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AQA FP1 2012 January Q2
5 marks Standard +0.3
2 Show that only one of the following improper integrals has a finite value, and find that value:
  1. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 2 } { 3 } } \mathrm {~d} x\);
  2. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 4 } { 3 } } \mathrm {~d} x\).
AQA FP1 2012 January Q3
8 marks Easy -1.2
3
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 9 = 0\);
    2. \(( x + 2 ) ^ { 2 } + 9 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + 2 \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Given that \(z = 1 + 2 \mathrm { i }\), find the value of $$z ^ { * } - z ^ { 3 }$$
AQA FP1 2012 January Q4
7 marks Standard +0.3
4
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 4 r - 3 ) = k n ( n + 1 ) \left( 2 n ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
  2. Hence evaluate $$\sum _ { r = 20 } ^ { 40 } r ^ { 2 } ( 4 r - 3 )$$ (2 marks)
AQA FP1 2012 January Q5
7 marks Moderate -0.3
5 The diagram below (not to scale) shows a part of a curve \(y = \mathrm { f } ( x )\) which passes through the points \(A ( 2 , - 10 )\) and \(B ( 5,22 )\).
    1. On the diagram, draw a line which illustrates the method of linear interpolation for solving the equation \(\mathrm { f } ( x ) = 0\). The point of intersection of this line with the \(x\)-axis should be labelled \(P\).
    2. Calculate the \(x\)-coordinate of \(P\). Give your answer to one decimal place.
    1. On the same diagram, draw a line which illustrates the Newton-Raphson method for solving the equation \(\mathrm { f } ( x ) = 0\), with initial value \(x _ { 1 } = 2\). The point of intersection of this line with the \(x\)-axis should be labelled \(Q\).
    2. Given that the gradient of the curve at \(A\) is 8 , calculate the \(x\)-coordinate of \(Q\). Give your answer as an exact decimal. \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-3_876_1063_1779_523}
AQA FP1 2012 January Q6
7 marks Standard +0.3
6 Find the general solution of each of the following equations:
  1. \(\quad \tan \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { \sqrt { 3 } }\);
  2. \(\quad \tan ^ { 2 } \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { 3 }\).
AQA FP1 2012 January Q7
12 marks Standard +0.3
7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$
  1. Find the equations of the asymptotes of \(H\).
  2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
  3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3 \\ 0 \end{array} \right]\).
    1. Write down the equation of the translated curve.
    2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
  4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}
AQA FP1 2012 January Q8
8 marks Moderate -0.3
8 The diagram below shows a rectangle \(R _ { 1 }\) which has vertices \(( 0,0 ) , ( 3,0 ) , ( 3,2 )\) and \(( 0,2 )\).
  1. On the diagram, draw:
    1. the image \(R _ { 2 }\) of \(R _ { 1 }\) under a rotation through \(90 ^ { \circ }\) clockwise about the origin;
    2. the image \(R _ { 3 }\) of \(R _ { 2 }\) under the transformation which has matrix $$\left[ \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right]$$
  2. Find the matrix of:
    1. the rotation which maps \(R _ { 1 }\) onto \(R _ { 2 }\);
    2. the combined transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}
AQA FP1 2012 January Q9
12 marks Standard +0.3
9 A curve has equation $$y = \frac { x } { x - 1 }$$
  1. Find the equations of the asymptotes of this curve.
  2. Given that the line \(y = - 4 x + c\) intersects the curve, show that the \(x\)-coordinates of the points of intersection must satisfy the equation $$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
  3. It is given that the line \(y = - 4 x + c\) is a tangent to the curve.
    1. Find the two possible values of \(c\).
      (No credit will be given for methods involving differentiation.)
    2. For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.
AQA FP1 2013 January Q1
5 marks Moderate -0.3
1 A curve passes through the point (1,3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { 1 + x ^ { 3 } }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 1.2\). Give your answer to four decimal places.
AQA FP1 2013 January Q2
9 marks Moderate -0.3
2
  1. Solve the equation \(w ^ { 2 } + 6 w + 34 = 0\), giving your answers in the form \(p + q \mathrm { i }\), where \(p\) and \(q\) are integers.
  2. It is given that \(z = \mathrm { i } ( 1 + \mathrm { i } ) ( 2 + \mathrm { i } )\).
    1. Express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are integers.
    2. Find integers \(m\) and \(n\) such that \(z + m z ^ { * } = n \mathrm { i }\).
AQA FP1 2013 January Q3
8 marks Standard +0.3
3
  1. Find the general solution of the equation $$\sin \left( 2 x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
  2. Use your general solution to find the exact value of the greatest solution of this equation which is less than \(6 \pi\).
