AQA FP1 (Further Pure Mathematics 1) 2015 June

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 \(\beta ^ { 2 }\).

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]
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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 \(z ^ { 3 } + p z + q = 0\).
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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]

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 $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

1. Sketch the curve \(C _ { 1 }\), stating the values of its intercepts with the coordinate axes.
2. The curve \(C _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } k
   0 \end{array} \right]\), where \(k < 0\), to give a curve \(C _ { 2 }\). Given that \(C _ { 2 }\) passes through the origin \(( 0,0 )\), find the equations of the asymptotes of \(C _ { 2 }\).
   [0pt] [3 marks]

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.
3. 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]

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.)
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