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Edexcel M5 2018 June Q7
16 marks Challenging +1.8
7. A pendulum consists of a uniform circular disc, of radius \(a\) and mass \(4 m\), whose centre is fixed to the end \(B\) of a uniform \(\operatorname { rod } A B\). The rod has mass \(3 m\) and length \(4 l\), where \(2 l > a\). The rod lies in the same plane as the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the disc. The moment of inertia of the pendulum about \(L\) is \(2 m \left( a ^ { 2 } + 40 l ^ { 2 } \right)\).
  1. Find the approximate period of small oscillations of the pendulum about its position of stable equilibrium. The pendulum is held with \(B\) vertically above \(A\) and is then slightly displaced from rest. In the subsequent motion the midpoint of \(A B\) strikes a small peg, which is fixed at the same horizontal level as \(A\), and the pendulum rebounds upwards. Immediately before it strikes the peg, the angular speed of the pendulum is \(\omega\).
  2. Show that \(\omega ^ { 2 } = \frac { 22 g l } { \left( a ^ { 2 } + 40 l ^ { 2 } \right) }\) Immediately after it strikes the peg, the angular speed of the pendulum is \(\frac { 1 } { 2 } \omega\).
  3. Find, in terms of \(m , g , a\) and \(l\), the magnitude of the impulse exerted on the peg by the pendulum.
  4. Show that the size of the angle turned through by the pendulum, between it hitting the peg and it next coming to rest, is \(\arcsin \frac { 1 } { 4 }\).
    \includegraphics[max width=\textwidth, alt={}]{1242d28a-a4bd-4754-ac49-9b48de95b880-24_2632_1830_121_121}
Edexcel M5 Q1
7 marks Standard +0.3
A bead of mass 0.5 kg is threaded on a smooth straight wire. The forces acting on the bead are a constant force \(( 2 \mathbf { i } + 3 \mathbf { j } + \chi \mathbf { k } ) \mathrm { N }\), its weight \(( - 4.9 \mathbf { k } ) \mathrm { N }\), and the reaction on the bead from the wire.
  1. Explain why the reaction on the bead from the wire does no work as the bead moves along the wire. The bead moves from the point \(A\) with position vector \(( \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\) relative to a fixed origin \(O\) to the point \(B\) with position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\). The speed of the bead at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of the bead at \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(x\).
Edexcel M5 Q2
7 marks Challenging +1.8
2. A rod \(A B\) has mass \(m\) and length \(4 a\). It is free to rotate about a fixed smooth horizontal axis through the point \(O\) of the rod, where \(A O = a\). The rod is hanging in equilibrium with \(B\) below \(O\) when it is struck by a particle \(P\), of mass \(3 m\), moving horizontally with speed \(v\). When \(P\) strikes the rod, it adheres to it. Immediately after striking the rod, \(P\) has speed \(\frac { 2 } { 3 } v\). Find the distance from \(O\) of the point where \(P\) strikes the rod.
(7 marks)
Edexcel M5 Q3
10 marks Standard +0.8
3. At time \(t\) seconds, the position vector \(\mathbf { r }\) metres of a particle \(P\), relative to a fixed origin \(O\), satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } + 3 \mathbf { r } = \mathbf { 0 }$$ At time \(t = 0 , P\) is at the point with position vector \(2 \mathbf { i } \mathrm {~m}\) and is moving with velocity \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the position vector of \(P\) when \(t = \ln 2\).
(10 marks)
Edexcel M5 Q4
11 marks Standard +0.8
4. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \mathbf { i } + 4 \mathbf { k } ) \mathrm { N }\). The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(( \mathbf { i } + \mathbf { k } ) \mathrm { m }\) relative to a fixed origin \(O\). The force \(\mathbf { F } _ { 2 }\) acts through the point with position vector ( \(2 \mathbf { j }\) ) m . The two forces are equivalent to a single force \(\mathbf { F }\).
  1. Find the magnitude of \(\mathbf { F }\).
  2. Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), a vector equation of the line of action of \(\mathbf { F }\).
