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Edexcel FP3 2017 June Q8
10 marks Challenging +1.8
8. The curve \(C\) has equation $$y = \ln \left( \frac { \mathrm { e } ^ { x } + 1 } { \mathrm { e } ^ { x } - 1 } \right) , \quad \ln 2 \leqslant x \leqslant \ln 3$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$$
  2. Find the length of the curve \(C\), giving your answer in the form \(\ln a\), where \(a\) is a rational number.
    (6)
Edexcel FP3 2018 June Q1
5 marks Standard +0.3
  1. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that, for \(x \in \mathbb { R }\)
$$\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }$$ (b) Hence, given that \(- 1 < \theta < 1\), prove that $$\operatorname { artanh } \theta = \frac { 1 } { 2 } \ln \left( \frac { 1 + \theta } { 1 - \theta } \right)$$ uestion 1 continued \(\_\_\_\_\) 7
Edexcel FP3 2018 June Q2
10 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{38487750-8c0f-4c3d-a019-5213ed2866eb-04_616_764_246_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = 5 \cosh x - 6 \sinh x$$ The curve crosses the \(x\)-axis at the point \(A\).
  1. Find the exact value of the \(x\) coordinate of the point \(A\), giving your answer as a natural logarithm.
  2. Show that $$( 5 \cosh x - 6 \sinh x ) ^ { 2 } \equiv a \cosh 2 x + b \sinh 2 x + c$$ where \(a , b\) and \(c\) are constants to be found. The finite region \(R\), bounded by the curve and the coordinate axes, is shown shaded in Figure 1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Use calculus to find the volume of the solid generated, giving your answer as an exact multiple of \(\pi\).
Edexcel FP3 2018 June Q3
9 marks Standard +0.3
3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\),
  1. find the value of \(k\).
  2. Hence find the other two eigenvalues of \(\mathbf { M }\).
  3. Find an eigenvector corresponding to the eigenvalue 3
    3. \(\quad \mathbf { M } = \left( \begin{array} { r c c } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\), (a) find the value of \(k\).
Edexcel FP3 2018 June Q4
12 marks Challenging +1.3
4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
  2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
  3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.
Edexcel FP3 2018 June Q5
11 marks Challenging +1.2
5. Given that $$I _ { n } = \int x ^ { n } \sqrt { ( x + 8 ) } \mathrm { d } x , \quad n \geqslant 0 , x \geqslant 0$$
  1. show that, for \(n \geqslant 1\) $$I _ { n } = \frac { p x ^ { n } ( x + 8 ) ^ { \frac { 3 } { 2 } } } { 2 n + 3 } - \frac { q n } { 2 n + 3 } I _ { n - 1 }$$ where \(p\) and \(q\) are constants to be found.
  2. Use part (a) to find the exact value of $$\int _ { 0 } ^ { 10 } x ^ { 2 } \sqrt { ( x + 8 ) } d x$$ giving your answer in the form \(k \sqrt { 2 }\), where \(k\) is rational.
Edexcel FP3 2018 June Q6
13 marks Standard +0.8
6. The line \(l _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation $$\frac { x + 1 } { 1 } = \frac { y - 4 } { 1 } = \frac { z - 1 } { 3 }$$
  1. Prove that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
  2. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The plane \(\Pi\) contains \(l _ { 1 }\) and intersects \(l _ { 2 }\) at the point \(( 3,8,13 )\).
  3. Find a cartesian equation for the plane \(\Pi\).
Edexcel FP3 2018 June Q7
15 marks Challenging +1.2
7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)
  1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
  2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
  3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
  4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
  5. Find the minimum area of triangle \(O A B\).
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    Q7

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Edexcel FP3 Specimen Q1
4 marks Standard +0.3
  1. Find the eigenvalues of the matrix \(\left( \begin{array} { l l } 7 & 6 \\ 6 & 2 \end{array} \right)\)
  2. Find the values of \(x\) for which
$$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms.
