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Edexcel M4 Q7
12 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-38_451_1077_315_370} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).
  1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
  2. Find the value of \(\theta\) for which the system is in equilibrium.
  3. Determine the stability of this position of equilibrium. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).
  1. Find, in terms of \(V\), the speed of \(S\)
    1. immediately before the collision,
    2. immediately after the collision.
  2. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
AQA FP1 2005 January Q1
7 marks Standard +0.3
1 The equation $$x ^ { 2 } - 5 x - 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } \beta + \alpha \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots $$\alpha ^ { 2 } \beta \quad \text { and } \quad \alpha \beta ^ { 2 }$$
AQA FP1 2005 January Q2
8 marks Moderate -0.5
2 A curve has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
  1. Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
  2. Calculate the \(y\)-coordinates of the points of intersection of the curve with the line \(x = 1\). Give your answers in the form \(p \sqrt { 2 }\), where \(p\) is a rational number.
  3. The curve is translated one unit in the positive \(x\) direction. Write down the equation of the curve after the translation.
AQA FP1 2005 January Q3
6 marks Easy -1.2
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Write down, in terms of \(x\) and \(y\), an expression for \(z ^ { * }\), the complex conjugate of \(z\).
  2. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$2 z - \mathrm { i } z ^ { * }$$
  3. Find the complex number \(z\) such that $$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$
AQA FP1 2005 January Q4
7 marks Standard +0.3
4 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x\);
    (3 marks)
  2. \(\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x\);
  3. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x\).
AQA FP1 2005 January Q5
8 marks Moderate -0.3
5
  1. The transformation \(T _ { 1 }\) is defined by the matrix $$\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]$$ Describe this transformation geometrically.
  2. The transformation \(T _ { 2 }\) is an anticlockwise rotation about the origin through an angle of \(60 ^ { \circ }\). Find the matrix of the transformation \(T _ { 2 }\). Use surds in your answer where appropriate.
    (3 marks)
  3. Find the matrix of the transformation obtained by carrying out \(T _ { 1 }\) followed by \(T _ { 2 }\).
    (3 marks)
AQA FP1 2005 January Q6
8 marks Standard +0.3
6 The angle \(x\) radians satisfies the equation $$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$
  1. Find the general solution of this equation, giving the roots as exact values in terms of \(\pi\).
  2. Find the number of roots of the equation which lie between 0 and \(2 \pi\).
AQA FP1 2005 January Q7
Moderate -0.3
7 [Figure 1, printed on the insert, is provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y ^ { 3 } = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants. Experimental evidence has provided the following approximate values:
\(x\)1.54.05.06.58.0
\(y\)5.06.37.08.09.0
  1. On Figure 1, draw a linear graph connecting the variables \(X\) and \(Y\), where $$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
  2. From your graph, find approximate values for the constants \(a\) and \(b\).
AQA FP1 2005 January Q8
Moderate -0.3
8 [Figure 2, printed on the insert, is provided for use in this question.]
The diagram shows a part of the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 3 } - 2 x - 1$$ The point \(P\) has coordinates \(( 1 , - 2 )\). \includegraphics[max width=\textwidth, alt={}, center]{a77cc9c3-5ff6-4abc-931e-e811740267f2-05_606_565_717_740}
  1. Taking \(x _ { 1 } = 1\) as a first approximation to a root of the equation \(\mathrm { f } ( x ) = 0\), use the NewtonRaphson method to find a second approximation, \(x _ { 2 }\), to the root.
  2. On Figure 2, draw a straight line to illustrate the Newton-Raphson method as used in part (a). Mark \(x _ { 1 }\) and \(x _ { 2 }\) on Figure 2
  3. By considering \(f ( 2 )\), show that the second approximation found in part (a) is not as good as the first approximation.
  4. Taking \(x _ { 1 } = 1.6\) as a first approximation to the root, use the Newton-Raphson method to find a second approximation to the root. Give your answer to three decimal places.
    (2 marks)
AQA FP1 2005 January Q9
11 marks Standard +0.3
9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$
  1. Write down the equations of the two asymptotes to the curve \(y = \mathrm { f } ( x )\).
  2. By considering the expression \(x ^ { 2 } + 2 x + 2\) :
    1. show that the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis;
    2. find the non-real roots of the equation \(\mathrm { f } ( x ) = 0\).
    1. Show that, if the equation \(\mathrm { f } ( x ) = k\) has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
    2. Deduce that the graph of \(y = \mathrm { f } ( x )\) has exactly one stationary point and find its coordinates.
AQA FP1 2008 January Q1
4 marks Moderate -0.8
1 It is given that \(z _ { 1 } = 2 + \mathrm { i }\) and that \(z _ { 1 } { } ^ { * }\) is the complex conjugate of \(z _ { 1 }\).
Find the real numbers \(x\) and \(y\) such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$
AQA FP1 2008 January Q2
5 marks Moderate -0.3
2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x }$$ Starting at the point \(( 1,4 )\) on the curve, use a step-by-step method with a step length of 0.01 to estimate the value of \(y\) at \(x = 1.02\). Give your answer to six significant figures.