Edexcel M5 Q5
11 marks Challenging +1.8
5. A spaceship is moving in deep space with no external forces acting on it. Initially it has total mass \(M\) and is moving with speed \(V\). The spaceship reduces its speed to \(\frac { 2 } { 3 } V\) by ejecting fuel from its front end with a speed of \(c\) relative to itself and in the same direction as its own motion. Find the mass of fuel ejected.
(11 marks)
Edexcel M5 Q6
12 marks Challenging +1.2
6.
  1. Show by integration that the moment of inertia of a uniform disc, of mass \(m\) and radius \(a\), about an axis through the centre of disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
    (3 marks) \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4e874199-105a-460f-af7c-da0ef1603933-4_887_591_997_812}
    \end{figure} A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). A uniform disc, of mass \(4 m\) and radius \(\frac { 1 } { 2 } a\), is attached to the rod with the centre of the disc lying on the rod a distance \(\frac { 3 } { 2 } a\) from \(A\). The rod lies in the plane of the disc, as shown in Fig. 1. The disc and rod together form a pendulum which is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the pendulum.
  2. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 27 } { 2 } m a ^ { 2 }\). The pendulum makes small oscillations about its position of stable equilibrium.
  3. Show that the motion of the pendulum is approximately simple harmonic, and find the period of the oscillations.
    (6 marks)
Edexcel M5 Q7
17 marks Challenging +1.8
7. A uniform sphere, of mass \(m\) and radius \(a\), is free to rotate about a smooth fixed horizontal axis \(L\) which forms a tangent to the sphere. The sphere is hanging in equilibrium below the axis when it receives an impulse, causing it to rotate about \(L\) with an initial angular velocity of \(\sqrt { \frac { 18 g } { 7 a } }\). Show that, when the sphere has turned through an angle \(\theta\),
  1. the angular speed \(\omega\) of the sphere is given by \(\omega ^ { 2 } = \frac { 2 g } { 7 a } ( 4 + 5 \cos \theta )\),
  2. the angular acceleration of the sphere has magnitude \(\frac { 5 g } { 7 a } \sin \theta\).
  3. Hence find the magnitude of the force exerted by the axis on the sphere when the sphere comes to instantaneous rest for the first time. END
AQA FP1 2006 January Q1
5 marks Moderate -0.8
1
  1. Show that the equation $$x ^ { 3 } + 2 x - 2 = 0$$ has a root between 0.5 and 1 .
  2. Use linear interpolation once to find an estimate of this root. Give your answer to two decimal places.
AQA FP1 2006 January Q2
7 marks Standard +0.8
2
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
    2. \(\int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).
  2. Explain briefly why the integrals in part (a) are improper integrals.
AQA FP1 2006 January Q3
5 marks Moderate -0.8
3 Find the general solution, in degrees, for the equation $$\sin \left( 4 x + 10 ^ { \circ } \right) = \sin 50 ^ { \circ }$$
AQA FP1 2006 January Q4
10 marks Standard +0.3
4 A curve has equation $$y = \frac { 6 x } { x - 1 }$$
  1. Write down the equations of the two asymptotes to the curve.
  2. Sketch the curve and the two asymptotes.
  3. Solve the inequality $$\frac { 6 x } { x - 1 } < 3$$
AQA FP1 2006 January Q5
11 marks Standard +0.3
5
    1. Calculate \(( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )\).
    2. Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation $$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$ where \(z ^ { * }\) is the conjugate of \(z\).
  2. The quadratic equation $$x ^ { 2 } + p x + q = 0$$ in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
    1. Write down the other root of the equation.
    2. Find the sum and product of the two roots of the equation.
    3. Hence state the values of \(p\) and \(q\).
AQA FP1 2006 January Q6
11 marks Moderate -0.5
6 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = k x ^ { n }$$ where \(k\) and \(n\) are constants.
Experimental evidence has provided the following approximate values:
\(x\)417150300
\(y\)1.85.03050
  1. Complete the table in Figure 1, showing values of \(X\) and \(Y\), where $$X = \log _ { 10 } x \quad \text { and } \quad Y = \log _ { 10 } y$$ Give each value to two decimal places.
