 )\).
Show that the area of triangle \(P Q R\) is 1331
Edexcel FP1 2019 June Q5
8 marks Challenging +1.2
5. $$I = \int \frac { 1 } { 4 \cos x - 3 \sin x } \mathrm {~d} x \quad 0 < x < \frac { \pi } { 4 }$$ Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that $$I = \frac { 1 } { 5 } \ln \left( \frac { 2 + \tan \left( \frac { x } { 2 } \right) } { 1 - 2 \tan \left( \frac { x } { 2 } \right) } \right) + k$$ where \(k\) is an arbitrary constant.
Edexcel FP1 2019 June Q6
17 marks Challenging +1.2
  1. The concentration of a drug in the bloodstream of a patient, \(t\) hours after the drug has been administered, where \(t \leqslant 6\), is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } C } { \mathrm {~d} t ^ { 2 } } - 5 t \frac { \mathrm {~d} C } { \mathrm {~d} t } + 8 C = t ^ { 3 }$$ where \(C\) is measured in micrograms per litre.
  1. Show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} C } { \mathrm {~d} x } + 8 C = \mathrm { e } ^ { 3 x }$$
  2. Hence find the general solution for the concentration \(C\) at time \(t\) hours. Given that when \(t = 6 , C = 0\) and \(\frac { \mathrm { d } C } { \mathrm {~d} t } = - 36\)
  3. find the maximum concentration of the drug in the bloodstream of the patient.
Edexcel FP1 2019 June Q7
10 marks Challenging +1.2
  1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates \(( 3,4,5 ) , ( 10 , - 1,5 )\) and ( \(4,7 , - 9\) ) respectively.
The plane \(\Pi\) has equation \(4 x - 8 y + z = 2\) The line segment \(A B\) meets the plane \(\Pi\) at the point \(P\) and the line segment \(B C\) meets the plane \(\Pi\) at the point \(Q\).
  1. Show that, to 3 significant figures, the area of quadrilateral \(A P Q C\) is 38.5 The point \(D\) has coordinates \(( k , 4 , - 1 )\), where \(k\) is a constant.
    Given that the vectors \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) form three edges of a parallelepiped of volume 226
  2. find the possible values of the constant \(k\).
Edexcel FP1 2019 June Q8
14 marks Challenging +1.8
  1. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \cosh \theta , 3 \sinh \theta )\).
The line \(l _ { 1 }\) meets the \(x\)-axis at the point \(A\).
The line \(l _ { 2 }\) is the tangent to \(H\) at the point \(( 4,0 )\).
The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(B\) and the midpoint of \(A B\) is the point \(M\).
  1. Show that, as \(\theta\) varies, a Cartesian equation for the locus of \(M\) is $$y ^ { 2 } = \frac { 9 ( 4 - x ) } { 4 x } \quad p < x < q$$ where \(p\) and \(q\) are values to be determined. Let \(S\) be the focus of \(H\) that lies on the positive \(x\)-axis.
  2. Show that the distance from \(M\) to \(S\) is greater than 1
Edexcel FP1 2020 June Q1
5 marks Standard +0.3
  1. Use l'Hospital's Rule to show that
$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$
Edexcel FP1 2020 June Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f127ab1-0e03-4f9f-87c2-01c553c54ee9-04_807_649_251_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 3 \cos \left( \frac { \pi } { 2 } x ^ { 2 } \right) \quad x \in [ - 1,1 ]$$ A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.
Use Simpson's rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in \(\mathrm { m } ^ { 3 }\) to 4 significant figures.
Edexcel FP1 2020 June Q3
9 marks Standard +0.3
  1. The points \(A , B\) and \(C\), with position vectors \(\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \mathbf { b } = \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }\) and \(\mathbf { c } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) respectively, lie on the plane \(\Pi\)
    1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\)
    2. Find an equation for \(\Pi\) in the form r.n \(= p\)
    The point \(D\) has position vector \(8 \mathbf { i } + 7 \mathbf { j } + 5 \mathbf { k }\)
  2. Determine the volume of the tetrahedron \(A B C D\)
Edexcel FP1 2020 June Q4
8 marks Challenging +1.8
4. $$f ( x ) = x ^ { 4 } \sin ( 2 x )$$ Use Leibnitz's theorem to show that the coefficient of \(( x - \pi ) ^ { 8 }\) in the Taylor series expansion of \(\mathrm { f } ( x )\) about \(\pi\) is $$\frac { a \pi + b \pi ^ { 3 } } { 315 }$$ where \(a\) and \(b\) are integers to be determined. The Taylor series expansion of \(\mathrm { f } ( \mathrm { x } )\) about \(\mathrm { x } = \mathrm { k }\) is given by $$f ( x ) = f ( k ) + ( x - k ) f ^ { \prime } ( k ) + \frac { ( x - k ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( k ) + \ldots + \frac { ( x - k ) ^ { r } } { r ! } f ^ { ( r ) } ( k ) + \ldots$$
Edexcel FP1 2020 June Q5
7 marks Standard +0.8
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 16 } = 1$$ The points \(S\) and \(S ^ { \prime }\) are the foci of \(E\).
  1. Find the coordinates of \(S\) and \(S ^ { \prime }\)
  2. Show that for any point \(P\) on \(E\), the triangle \(P S S ^ { \prime }\) has constant perimeter and determine its value.
Edexcel FP1 2020 June Q6
10 marks Challenging +1.2
  1. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by
$$\begin{aligned} & d _ { A } = | 5 t - 31 | \\ & d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right| \end{aligned}$$ respectively, where \(t\) is the time in seconds after motion begins.
  1. Use algebra to find the range of time for which particle \(A\) is further away from \(O\) than particle \(B\) is from \(O\). It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
  2. Use this information to assess the validity of the model.
Edexcel FP1 2020 June Q7
14 marks Challenging +1.8
  1. The points \(P \left( 9 p ^ { 2 } , 18 p \right)\) and \(Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q\), lie on the parabola \(C\) with equation
$$y ^ { 2 } = 36 x$$ The line \(l\) passes through the points \(P\) and \(Q\)
  1. Show that an equation for the line \(l\) is $$( p + q ) y = 2 ( x + 9 p q )$$ The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(A\).
  2. Show that the coordinates of \(A\) are $$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$ Given that the points \(P\) and \(Q\) vary such that \(l\) always passes through the point \(( 12,0 )\)
  3. find, in the form \(y ^ { 2 } = \mathrm { f } ( x )\), an equation for the locus of \(A\), giving \(\mathrm { f } ( x )\) in simplest form.