AQA FP1 (Further Pure Mathematics 1) 2014 June

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]

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 \(\alpha \beta\).
2. 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 $$2 \alpha ^ { 4 } + \frac { 1 } { \beta ^ { 2 } } \text { and } 2 \beta ^ { 4 } + \frac { 1 } { \alpha ^ { 2 } }$$
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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 $$\frac { 1 } { x ( x + 2 ) } \leqslant \frac { 1 } { 8 }$$

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]

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]

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]
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