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Edexcel M5 Specimen Q5
10 marks Challenging +1.3
5. A uniform square lamina \(A B C D\) of side \(a\) and mass \(m\) is free to rotate in vertical plane about a horizontal axis through \(A\). The axis is perpendicular to the plane of the lamina. The lamina is released from rest when \(t = 0\) and \(A C\) makes a small angle with the downward vertical through \(A\).
  1. Show that the moment of inertia of the lamina about the axis is \(\frac { 2 } { 3 } m a ^ { 2 }\).
  2. Show that the motion of the lamina is approximately simple harmonic.
  3. Find the time \(t\) when \(A C\) is first vertical.
Edexcel M5 Specimen Q6
11 marks Challenging +1.2
6. A uniform rod \(A B\) of mass \(m\) and length \(4 a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(O A = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).
  1. Find the magnitude of the angular acceleration of the rod when it is horizontal.
  2. Find the angular speed of the rod when it is horizontal.
  3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal.
    (5)
Edexcel M5 Specimen Q7
12 marks Challenging +1.8
7. As a hailstone falls under gravity in still air, its mass increases. At time \(t\) the mass of the hailstone is \(m\). The hailstone is modelled as a uniform sphere of radius \(r\) such that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = k r$$ where \(k\) is a positive constant.
  1. Show that \(\frac { \mathrm { d } m } { \mathrm {~d} t } = 3 \mathrm {~km}\). Assuming that there is no air resistance,
  2. show that the speed \(v\) of the hailstone at time \(t\) satisfies $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - 3 k v$$ Given that the speed of the hailstone at time \(t = 0\) is \(u\),
  3. find an expression for \(v\) in terms of \(t\).
  4. Hence show that the speed of the hailstone approaches the limiting value \(\frac { g } { 3 k }\).
Edexcel M5 Specimen Q8
13 marks Challenging +1.2
8. A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\) at time \(t \mathrm {~s}\). Given that \(\mathbf { r }\) satisfies the vector differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 9 \mathbf { r } = 8 \sin t \mathbf { i }$$ and that when \(t = 0 \mathrm {~s} , P\) is at \(O\) and moving with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\),
  1. find \(\mathbf { r }\) at time \(t\).
  2. Hence find when \(P\) next returns to \(O\).
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.
AQA FP1 2009 January Q2
5 marks Moderate -0.8
2 The complex number \(2 + 3 \mathrm { i }\) is a root of the quadratic equation $$x ^ { 2 } + b x + c = 0$$ where \(b\) and \(c\) are real numbers.
  1. Write down the other root of this equation.
  2. Find the values of \(b\) and \(c\).
AQA FP1 2009 January Q3
5 marks Moderate -0.5
3 Find the general solution of the equation $$\tan \left( \frac { \pi } { 2 } - 3 x \right) = \sqrt { 3 }$$
AQA FP1 2009 January Q4
7 marks Standard +0.3
4 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right)$$
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = n ^ { 3 }\).
  2. Hence show that \(\sum _ { r = n + 1 } ^ { 2 n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = k n ^ { 3 }\) for some integer \(k\).
AQA FP1 2009 January Q5
12 marks Standard +0.3
5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k \\ k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k \\ k & k \end{array} \right]$$ where \(k\) is a constant.
  1. Find, in terms of \(k\) :
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { A } ^ { 2 }\).
  2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
  3. It is now given that \(k = 1\).
    1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
    2. The matrix \(\mathbf { A }\) represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.