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Edexcel M2 2023 January Q1
8 marks Standard +0.3
  1. A truck of mass 1500 kg is moving on a straight horizontal road.
The engine of the truck is working at a constant rate of 30 kW .
The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the truck is moving at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Find the value of \(R\). Later on, the truck is moving up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 8 }\) The resistance to the motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 500 N .
    The engine of the truck is again working at a constant rate of 30 kW . At the instant when the speed of the truck is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  2. Find the value of \(V\)
Edexcel M2 2023 January Q2
7 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) The particle receives an impulse \(( - 2 \mathbf { i } + \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after receiving the impulse, the velocity of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The kinetic energy gained by \(P\) as a result of receiving the impulse is 22 J .
Find the possible values of \(\lambda\).
Edexcel M2 2023 January Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-06_618_803_244_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B D E\) is in the shape of a rectangle with \(A B = 8 a\) and \(B D = 6 a\). The triangle \(B C D\) is isosceles and has base \(6 a\) and perpendicular height \(6 a\). The template \(A B C D E\), shown shaded in Figure 1, is formed by removing the triangular lamina \(B C D\) from the lamina \(A B D E\).
  1. Show that the centre of mass of the template is \(\frac { 14 } { 5 } a\) from \(A E\). The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.
  2. Find the value of \(\theta\), giving your answer to the nearest whole number.
Edexcel M2 2023 January Q4
10 marks Standard +0.3
  1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.]
A particle \(Q\) of mass 1.5 kg is moving on a smooth horizontal plane under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds ( \(t \geqslant 0\) ), the position vector of \(Q\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(Q\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) It is given that $$\mathbf { v } = \left( 3 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t ^ { 3 } + k t \right) \mathbf { j }$$ where \(k\) is a constant.
Given that when \(t = 2\) particle \(Q\) is moving in the direction of the vector \(\mathbf { i } + \mathbf { j }\)
  1. show that \(k = 4\)
  2. find the magnitude of \(\mathbf { F }\) when \(t = 2\) Given that \(\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }\) when \(t = 0\)
  3. find \(\mathbf { r }\) when \(t = 2\)
Edexcel M2 2023 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-12_296_1125_246_470} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The points \(A\) and \(B\) are on a line of greatest slope of the ramp, with \(A B = 2.5 \mathrm {~m}\) and \(B\) above \(A\), as shown in Figure 2. A package of mass 1.5 kg is projected up the ramp from \(A\) with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and first comes to instantaneous rest at \(B\). The coefficient of friction between the package and the ramp is \(\frac { 2 } { 7 }\) The package is modelled as a particle.
  1. Find the work done against friction as the package moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\). After coming to instantaneous rest at \(B\), the package slides back down the slope.
  3. Use the work-energy principle to find the speed of the package at the instant it returns to \(A\).
Edexcel M2 2023 January Q6
10 marks Standard +0.3
6. Figure 3 A uniform pole \(A B\), of weight 50 N and length 6 m , has a particle of weight \(W\) newtons attached at its end \(B\). The pole has its end \(A\) freely hinged to a vertical wall.
A light rod holds the particle and pole in equilibrium with the pole at \(60 ^ { \circ }\) to the wall. One end of the light rod is attached to the pole at \(C\), where \(A C = 4 \mathrm {~m}\).
The other end of the light rod is attached to the wall at the point \(D\).
The point \(D\) is vertically below \(A\) with \(A D = 4 \mathrm {~m}\), as shown in Figure 3.
The pole and the light rod lie in a vertical plane which is perpendicular to the wall.
The pole is modelled as a rod.
Given that the thrust in the light rod is \(60 \sqrt { 3 } \mathrm {~N}\),
  1. show that \(W = 15\)
  2. find the magnitude of the resultant force acting on the pole at \(A\).
Edexcel M2 2023 January Q7
10 marks Standard +0.3
  1. Particle \(P\) has mass \(3 m\) and particle \(Q\) has mass \(k m\). The particles are moving towards each other on the same straight line on a smooth horizontal surface.
