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AQA FP1 2013 June Q8
6 marks Standard +0.3
8 The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-09_972_967_358_589}
  1. Find the matrix which represents the stretch that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 2 }\).
  2. The triangle \(T _ { 2 }\) is reflected in the line \(y = \sqrt { 3 } x\) to give a third triangle, \(T _ { 3 }\). Find, using surd forms where appropriate:
    1. the matrix which represents the reflection that maps triangle \(T _ { 2 }\) onto triangle \(T _ { 3 }\);
    2. the matrix which represents the combined transformation that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 3 }\).
      (2 marks)
AQA FP1 2013 June Q9
14 marks Challenging +1.2
9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
  1. Find the equations of the three asymptotes of the curve.
    1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
    2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
    3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve and its asymptotes.
AQA FP1 2014 June Q1
5 marks Standard +0.3
1 A curve passes through the point \(( 9,6 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 + \sqrt { x } }$$ Use a step-by-step method with a step length of 0.25 to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places.
[0pt] [5 marks]
AQA FP1 2014 June Q2
11 marks Standard +0.8
2 The quadratic equation $$2 x ^ { 2 } + 8 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
    1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
    2. Hence, or otherwise, show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \frac { 449 } { 2 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots $$2 \alpha ^ { 4 } + \frac { 1 } { \beta ^ { 2 } } \text { and } 2 \beta ^ { 4 } + \frac { 1 } { \alpha ^ { 2 } }$$
    \includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-04_2443_1707_260_153}
AQA FP1 2014 June Q4
6 marks Moderate -0.5
4 Find the complex number \(z\) such that $$5 \mathrm { i } z + 3 z ^ { * } + 16 = 8 \mathrm { i }$$ Give your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.
[0pt] [6 marks] \(5 \quad\) A curve \(C\) has equation \(y = x ( x + 3 )\).
  1. Find the gradient of the line passing through the point ( \(- 5,10\) ) and the point on \(C\) with \(x\)-coordinate \(- 5 + h\). Give your answer in its simplest form.
  2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \(( - 5,10 )\). State the value of this gradient.
    [0pt] [2 marks] \(6 \quad\) A curve \(C\) has equation \(y = \frac { 1 } { x ( x + 2 ) }\).
  3. Write down the equations of all the asymptotes of \(C\).
  4. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is - 1 .
    1. Find the \(y\)-coordinate of the stationary point.
    2. Sketch the curve \(C\).
  5. Solve the inequality $$\frac { 1 } { x ( x + 2 ) } \leqslant \frac { 1 } { 8 }$$
AQA FP1 2014 June Q7
10 marks Moderate -0.3
7
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a reflection in the line \(y = - x\);
    2. a stretch parallel to the \(y\)-axis of scale factor 7 .
  2. Hence find the matrix corresponding to the combined transformation of a reflection in the line \(y = - x\) followed by a stretch parallel to the \(y\)-axis of scale factor 7 .
  3. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 3 & - \sqrt { 3 } \\ - \sqrt { 3 } & 3 \end{array} \right]\).
    1. Show that \(\mathbf { A } ^ { 2 } = k \mathbf { I }\), where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    2. Show that the matrix \(\mathbf { A }\) corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form \(y = ( \tan \theta ) x\).
      [0pt] [5 marks]
AQA FP1 2014 June Q8
9 marks Standard +0.8
8
  1. Find the general solution of the equation $$\cos \left( \frac { 5 } { 4 } x - \frac { \pi } { 3 } \right) = \frac { \sqrt { 2 } } { 2 }$$ giving your answer for \(x\) in terms of \(\pi\).
  2. Use your general solution to find the sum of all the solutions of the equation \(\cos \left( \frac { 5 } { 4 } x - \frac { \pi } { 3 } \right) = \frac { \sqrt { 2 } } { 2 }\) that lie in the interval \(0 \leqslant x \leqslant 20 \pi\). Give your answer in the form \(k \pi\), stating the exact value of \(k\).
    [0pt] [4 marks]
AQA FP1 2014 June Q9
15 marks Standard +0.8
9 An ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$
  1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes.
    [0pt] [2 marks]
  2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(- 5 < k < 5\).
    [0pt] [5 marks]
  3. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { l } a \\ b \end{array} \right]\) to form another ellipse whose equation is \(9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c\). Find the values of the constants \(a , b\) and \(c\).
    [0pt] [5 marks]
  4. Hence find an equation for each of the two tangents to the ellipse \(9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c\) that are parallel to the line \(y = x\).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-10_1438_1707_1265_153}
AQA FP1 2015 June Q1
9 marks Standard +0.3
1 The quadratic equation \(2 x ^ { 2 } + 6 x + 7 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 } - 1\) and \(\beta ^ { 2 } - 1\).