  2. Show that if \(y = k x ^ { n }\), then \(X\) and \(Y\) must satisfy an equation of the form $$Y = a X + b$$
  3. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
  4. Find an estimate for the value of \(n\).
AQA FP1 2006 January Q7
11 marks Moderate -0.8
7
  1. The transformation T is defined by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]$$
    1. Describe the transformation T geometrically.
    2. Calculate the matrix product \(\mathbf { A } ^ { 2 }\).
    3. Explain briefly why the transformation T followed by T is the identity transformation.
  2. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$$
    1. Calculate \(\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }\).
    2. Calculate \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).
AQA FP1 2006 January Q8
15 marks Standard +0.3
8 A curve has equation \(y ^ { 2 } = 12 x\).
  1. Sketch the curve.
    1. The curve is translated by 2 units in the positive \(y\) direction. Write down the equation of the curve after this translation.
    2. The original curve is reflected in the line \(y = x\). Write down the equation of the curve after this reflection.
    1. Show that if the straight line \(y = x + c\), where \(c\) is a constant, intersects the curve \(y ^ { 2 } = 12 x\), then the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
    2. Hence find the value of \(c\) for which the straight line is a tangent to the curve.
    3. Using this value of \(c\), find the coordinates of the point where the line touches the curve.
    4. In the case where \(c = 4\), determine whether the line intersects the curve or not.
AQA FP1 2007 January Q1
10 marks Easy -1.2
1
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 16 = 0\);
    2. \(x ^ { 2 } - 2 x + 17 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Hence, or otherwise, verify that \(x = 1 + \mathrm { i }\) satisfies the equation $$x ^ { 3 } + 2 x - 4 \mathrm { i } = 0$$
AQA FP1 2007 January Q2
11 marks Moderate -0.3
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , \mathbf { B } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$
  1. Calculate:
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { B A }\).
  2. Describe fully the geometrical transformation represented by each of the following matrices:
    1. \(\mathbf { A }\);
    2. \(\mathbf { B }\);
    3. \(\mathbf { B A }\).
AQA FP1 2007 January Q3
8 marks Moderate -0.3
3 The quadratic equation $$2 x ^ { 2 } + 4 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 1\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 }\).
AQA FP1 2007 January Q4
6 marks Moderate -0.5
4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { b }$$ where \(a\) and \(b\) are constants.
  1. Using logarithms to base 10 , reduce the relation \(y = a x ^ { b }\) to a linear law connecting \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
  2. The diagram shows the linear graph that results from plotting \(\log _ { 10 } y\) against \(\log _ { 10 } x\). \includegraphics[max width=\textwidth, alt={}, center]{49539feb-f842-49f4-b809-72e8147072e7-3_711_1223_1503_411} Find the values of \(a\) and \(b\).
AQA FP1 2007 January Q5
10 marks Standard +0.3
5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$
  1. Write down the equations of the three asymptotes to the curve.
  2. Sketch the curve.
    (You are given that the curve has no stationary points.)
  3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$
AQA FP1 2007 January Q6
10 marks Moderate -0.5
6
    1. Expand \(( 2 r - 1 ) ^ { 2 }\).
    2. Hence show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
  2. Hence find the sum of the squares of the odd numbers between 100 and 200 .
AQA FP1 2007 January Q7
8 marks Moderate -0.3
7 The function f is defined for all real numbers by $$f ( x ) = \sin \left( x + \frac { \pi } { 6 } \right)$$
  1. Find the general solution of the equation \(\mathrm { f } ( x ) = 0\).
  2. The quadratic function g is defined for all real numbers by $$\mathrm { g } ( x ) = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 4 } x ^ { 2 }$$ It can be shown that \(\mathrm { g } ( x )\) gives a good approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
    1. Show that \(\mathrm { g } ( 0.05 )\) and \(\mathrm { f } ( 0.05 )\) are identical when rounded to four decimal places.
    2. A chord joins the points on the curve \(y = \mathrm { g } ( x )\) for which \(x = 0\) and \(x = h\). Find an expression in terms of \(h\) for the gradient of this chord.
    3. Using your answer to part (b)(ii), find the value of \(\mathrm { g } ^ { \prime } ( 0 )\).
AQA FP1 2007 January Q8
12 marks Standard +0.3
8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$
  1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
  3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
    1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
    2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).
AQA FP1 2009 January Q1
5 marks Moderate -0.5
1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.