AQA Further AS Paper 1 (Further AS Paper 1) Specimen

1 A reflection is represented by the matrix \(\left[ \begin{array} { c c } 1 & 0
0 & - 1 \end{array} \right]\)
State the equation of the line of invariant points. Circle your answer.
[0pt] [1 mark] $$x = 0 \quad y = 0 \quad y = x \quad y = - x$$

2 Find the mean value of \(3 x ^ { 2 }\) over the interval \(1 \leq x \leq 3\) Circle your answer.
[0pt] [1 mark] $$8 \frac { 2 } { 3 } \quad 10 \quad 13 \quad 26$$

3 Find the equations of the asymptotes of the curve \(x ^ { 2 } - 3 y ^ { 2 } = 1\) Circle your answer.
[0pt] [1 mark] $$y = \pm 3 x \quad y = \pm \frac { 1 } { 3 } x \quad y = \pm \sqrt { 3 } x \quad y = \pm \frac { 1 } { \sqrt { 3 } } x$$ Turn over for the next question
\(\mathbf { 4 } \quad \mathbf { A } = \left[ \begin{array} { l l } 1 & 2
1 & k \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { c c } - 1 & 0
0 & 1 \end{array} \right]\)

4

1. Find the value of \(k\) for which matrix \(\mathbf { A }\) is singular. 4
2. Describe the transformation represented by matrix \(\mathbf { B }\). 4
3. 1. Given that \(\mathbf { A }\) and \(\mathbf { B }\) are both non-singular, verify that \(\mathbf { A } ^ { \mathbf { - 1 } } \mathbf { B } ^ { \mathbf { - 1 } } = ( \mathbf { B A } ) ^ { \mathbf { - 1 } }\).
      [0pt] [4 marks]
      4
4. (ii) Prove the result \(\mathbf { M } ^ { - \mathbf { 1 } } \mathbf { N } ^ { - \mathbf { 1 } } = ( \mathbf { N M } ) ^ { - \mathbf { 1 } }\) for all non-singular square matrices \(\mathbf { M }\) and \(\mathbf { N }\) of the same size.
   [0pt] [4 marks]

5 The region bounded by the curve with equation \(y = 3 + \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use integration to show that the volume generated is \(\frac { 125 \pi } { 2 }\)
[0pt] [5 marks]

6

1. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \frac { 1 } { 2 } \ln \left( \frac { 1 + t } { 1 - t } \right)\) where \(t = \tanh x\)
   [0pt] [4 marks]
   6
2. 1. Prove \(\cosh ^ { 3 } x = \frac { 1 } { 4 } \cosh 3 x + \frac { 3 } { 4 } \cosh x\)
      [0pt] [4 marks] 6
3. (ii) Show that the equation \(\cosh 3 x = 13 \cosh x\) has only one positive solution.
   Find this solution in exact logarithmic form.
   [0pt] [4 marks]

7 A lighting engineer is setting up part of a display inside a large building. The diagram shows a plan view of the area in which he is working. He has two lights, which project narrow beams of light. One is set up at a point 3 metres above the point \(A\) and the beam from this light hits the wall 23 metres above the point \(D\). The other is set up 1 metre above the point \(B\) and the beam from this light hits the wall 29 metres above the point \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-10_776_1301_826_392} 7

1. By creating a suitable model, show that the beams of light intersect. 7
2. Find the angle between the two beams of light.
   [0pt] [3 marks]
   7
3. State one way in which the model you created in part (a) could be refined.
   [0pt] [1 mark]

8 A curve has polar equation \(r = 3 + 2 \cos \theta\), where \(0 \leq \theta < 2 \pi\)
8

1. 1. State the maximum and minimum values of \(r\).
      [0pt] [2 marks]
      L
      8
2. (ii) Sketch the curve.
   \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-12_77_832_2037_651} 8
3. The curve \(r = 3 + 2 \cos \theta\) intersects the curve with polar equation \(r = 8 \cos ^ { 2 } \theta\), where \(0 \leq \theta < 2 \pi\) Find all of the points of intersection of the two curves in the form \([ r , \theta ]\).
   [0pt] [6 marks]

9

1. Sketch on the Argand diagram below, the locus of points satisfying the equation \(| z - 2 | = 2\)
   \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-14_1417_1475_790_399} 9
2. Given that \(| z - 2 | = 2\) and \(\arg ( z - 2 ) = - \frac { \pi } { 3 }\), express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
   [0pt] [3 marks]

10

1. Prove that $$6 + 3 \sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 2 ) = ( n + 1 ) ( n + 2 ) ( n + 3 )$$ [6 marks]
   10
2. Alex substituted a few values of \(n\) into the expression \(( n + 1 ) ( n + 2 ) ( n + 3 )\) and made the statement:
   "For all positive integers n, $$6 + 3 \sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 2 )$$ is divisible by \(12 . "\) Disprove Alex's statement.
   [0pt] [2 marks]

11 The equation \(x ^ { 3 } - 8 x ^ { 2 } + c x + d = 0\) where \(c\) and \(d\) are real numbers, has roots \(\alpha , \beta , \gamma\).
When plotted on an Argand diagram, the triangle with vertices at \(\alpha , \beta , \gamma\) has an area of 8 . Given \(\alpha = 2\), find the values of \(c\) and \(d\). Fully justify your solution.
[0pt] [5 marks]

12 A curve, \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }\)
The line \(y = k\) intersects the curve, \(C _ { 1 }\) 12

1. 1. Show that \(( k + 3 ) ( k - 1 ) \geq 0\)
      [0pt] [5 marks]
      12
2. (ii) Hence find the coordinates of the stationary point of \(C _ { 1 }\) that is a maximum point.
   [0pt] [4 marks] 12
3. Show that the curve \(C _ { 2 }\) whose equation is \(y = \frac { 1 } { \mathrm { f } ( x ) }\), has no vertical asymptotes.
   [0pt] [2 marks]
   12
4. State the equation of the line that is a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
   [0pt] [1 mark]