  3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
AQA FP1 2015 June Q2
5 marks Challenging +1.2
2
  1. Explain why \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) is an improper integral.
  2. Either find the value of the integral \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) or explain why it does not have a finite value.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-04_1970_1712_737_150}
AQA FP1 2015 June Q3
11 marks Standard +0.3
3
  1. Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) can be expressed in the form \(2 + b \mathrm { i }\), where \(b\) is an integer.
  2. It is given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 3 } + p z + q = 0$$ where \(p\) and \(q\) are real numbers.
    1. Show that \(p = - 11\) and find the value of \(q\).
    2. Given that \(2 - \mathrm { i }\) is also a root of \(z ^ { 3 } + p z + q = 0\), find a quadratic factor of \(z ^ { 3 } + p z + q\) with real coefficients.
    3. Find the real root of the equation \(z ^ { 3 } + p z + q = 0\).
      \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-06_1568_1707_1139_155}
AQA FP1 2015 June Q4
6 marks Moderate -0.3
4
  1. Find the general solution, in degrees, of the equation $$2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1$$
  2. Use your general solution to find the solution of \(2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1\) that is closest to \(200 ^ { \circ }\).
    [0pt] [1 mark]
AQA FP1 2015 June Q5
13 marks
5
  1. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]\).
    Given that the image of the point \(( 5,2 )\) under the transformation represented by \(\mathbf { A }\) is \(( - 2,1 )\), find the value of \(c\) and the value of \(d\).
    [0pt] [4 marks]
  2. The matrix \(\mathbf { B }\) is defined by \(\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]\).
    1. Show that \(\mathbf { B } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    2. Describe the transformation represented by the matrix \(\mathbf { B }\) as a combination of two geometrical transformations.
    3. Find the matrix \(\mathbf { B } ^ { 17 }\). \(6 \quad \mathrm {~A}\) curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$
AQA FP1 2015 June Q7
15 marks Moderate -0.3
7
  1. The equation \(2 x ^ { 3 } + 5 x ^ { 2 } + 3 x - 132000 = 0\) has exactly one real root \(\alpha\).
    1. Show that \(\alpha\) lies in the interval \(39 < \alpha < 40\).
    2. Taking \(x _ { 1 } = 40\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to two decimal places.
  2. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } 2 r ( 3 r + 2 ) = n ( n + p ) ( 2 n + q )$$ where \(p\) and \(q\) are integers.
    1. Express \(\log _ { 8 } 4 ^ { r }\) in the form \(\lambda r\), where \(\lambda\) is a rational number.
    2. By first finding a suitable cubic inequality for \(k\), find the greatest value of \(k\) for which \(\sum _ { r = k + 1 } ^ { 60 } ( 3 r + 2 ) \log _ { 8 } 4 ^ { r }\) is greater than 106060.
      [0pt] [4 marks]
AQA FP1 2015 June Q8
11 marks Challenging +1.2
8 A curve \(C\) has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$
  1. State the equation of the asymptote of \(C\).
  2. The line \(y = k\) intersects the curve \(C\). Show that \(4 k ^ { 2 } - 4 k - 3 \leqslant 0\).
  3. Hence find the coordinates of the stationary points of the curve \(C\). (No credit will be given for solutions based on differentiation.) \includegraphics[max width=\textwidth, alt={}, center]{e45b07a3-e303-4caf-8f3a-5341bad7560a-24_2488_1728_219_141}
AQA FP1 2016 June Q1
7 marks
1 The quadratic equation \(x ^ { 2 } - 6 x + 14 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
    [0pt] [5 marks] \(2 \quad\) A curve \(C\) has equation \(y = ( 2 - x ) ( 1 + x ) + 3\).
  3. A line passes through the point \(( 2,3 )\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form.
  4. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \(( 2,3 )\). State the value of this gradient.
    [0pt] [2 marks]
AQA FP1 2016 June Q3
4 marks Moderate -0.5
3 The variables \(y\) and \(x\) are related by an equation of the form $$y = a \left( b ^ { x } \right)$$ where \(a\) and \(b\) are positive constants.
Let \(Y = \log _ { 10 } y\).
  1. Show that there is a linear relationship between \(Y\) and \(x\).
  2. The graph of \(Y\) against \(x\), shown below, passes through the points ( \(0,2.5\) ) and (5, 0.5). \includegraphics[max width=\textwidth, alt={}, center]{7e7eaea5-22ca-4418-8ac6-351ce9ac09ea-06_433_506_904_776}
    1. Find the gradient of the line.
    2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]
AQA FP1 2016 June Q4
2 marks Standard +0.3
4
  1. Given that \(\sin \frac { \pi } { 3 } = \cos \frac { \pi } { k }\), state the value of the integer \(k\).
  2. Hence, or otherwise, find the general solution of the equation $$\cos \left( 2 x - \frac { 5 \pi } { 6 } \right) = \sin \frac { \pi } { 3 }$$ giving your answer, in its simplest form, in terms of \(\pi\).