(Total 6 marks)
Edexcel FP3 Specimen Q3
6 marks Challenging +1.2
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6c256e1b-455d-42fb-81f2-a9a8ed1148bc-2_503_801_998_566}
\end{figure} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a ( t - \sin t ) , \quad y = a ( 1 - \cos t ) , \quad 0 \leq t \leq 2 \pi$$ Find, by using integration, the length of \(C\).
Edexcel FP3 Specimen Q4
7 marks Challenging +1.2
4. Find \(\int \sqrt { } \left( x ^ { 2 } + 4 \right) \mathrm { d } x\).
Edexcel FP3 Specimen Q5
7 marks Standard +0.3
5. Given that \(y = \arcsin x\) prove that
  1. \(\quad \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { \left( 1 - x ^ { 2 } \right) } }\)
  2. \(\quad \left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0\)
Edexcel FP3 Specimen Q6
8 marks Challenging +1.2
6. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \sin x \mathrm {~d} x$$
  1. Show that for \(n \geq 2\) $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
  2. Hence obtain \(I _ { 3 }\), giving your answers in terms of \(\pi\).
Edexcel FP3 Specimen Q7
14 marks Standard +0.3
7. $$\mathbf { A } ( x ) = \left( \begin{array} { c c c } 1 & x & - 1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{array} \right) , x \neq \frac { 5 } { 2 }$$
  1. Calculate the inverse of \(\mathbf { A } ( x )\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & - 1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{array} \right)$$ The image of the vector \(\left( \begin{array} { c } p \\ q \\ r \end{array} \right)\) when transformed by \(\mathbf { B }\) is \(\left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)\)
  2. Find the values of \(p , q\) and \(r\).
Edexcel FP3 Specimen Q8
12 marks Standard +0.8
8. The points \(A , B , C\), and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } + \mathbf { k } , \mathrm { b } = \mathbf { i } + 3 \mathbf { j } , \mathbf { c } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \mathbf { d } = 4 \mathbf { j } + \mathbf { k }$$ respectively.
  1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\) and hence find the area of triangle \(A B C\).
  2. Find the volume of the tetrahedron \(A B C D\).
  3. Find the perpendicular distance of \(D\) from the plane containing \(A , B\) and \(C\).
Edexcel FP3 Specimen Q9
13 marks Challenging +1.8
9. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
  1. Show that an equation of the normal to \(C\) at \(P ( a \sec \theta , b \tan \theta )\) is $$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(A B\) is \(M\).
  2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies.
    (Total 13 marks)
Edexcel M1 2014 January Q1
6 marks Moderate -0.8
  1. A truck \(P\) of mass \(2 M\) is moving with speed \(U\) on smooth straight horizontal rails. It collides directly with another truck \(Q\) of mass \(3 M\) which is moving with speed \(4 U\) in the opposite direction on the same rails. The trucks join so that immediately after the collision they move together. By modelling the trucks as particles, find
    1. the speed of the trucks immediately after the collision,
    2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2014 January Q2
6 marks Moderate -0.8
2. A particle \(P\) is moving with constant velocity ( \(2 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the speed of \(P\). The particle \(P\) passes through the point \(A\) and 4 seconds later passes through the point with position vector ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
  2. Find the position vector of \(A\).
Edexcel M1 2014 January Q3
5 marks Standard +0.3
3. A beam \(A B\) has length 15 m and mass 25 kg . The beam is smoothly supported at the point \(P\), where \(A P = 8 \mathrm {~m}\). A man of mass 100 kg stands on the beam at a distance of 2 m from \(A\) and another man stands on the beam at a distance of 1 m from \(B\). The beam is modelled as a non-uniform rod and the men are modelled as particles. The beam is in equilibrium in a horizontal position with the reaction on the beam at \(P\) having magnitude 2009 N. Find the distance of the centre of mass of the beam from \(A\).