    The particles collide directly.
    Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(v\).
The direction of motion of \(P\) is unchanged by the collision.
  1. Show that \(v = \frac { ( 3 - 3 k ) } { k } u\)
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    Given that \(v \neq u\)
  3. find the range of possible values of \(k\).
Edexcel M2 2023 January Q8
12 marks Standard +0.3
  1. A particle \(P\) is projected from a fixed point \(O\). The particle is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\alpha\) above the horizontal. The particle moves freely under gravity. At the instant when the horizontal distance of \(P\) from \(O\) is \(x\) metres, \(P\) is \(y\) metres vertically above the level of \(O\).
    1. Show that \(y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)\)
    A small ball is projected from a fixed point \(A\) with speed \(U \mathrm {~ms} ^ { - 1 }\) at \(\theta ^ { \circ }\) above the horizontal.
    The point \(B\) is on horizontal ground and is vertically below the point \(A\), with \(A B = 20 \mathrm {~m}\).
    The ball hits the ground at the point \(C\), where \(B C = 30 \mathrm {~m}\), as shown in Figure 4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ee5f81bc-1bdb-47a1-81e7-7e3cb8219e91-24_556_961_904_552} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The speed of the ball immediately before it hits the ground is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The motion of the ball is modelled as that of a particle moving freely under gravity.
  2. Use the principle of conservation of mechanical energy to find the value of \(U\).
  3. Find the value of \(\theta\)
Edexcel F3 2020 June Q1
7 marks Standard +0.8
  1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that
$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.
Edexcel F3 2020 June Q2
8 marks Standard +0.3
2. Determine
  1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
  2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)
Edexcel F3 2020 June Q3
9 marks Standard +0.3
3. $$\mathbf { M } = \left( \begin{array} { c c c } 3 & - 4 & k \\ 1 & - 2 & k \\ 1 & - 5 & 5 \end{array} \right) \text { where } k \text { is a constant }$$ 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 a normalised eigenvector corresponding to the eigenvalue 3
    .
    VIIIV SIHI NI JIIHM ION OCVARV SHAL NI ALIAM LON OOVERV SIHI NI JIIIM ION OO
Edexcel F3 2020 June Q4
9 marks Challenging +1.2
4.
  1. Show that, for \(n \geqslant 2\)
  2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that $$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$ where \(c\) is an arbitrary constant. $$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
  3. Show that, for \(n \geqslant 2\) $$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
  4. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that
Edexcel F3 2020 June Q5
12 marks Challenging +1.2
5. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1\) The line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants. Given that \(l\) is a tangent to \(H\),
  1. show that \(25 m ^ { 2 } = 4 + c ^ { 2 }\)
  2. Hence find the equations of the tangents to \(H\) that pass through the point ( 1,2 ).
  3. Find the coordinates of the point of contact each of these tangents makes with \(H\).
Edexcel F3 2020 June Q6
8 marks Challenging +1.2
6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{array} \right) \quad a \neq 1$$
  1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
    . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
  2. Find a vector equation for the line \(l _ { 1 }\)
Edexcel F3 2020 June Q7
12 marks
7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
  2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
  3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.
Edexcel F3 2020 June Q8
10 marks Standard +0.8
8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).
  1. Find a vector equation for \(l\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\) The line \(n\), which passes through \(Q\), is also parallel to \(l\).