AQA FP1 2013 January Q4
4 marks Standard +0.3
4 Show that the improper integral \(\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\) has a finite value and find that value.
AQA FP1 2013 January Q5
9 marks Standard +0.8
5 The roots of the quadratic equation $$x ^ { 2 } + 2 x - 5 = 0$$ are \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Calculate the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(\alpha ^ { 3 } \beta + 1\) and \(\alpha \beta ^ { 3 } + 1\).
AQA FP1 2013 January Q6
12 marks Moderate -0.3
6
  1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2 \\ 3 & 0 \end{array} \right]\).
    1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2 \\ 3 & 6 \end{array} \right]\), find the value of \(m\).
    2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right]\).
    1. Describe the geometrical transformation represented by \(\mathbf { A }\).
    2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1 \\ 1 & 1 \end{array} \right]\), where \(k\) is a surd.
    3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\). \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).
AQA FP1 2013 January Q8
8 marks Standard +0.8
8
  1. Show that $$\sum _ { r = 1 } ^ { n } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right) = n ( n + p ) ( n + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
  2. Hence find the value of $$\sum _ { r = 11 } ^ { 20 } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right)$$ (2 marks)
AQA FP1 2013 January Q9
13 marks Standard +0.8
9 An ellipse is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699} The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is $$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$
  1. Find the \(x\)-coordinates of \(A\) and \(B\).
  2. The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
    1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
    2. Hence find the exact value of \(m\).
    3. Find the coordinates of \(P\).
AQA FP1 2007 June Q1
6 marks Moderate -0.3
1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 1 \\ 3 & 8 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right]$$ The matrix \(\mathbf { M } = \mathbf { A } - 2 \mathbf { B }\).
  1. Show that \(\mathbf { M } = n \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]\), where \(n\) is a positive integer.
    (2 marks)
  2. The matrix \(\mathbf { M }\) represents a combination of an enlargement of scale factor \(p\) and a reflection in a line \(L\). State the value of \(p\) and write down the equation of \(L\).
  3. Show that $$\mathbf { M } ^ { 2 } = q \mathbf { I }$$ where \(q\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
AQA FP1 2007 June Q2
7 marks Easy -1.2
2
  1. Show that the equation $$x ^ { 3 } + x - 7 = 0$$ has a root between 1.6 and 1.8.
  2. Use interval bisection twice, starting with the interval in part (a), to give this root to one decimal place.
AQA FP1 2007 June Q3
6 marks Moderate -0.5
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$z - 3 \mathbf { i } z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
  2. Find the complex number \(z\) such that $$z - 3 \mathrm { i } z ^ { * } = 16$$
AQA FP1 2007 June Q4
7 marks Standard +0.3
4 The quadratic equation $$2 x ^ { 2 } - x + 4 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 1 } { 4 }\).
  3. Find a quadratic equation with integer coefficients such that the roots of the equation are $$\frac { 4 } { \alpha } \text { and } \frac { 4 } { \beta }$$ (3 marks)
AQA FP1 2007 June Q5
11 marks Moderate -0.3
5 [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 $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants. The following approximate values of \(x\) and \(y\) have been found.
\(x\)1234
\(y\)3.846.149.8215.7
  1. Complete the table in Figure 1, showing values of \(x\) and \(Y\), where \(Y = \log _ { 10 } y\). Give each value of \(Y\) to three decimal places.
  2. Show that, if \(y = a b ^ { x }\), then \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
  3. Draw on Figure 2 a linear graph relating \(x\) and \(Y\).
  4. Hence find estimates for the values of \(a\) and \(b\).
AQA FP1 2007 June Q6
6 marks Moderate -0.3
6 Find the general solution of the equation $$\sin \left( 2 x - \frac { \pi } { 2 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
AQA FP1 2007 June Q7
9 marks Moderate -0.3
7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$
  1. Write down the equations of the two asymptotes to the curve.
  2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
  3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$
AQA FP1 2007 June Q8
8 marks Challenging +1.2
8 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x\);
  2. \(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).
AQA FP1 2007 June Q9
15 marks Standard +0.3
9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation $$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$ and the straight line with equation $$x + y = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}
  1. Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
  2. The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
  3. Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation $$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
  4. Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
  5. On Figure 3, show the translated curves corresponding to these two values of \(k\). \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2 (for use in Question 5)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3 (for use in Question 9)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
    \end{figure}