Edexcel M1 2014 January Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fade35da-8dca-4d98-a07c-ed3a173fccda-08_396_483_214_735} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A fixed rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) A small box of mass \(m\) is at rest on the plane. A force of magnitude \(k m g\), where \(k\) is a constant, is applied to the box. The line of action of the force is at angle \(\alpha\) to the line of greatest slope of the plane through the box, as shown in Figure 1, and lies in the same vertical plane as this line of greatest slope. The coefficient of friction between the box and the plane is \(\mu\). The box is on the point of slipping up the plane. By modelling the box as a particle, find \(k\) in terms of \(\mu\).
Edexcel M1 2014 January Q5
7 marks Moderate -0.3
5. A racing car is moving along a straight horizontal track with constant acceleration. There are three checkpoints, \(P , Q\) and \(R\), on the track, where \(P Q = 48 \mathrm {~m}\) and \(Q R = 200 \mathrm {~m}\). The car takes 3 s to travel from \(P\) to \(Q\) and 5 s to travel from \(Q\) to \(R\). Find
  1. the acceleration of the car,
  2. the speed of the car as it passes \(P\).
Edexcel M1 2014 January Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fade35da-8dca-4d98-a07c-ed3a173fccda-16_398_860_210_543} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(P\) and \(Q\) have masses 0.1 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth pulley which is fixed to the edge of the table. Particle \(Q\) is at rest on a smooth plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\) The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 2. Particle \(P\) is released from rest with the string taut. During the first 0.5 s of the motion \(P\) does not reach the pulley and \(Q\) moves 0.75 m down the plane.
  1. Find the tension in the string during the first 0.5 s of the motion.
  2. Find the coefficient of friction between \(P\) and the table. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-19_72_59_2613_1886}
Edexcel M1 2014 January Q7
12 marks Moderate -0.3
7. A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 9 \mathbf { i } + 13 \mathbf { j } )\) N.
  1. Find the size of the angle between the direction of \(\mathbf { F }\) and the vector \(\mathbf { j }\). The force \(\mathbf { F }\) is the resultant of two forces \(\mathbf { P }\) and \(\mathbf { Q }\). The line of action of \(\mathbf { P }\) is parallel to the vector ( \(2 \mathbf { i } - \mathbf { j }\) ). The line of action of \(\mathbf { Q }\) is parallel to the vector ( \(\mathbf { i } + 3 \mathbf { j }\) ).
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
    1. the force \(\mathbf { P }\),
    2. the force \(\mathbf { Q }\).
Edexcel M1 2014 January Q8
17 marks Moderate -0.3
8. Two trains, \(A\) and \(B\), start together from rest, at time \(t = 0\), at a station and move along parallel straight horizontal tracks. Both trains come to rest at the next station after 180 s . Train \(A\) moves with constant acceleration \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 30 s , then moves at constant speed for 120 s and then moves with constant deceleration for the final 30 s . Train \(B\) moves with constant acceleration for 90 s and then moves with constant deceleration for the final 90 s .
  1. Sketch, on the same axes, the speed-time graphs for the motion of the two trains between the two stations.
  2. Find the acceleration of train \(B\) for the first half of its journey.
  3. Find the times when the two trains are moving at the same speed.
  4. Find the distance between the trains 96 s after they start. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-28_43_58_2457_1893}
Edexcel M1 2015 January Q1
7 marks Moderate -0.8
  1. A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4 m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3 u\) and the speed of truck \(B\) is \(2 u\). In the collision the trucks join together. Modelling the trucks as particles, find
    1. the speed of \(A\) immediately after the collision,
    2. the direction of motion of \(A\) immediately after the collision,
    3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{aaa8b297-347c-4a9b-a2c2-c4bd70d56912-03_534_1065_118_445} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A block of mass 50 kg lies on a rough plane which is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\). The block is held at rest by a vertical rope, as shown in Figure 1, and is on the point of sliding down the plane. The block is modelled as a particle and the rope is modelled as a light inextensible string. Given that the friction force acting on the block has magnitude 65.8 N, find
  2. the tension in the rope,
  3. the coefficient of friction between the block and the plane.