Questions — Edexcel (10514 questions)

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Edexcel CP2 2024 June Q6
14 marks Challenging +1.2
  1. The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation
$$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 4 t + 12$$ where \(P\) is \(x\) metres from the origin \(O\) at time \(t\) seconds, \(t \geqslant 0\)
  1. Determine the general solution of the differential equation.
  2. Hence determine the particular solution for which \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2\) when \(t = 0\)
    1. Show that, according to the model, the minimum distance between \(O\) and \(P\) is \(( 2 + \ln 2 )\) metres.
    2. Justify that this distance is a minimum. For large values of \(t\) the particle is expected to move with constant speed.
  4. Comment on the suitability of the model in light of this information.
Edexcel CP2 2024 June Q7
9 marks Moderate -0.8
  1. Determine the roots of the equation $$z ^ { 6 } = 1$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)
  2. Show the roots of the equation in part (a) on a single Argand diagram.
  3. Show that $$( \sqrt { 3 } + i ) ^ { 6 } = - 64$$
  4. Hence, or otherwise, solve the equation $$z ^ { 6 } + 64 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)
Edexcel CP2 2024 June Q8
7 marks Standard +0.3
8. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1 \\ 1 & 1 & 1 \\ k & 3 & 6 \end{array} \right) \quad k \neq 0$$
  1. Find, in terms of \(k , \mathbf { A } ^ { - 1 }\)
  2. Determine, in simplest form in terms of \(k\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 3 x + y - z = 3 \\ x + y + z = 1 \\ k x + 3 y + 6 z = 6 \end{array}$$
Edexcel CP2 2024 June Q9
10 marks Challenging +1.2
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f2d33c3-eb35-4b50-9a4d-54f43c514f49-28_586_560_246_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f2d33c3-eb35-4b50-9a4d-54f43c514f49-28_606_542_269_1110} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the central vertical cross-section \(A B C D E F A\) of a vase together with measurements that have been taken from the vase. The horizontal cross-section between \(A B\) and \(F C\) is a circle with diameter 4 cm .
The base of the vase \(E D\) is horizontal and the point \(E\) is vertically below \(F\) and the point \(D\) is vertically below \(C\). Using these measurements, the curve \(C D\) is modelled by the parametric equations $$x = a + 3 \sin 2 t \quad y = b \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
  1. Determine the value of \(a\) and the value of \(b\) according to the model.
  2. Using algebraic integration and showing all your working, determine, according to the model, the volume of the vase, giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\)
  3. State a limitation of the model.
Edexcel CP2 Specimen Q1
8 marks Moderate -0.3
  1. The roots of the equation
$$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$ are \(\alpha , \beta\) and \(\gamma\) Without solving the equation, find the value of
  1. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
  2. \(( \alpha + 2 ) ( \beta + 2 ) ( \gamma + 2 )\)
  3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)
Edexcel CP2 Specimen Q2
8 marks Standard +0.3
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$
  1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  2. Show that the vector \(- \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\) is perpendicular to \(\Pi _ { 2 }\)
  3. Show that the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(52 ^ { \circ }\) to the nearest degree.
Edexcel CP2 Specimen Q3
12 marks Standard +0.3
  1. (i)
$$\mathbf { M } = \left( \begin{array} { c c c } 2 & a & 4 \\ 1 & - 1 & - 1 \\ - 1 & 2 & - 1 \end{array} \right)$$ where \(a\) is a constant.
  1. For which values of \(a\) does the matrix \(\mathbf { M }\) have an inverse? Given that \(\mathbf { M }\) is non-singular,
  2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(a\) (ii) Prove by induction that for all positive integers \(n\), $$\left( \begin{array} { l l } 3 & 0 \\ 6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0 \\ 3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)$$
Edexcel CP2 Specimen Q4
7 marks Standard +0.8
A complex number \(z\) has modulus 1 and argument \(\theta\).
  1. Show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta , \quad n \in \mathbb { Z } ^ { + }$$
  2. Hence, show that $$\cos ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta + 4 \cos 2 \theta + 3 )$$
Edexcel CP2 Specimen Q5
10 marks Challenging +1.2
5. $$y = \sin x \sinh x$$
  1. Show that \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = - 4 y\)
  2. Hence find the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in its simplest form.
  3. Find an expression for the \(n\)th non-zero term of the Maclaurin series for \(y\).
Edexcel CP2 Specimen Q6
13 marks Challenging +1.8
    1. Show on an Argand diagram the locus of points given by the values of \(z\) satisfying $$| z - 4 - 3 \mathbf { i } | = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that \(\theta \in [ \alpha , \alpha + \pi ]\), where \(\alpha = - \arctan \left( \frac { 4 } { 3 } \right)\),
    2. show that this locus of points can be represented by the polar curve with equation $$r = 8 \cos \theta + 6 \sin \theta$$ The set of points \(A\) is defined by $$A = \left\{ z : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\} \cap \{ z : | z - 4 - 3 \mathbf { i } | \leqslant 5 \}$$
    (b)
    1. Show, by shading on your Argand diagram, the set of points \(A\).
    2. Find the exact area of the region defined by \(A\), giving your answer in simplest form.
Edexcel CP2 Specimen Q7
17 marks Standard +0.8
  1. At the start of the year 2000, a survey began of the number of foxes and rabbits on an island.
At time \(t\) years after the survey began, the number of foxes, \(f\), and the number of rabbits, \(r\), on the island are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } f } { \mathrm {~d} t } = 0.2 f + 0.1 r \\ & \frac { \mathrm {~d} r } { \mathrm {~d} t } = - 0.2 f + 0.4 r \end{aligned}$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } f } { \mathrm {~d} t ^ { 2 } } - 0.6 \frac { \mathrm {~d} f } { \mathrm {~d} t } + 0.1 f = 0\)
  2. Find a general solution for the number of foxes on the island at time \(t\) years.
  3. Hence find a general solution for the number of rabbits on the island at time \(t\) years. At the start of the year 2000 there were 6 foxes and 20 rabbits on the island.
    1. According to this model, in which year are the rabbits predicted to die out?
    2. According to this model, how many foxes will be on the island when the rabbits die out?
    3. Use your answers to parts (i) and (ii) to comment on the model.
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
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 \(8 \mathbf { i } + 7 \mathbf { j } + 5 \mathbf { k }\)
  3. 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.