AQA FP1 (Further Pure Mathematics 1) 2016 June

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]

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]

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]

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]

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]

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 \(\mathbf { A } = \left[ \begin{array} { l l } 2 & 0
      0 & 1 \end{array} \right]\).
3. 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 \(\frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) } \geqslant \frac { 1 } { 2 } ( x - 1 )\).