  2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
    VIIV STHI NI JINM ION OCVIAV SIHI NI JMAM/ION OCVIAV SIHL NI JIIYM ION OO
Edexcel F3 2021 June Q1
6 marks Standard +0.3
  1. (a) Using the definitions of hyperbolic functions in terms of exponentials, show that
$$1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x$$ (b) Solve the equation $$2 \operatorname { sech } ^ { 2 } x + 3 \tanh x = 3$$ giving your answer as an exact logarithm. \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-03_2635_74_123_116}
Edexcel F3 2021 June Q2
7 marks Challenging +1.2
2. A curve has equation $$y = \sqrt { 9 - x ^ { 2 } } \quad 0 \leqslant x \leqslant 3$$
  1. Using calculus, show that the length of the curve is \(\frac { 3 \pi } { 2 }\) The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Using calculus, find the exact area of the surface generated. \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-07_2647_1840_118_111}
Edexcel F3 2021 June Q3
9 marks Standard +0.3
3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & p \\ 1 & 1 & 2 \\ - 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant (a) Find the exact values of \(p\) for which \(\mathbf { M }\) has no inverse. Given that \(\mathbf { M }\) does have an inverse, (b) find \(\mathbf { M } ^ { - 1 }\) in terms of \(p\).
3. \(\mathbf { M } = \left( \begin{array} { r c c } 3 & 1 & p \\ 1 & 1 & 2 \\ - 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-11_2647_1840_118_111}
Edexcel F3 2021 June Q4
8 marks Challenging +1.3
4. (i) $$f ( x ) = x \arccos x \quad - 1 \leqslant x \leqslant 1$$ Find the exact value of \(f ^ { \prime } ( 0.5 )\).
(ii) $$\mathrm { g } ( x ) = \arctan \left( \mathrm { e } ^ { 2 x } \right)$$ Show that $$\mathrm { g } ^ { \prime \prime } ( x ) = k \operatorname { sech } ( 2 x ) \tanh ( 2 x )$$ where \(k\) is a constant to be found. \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-15_2647_1840_119_114}
  1. Prove that for \(n \geqslant 2\) $$( n - 1 ) I _ { n } = \tan x \sec ^ { n - 2 } x + ( n - 2 ) I _ { n - 2 }$$
  2. Hence, showing each step of your working, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 6 } x d x$$ $$I _ { n } = \int \sec ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$ Prove that for \(n > 2\)
Edexcel F3 2021 June Q6
13 marks Standard +0.8
  1. The line \(l _ { 1 }\) has equation
$$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathrm { r } = 2 \mathbf { i } + s \mathbf { j } + \mu ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) both lie in a common plane \(\Pi _ { 1 }\)
  1. show that an equation for \(\Pi _ { 1 }\) is \(3 x + y - z = 3\)
  2. find the value of \(s\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 3\)
  3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
  4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in degrees to 3 significant figures.
Edexcel F3 2021 June Q7
8 marks Standard +0.8
  1. Using calculus, find the exact values of
    1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
    2. \(\int _ { \sqrt { 3 } } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 3 } } { x ^ { 2 } } \mathrm {~d} x\)
    \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-27_2647_1840_118_111}
Edexcel F3 2021 June Q8
14 marks Challenging +1.2
8. The hyperbola \(H\) has equation $$4 x ^ { 2 } - y ^ { 2 } = 4$$
  1. Write down the equations of the asymptotes of \(H\).
  2. Find the coordinates of the foci of \(H\). The point \(P ( \sec \theta , 2 \tan \theta )\) lies on \(H\).
  3. Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is $$y \tan \theta = 2 x \sec \theta - 2$$ The point \(V ( - 1,0 )\) and the point \(W ( 1,0 )\) both lie on \(H\).
    The point \(Q ( \sec \theta , - 2 \tan \theta )\) also lies on \(H\).
    Given that \(P , Q , V\) and \(W\) are distinct points on \(H\) and that the lines \(V P\) and \(W Q\) intersect at the point \(S\),
  4. show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are integers to be found.
    \includegraphics[max width=\textwidth, alt={}]{d7a92540-e36c-4e00-bbe4-b253e09962f8-32_2647_1835_118_116}
Edexcel F3 2022 June Q1
7 marks Standard +0.3
  1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that
$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)
Edexcel F3 2022 June Q2
9 marks Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Determine $$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
  2. Use the substitution \(x = 3 \sec \theta\) to determine the exact value of $$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ Give your answer in the form \(A + B \sqrt { 3 }\) where \(A\) and \(B\) are constants to be found.