  3. Hence, given that \(\cos \left( 2 x - \frac { 5 \pi } { 6 } \right) = \sin \frac { \pi } { 3 }\), show that there is only one finite value for \(\tan x\) and state its exact value.
    [0pt] [2 marks]
AQA FP1 2016 June Q5
4 marks Standard +0.8
5
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(\sum _ { r = 1 } ^ { n } ( 6 r - 3 ) ^ { 2 } = 3 n \left( 4 n ^ { 2 } - 1 \right)\).
  2. Hence express \(\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } ( 6 r - 3 ) ^ { 2 }\) as a product of four linear factors in terms of \(n\).
    [0pt] [4 marks]
AQA FP1 2016 June Q6
6 marks Standard +0.3
6 A parabola with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant, is translated by the vector \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right]\) to give the curve \(C\). The curve \(C\) passes through the point (4, 7).
  1. Show that \(a = 2\).
  2. Find the values of \(k\) for which the line \(k y = x\) does not meet the curve \(C\).
    [0pt] [6 marks]
AQA FP1 2016 June Q7
11 marks Standard +0.8
7
  1. Solve the equation \(x ^ { 2 } + 4 x + 20 = 0\), giving your answers in the form \(c + d \mathrm { i }\), where \(c\) and \(d\) are integers.
  2. The roots of the quadratic equation $$z ^ { 2 } + ( 4 + i + q i ) z + 20 = 0$$ are \(w\) and \(w ^ { * }\).
    1. In the case where \(q\) is real, explain why \(q\) must be - 1 .
    2. In the case where \(w = p + 2 \mathrm { i }\), where \(p\) is real, find the possible values of \(q\).
      [0pt] [5 marks] \(8 \quad\) The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right]\).
    1. Find the matrix \(\mathbf { A } ^ { 2 }\).
    2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
  4. Given that the matrix \(\mathbf { B }\) represents a reflection in the line \(x + \sqrt { 3 } y = 0\), find the matrix \(\mathbf { B }\), giving the exact values of any trigonometric expressions.
  5. Hence find the coordinates of the point \(P\) which is mapped onto \(( 0 , - 4 )\) under the transformation represented by \(\mathbf { A } ^ { 2 }\) followed by a reflection in the line \(x + \sqrt { 3 } y = 0\).
    [0pt] [6 marks] \(9 \quad\) A curve \(C\) has equation \(y = \frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) }\).
    The line \(L\) has equation \(y = \frac { 1 } { 2 } ( x - 1 )\).
  6. Write down the equations of the asymptotes of \(C\).
  7. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
  8. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes.
  9. Hence solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) } \geqslant \frac { 1 } { 2 } ( x - 1 )\).
AQA FP2 2010 January Q1
9 marks Standard +0.3
1
  1. Use the definitions \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) and \(\sinh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)\) to show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
    1. Express $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x$$ in terms of \(\cosh x\).
    2. Sketch the curve \(y = \cosh x\).
    3. Hence solve the equation $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x = 9.5$$ giving your answers in logarithmic form.
AQA FP2 2010 January Q2
8 marks Standard +0.3
2
  1. On the same Argand diagram, draw:
    1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 4\);
    2. the locus of points satisfying \(| z | = | z - 2 \mathrm { i } |\).
  2. Indicate on your sketch the set of points satisfying both $$| z - 4 + 2 i | \leqslant 4$$ and $$| z | \geqslant | z - 2 \mathrm { i } |$$
AQA FP2 2010 January Q3
14 marks Standard +0.8
3 The cubic equation $$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$ where \(p\) and \(q\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }\).
    1. Write down another root, \(\beta\), of the equation.
    2. Find the third root, \(\gamma\).
    3. Find the values of \(p\) and \(q\).
    1. Express \(\alpha\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. Show that $$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
    3. Show that $$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$ where \(n\) is an integer.
AQA FP2 2010 January Q4
10 marks Challenging +1.2
4 A curve \(C\) is given parametrically by the equations $$x = \frac { 1 } { 2 } \cosh 2 t , \quad y = 2 \sinh t$$
  1. Express $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 }$$ in terms of \(\cosh t\).
  2. The arc of \(C\) from \(t = 0\) to \(t = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
    1. Show that \(S\), the area of the curved surface generated, is given by $$S = 8 \pi \int _ { 0 } ^ { 1 } \sinh t \cosh ^ { 2 } t \mathrm {~d} t$$
    2. Find the exact value of \(S\).