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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 \(S\).
AQA FP2 2010 January Q5
8 marks Standard +0.8
5 The sum to \(r\) terms, \(S _ { r }\), of a series is given by $$S _ { r } = r ^ { 2 } ( r + 1 ) ( r + 2 )$$ Given that \(u _ { r }\) is the \(r\) th term of the series whose sum is \(S _ { r }\), show that:
    1. \(u _ { 1 } = 6\);
    2. \(u _ { 2 } = 42\);
    3. \(\quad u _ { n } = n ( n + 1 ) ( 4 n - 1 )\).
  2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } = 3 n ^ { 2 } ( n + 1 ) ( 5 n + 2 )$$
AQA FP2 2010 January Q6
6 marks Standard +0.8
6
  1. Show that the substitution \(t = \tan \theta\) transforms the integral $$\int \frac { \mathrm { d } \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$$ into $$\int \frac { \mathrm { d } t } { 9 + t ^ { 2 } }$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { d \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } = \frac { \pi } { 18 }$$
AQA FP2 2010 January Q7
8 marks Standard +0.8
7 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { k + 1 } = 2 u _ { k } + 1$$
  1. Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = 3 \times 2 ^ { n - 1 } - 1$$
  2. Show that $$\sum _ { r = 1 } ^ { n } u _ { r } = u _ { n + 1 } - ( n + 2 )$$
AQA FP2 2010 January Q8
12 marks Standard +0.8
8
    1. Show that \(\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }\) is a root of the equation \(z ^ { 7 } = 1\).
    2. Write down the five other non-real roots in terms of \(\omega\).
  2. Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
  3. Show that:
    1. \(\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }\);
    2. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\).
AQA FP2 2011 January Q1
5 marks Standard +0.3
1
  1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 4 + 3 \mathrm { i } | = 5$$
    1. Indicate on your diagram the point \(P\) representing \(z _ { 1 }\), where both $$\left| z _ { 1 } - 4 + 3 \mathrm { i } \right| = 5 \quad \text { and } \quad \arg z _ { 1 } = 0$$
    2. Find the value of \(\left| z _ { 1 } \right|\).
AQA FP2 2011 January Q2
6 marks Standard +0.3
2
  1. Given that $$u _ { r } = \frac { 1 } { 6 } r ( r + 1 ) ( 4 r + 11 )$$ show that $$u _ { r } - u _ { r - 1 } = r ( 2 r + 3 )$$
  2. Hence find the sum of the first hundred terms of the series $$1 \times 5 + 2 \times 7 + 3 \times 9 + \ldots + r ( 2 r + 3 ) + \ldots$$
AQA FP2 2011 January Q3
11 marks Standard +0.8
3
  1. Show that \(( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2\).
  2. The cubic equation $$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$ where \(k\) is a real constant, has roots \(\alpha , \beta\) and \(\gamma\).
    It is given that \(\alpha = 1 + \mathrm { i }\).
    1. Find the value of \(k\).
    2. Show that \(\beta + \gamma = 4\).
    3. Find the values of \(\beta\) and \(\gamma\).
AQA FP2 2011 January Q4
11 marks Standard +0.8
4
  1. Prove that the curve $$y = 12 \cosh x - 8 \sinh x - x$$ has exactly one stationary point.
  2. Given that the coordinates of this stationary point are \(( a , b )\), show that \(a + b = 9\).
AQA FP2 2011 January Q5
8 marks Standard +0.8
5
  1. Given that \(u = \sqrt { 1 - x ^ { 2 } }\), find \(\frac { \mathrm { d } u } { \mathrm {~d} x }\).
  2. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 2 } } \sin ^ { - 1 } x \mathrm {~d} x = a \sqrt { 3 } \pi + b$$ where \(a\) and \(b\) are rational numbers.
AQA FP2 2011 January Q6
10 marks Challenging +1.2
6
  1. Given that $$x = \ln ( \sec t + \tan t ) - \sin t$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
  2. A curve is given parametrically by the equations $$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$ The length of the arc of the curve between the points where \(t = 0\) and \(t = \frac { \pi } { 3 }\) is denoted by \(s\). Show that \(s = \ln p\), where \(p\) is an integer.
AQA FP2 2011 January Q7
7 marks Standard +0.8
7
  1. Given that $$\mathrm { f } ( k ) = 12 ^ { k } + 2 \times 5 ^ { k - 1 }$$ show that $$\mathrm { f } ( k + 1 ) - 5 \mathrm { f } ( k ) = a \times 12 ^ { k }$$ where \(a\) is an integer.
  2. Prove by induction that \(12 ^ { n } + 2 \times 5 ^ { n - 1 }\) is divisible by 7 for all integers \(n \geqslant 1\).
AQA FP2 2011 January Q8
17 marks Challenging +1.2
8
  1. Express in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) :
    1. \(\quad 4 ( 1 + i \sqrt { 3 } )\);
    2. \(4 ( 1 - i \sqrt { 3 } )\).
  2. The complex number \(z\) satisfies the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ Show that \(z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }\).
    1. Solve the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. Illustrate the roots on an Argand diagram.
    1. Explain why the sum of the roots of the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ is zero.
    2. Deduce that \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).
AQA FP2 2012 January Q1
8 marks Standard +0.8
1
  1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
  2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).