AQA FP1 2008 January Q3
5 marks Moderate -0.8
3 Find the general solution of the equation $$\tan 4 \left( x - \frac { \pi } { 8 } \right) = 1$$ giving your answer in terms of \(\pi\).
AQA FP1 2008 January Q4
7 marks Standard +0.3
4
  1. Find $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 6 r \right)$$ expressing your answer in the form $$k n ( n + 1 ) ( n + p ) ( n + q )$$ where \(k\) is a fraction and \(p\) and \(q\) are integers.
  2. It is given that $$S = \sum _ { r = 1 } ^ { 1000 } \left( r ^ { 3 } - 6 r \right)$$ Without calculating the value of \(S\), show that \(S\) is a multiple of 2008 .
AQA FP1 2008 January Q5
8 marks Standard +0.3
5 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$ and its asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}
  1. Write down the equations of the two asymptotes.
  2. The points on the hyperbola for which \(x = 4\) are denoted by \(A\) and \(B\). Find, in surd form, the \(y\)-coordinates of \(A\) and \(B\).
  3. The hyperbola and its asymptotes are translated by two units in the positive \(y\) direction. Write down:
    1. the \(y\)-coordinates of the image points of \(A\) and \(B\) under this translation;
    2. the equations of the hyperbola and the asymptotes after the translation.
AQA FP1 2008 January Q6
10 marks Standard +0.3
6 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3 \\ 3 & - \sqrt { 3 } \end{array} \right]$$
    1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    2. Show that the matrix \(\mathbf { M }\) can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ } \\ \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where \(q\) is a real number. Give the value of \(q\) in surd form.
  2. The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection. Find:
    1. the scale factor of the enlargement;
    2. the equation of the mirror line of the reflection.
  3. Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).
AQA FP1 2008 January Q7
12 marks Moderate -0.3
7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows the curve $$y = x ^ { 3 } - x + 1$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates - 1 and \(- 1 + h\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}
    1. Show that the \(y\)-coordinate of the point \(B\) is $$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
    2. Find the gradient of the chord \(A B\) in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
    3. Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at \(A\). State the value of this gradient.
  2. The equation \(x ^ { 3 } - x + 1 = 0\) has one real root, \(\alpha\).
    1. Taking \(x _ { 1 } = - 1\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\).
    2. On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points \(\left( x _ { 2 } , 0 \right)\) and \(( \alpha , 0 )\) on your diagram.
AQA FP1 2008 January Q8
12 marks Standard +0.8
8
    1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
    2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
  2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
  3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$
AQA FP1 2008 January Q9
12 marks Challenging +1.2
9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$
  1. Write down the equations of the three asymptotes of \(C\).
  2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
    (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve \(C\).
AQA FP1 2010 January Q1
9 marks Standard +0.8
1 The quadratic equation $$3 x ^ { 2 } - 6 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 6\).
  3. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\).
AQA FP1 2010 January Q2
6 marks Moderate -0.8
2 The complex number \(z\) is defined by $$z = 1 + \mathrm { i }$$
  1. Find the value of \(z ^ { 2 }\), giving your answer in its simplest form.
  2. Hence show that \(z ^ { 8 } = 16\).
  3. Show that \(\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }\).
AQA FP1 2010 January Q3
4 marks Easy -1.2
3 Find the general solution of the equation $$\sin \left( 4 x + \frac { \pi } { 4 } \right) = 1$$
AQA FP1 2010 January Q4
7 marks Standard +0.3
4 It is given that $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 4 \\ 3 & 1 \end{array} \right]$$ and that \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  1. Show that \(( \mathbf { A } - \mathbf { I } ) ^ { 2 } = k \mathbf { I }\) for some integer \(k\).
  2. Given further that $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 3 \\ p & 1 \end{array} \right]$$ find the integer \(p\) such that $$( \mathbf { A } - \mathbf { B } ) ^ { 2 } = ( \mathbf { A } - \mathbf { I } ) ^ { 2 }$$
AQA FP1 2010 January Q5
7 marks Standard +0.3
5
  1. Explain why \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\) is an improper integral.
  2. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\);
    2. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\).
AQA FP1 2010 January Q6
8 marks Moderate -0.3
6 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a rectangle \(R _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}
  1. The rectangle \(R _ { 1 }\) is mapped onto a second rectangle, \(R _ { 2 }\), by a transformation with matrix \(\left[ \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right]\).
    1. Calculate the coordinates of the vertices of the rectangle \(R _ { 2 }\).
    2. On Figure 1, draw the rectangle \(R _ { 2 }\).
  2. The rectangle \(R _ { 2 }\) is rotated through \(90 ^ { \circ }\) clockwise about the origin to give a third rectangle, \(R _ { 3 }\).
    1. On Figure 1, draw the rectangle \(R _ { 3 }\).
    2. Write down the matrix of the rotation which maps \(R _ { 2 }\) onto \(R _ { 3 }\).
  3. Find the matrix of the transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).