AQA Further AS Paper 1 (Further AS Paper 1) 2019 June

1 Which of the following matrices is an identity matrix?
Circle your answer. $$\left[ \begin{array} { l l } 1 & 1
1 & 1 \end{array} \right] \quad \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 0
0 & 0 \end{array} \right]$$

2 Which of the following expressions is the determinant of the matrix \(\left[ \begin{array} { l l } a & 2
b & 5 \end{array} \right]\) ?
Circle your answer.
\(5 a - 2 b\)
\(2 a - 5 b\)
\(5 b - 2 a\)
\(2 b - 5 a\)
\(3 \quad\) Point \(P\) has polar coordinates \(\left( 2 , \frac { 2 \pi } { 3 } \right)\).
Which of the following are the Cartesian coordinates of \(P\) ?
Circle your answer.
[0pt] [1 mark]
\(( 1 , - \sqrt { 3 } )\)
\(( - \sqrt { 3 } , 1 )\)
\(( \sqrt { 3 } , - 1 )\)
\(( - 1 , \sqrt { 3 } )\) $$r = \frac { k } { \sin \theta }$$ where \(k\) is a positive constant.

4

1. Sketch \(L\). The line \(L\) has polar equation 4 The line \(L\) has polar equation $$r = \frac { k } { \sin \theta }$$ where \(k\) is a positive constant.
   Sketch \(L\).
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-03_94_716_1037_662} 4
2. State the minimum distance between \(L\) and the point \(O\).

5

1. Write down the equations of the asymptotes of \(H\). 5
2. Sketch the hyperbola \(H\) on the axes below, indicating the coordinates of any points of intersection with the coordinate axes. The asymptotes have already been drawn.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-04_693_798_1306_623} 5
3. The finite region bounded by \(H\), the positive \(x\)-axis, the positive \(y\)-axis and the line \(y = a\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Show that the volume of the solid generated is \(m a ^ { 3 }\), where \(m = 3.40\) correct to three significant figures.

6

1. On the axes provided, sketch the graph of $$x = \cosh ( y + b )$$ where \(b\) is a positive constant.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-06_1148_1317_1347_358} 6
2. Determine the minimum distance between the graph of \(x = \cosh ( y + b )\) and the \(y\)-axis.

7

1. Show that $$\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { A } { r ^ { 2 } - 1 }$$ where \(A\) is a constant to be found. 7
2. Hence use the method of differences to show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } \equiv \frac { a n ^ { 2 } + b n + c } { 4 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be found.

8 Given that \(z _ { 1 } = 2 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)\) and \(z _ { 2 } = 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)\)
8

1. Find the value of \(\left| z _ { 1 } z _ { 2 } \right|\) 8
2. Find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) 8
3. Sketch \(z _ { 1 }\) and \(z _ { 2 }\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-10_764_869_1546_587} 8
4. A third complex number \(w\) satisfies both \(| w | = 2\) and \(- \pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(P \widehat { R } Q\). Fully justify your answer.

9

1. Saul is solving the equation $$2 \cosh x + \sinh ^ { 2 } x = 1$$ He writes his steps as follows: $$\begin{aligned} 2 \cosh x + \sinh ^ { 2 } x & = 1
   2 \cosh x + 1 - \cosh ^ { 2 } x & = 1
   2 \cosh x - \cosh ^ { 2 } x & = 0
   \cosh x \neq 0 \therefore 2 - \cosh x & = 0
   \cosh x & = 2
   x & = \pm \cosh ^ { - 1 } ( 2 ) \end{aligned}$$ Identify and explain the error in Saul's method. 9
2. Anna is solving the different equation
   g (b) Anna is solving the different equation $$\sinh ^ { 2 } ( 2 x ) - 2 \cosh ( 2 x ) = 1$$ and finds the correct answers in the form \(x = \frac { 1 } { p } \cosh ^ { - 1 } ( q + \sqrt { r } )\), where \(p , q\) and \(r\) are integers. Find the possible values of \(p , q\) and \(r\).
   Fully justify your answer.

10

1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(\mathrm { e } ^ { x }\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). 10
2. Hence find a trigonometric function for which the first three terms of its Maclaurin series are the same as the first three terms of the Maclaurin series for cosh (ix).
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-15_2488_1716_219_153}

11

1. Curve \(C\) has equation $$y = \frac { x ^ { 2 } + p x - q } { x ^ { 2 } - r }$$ where \(p , q\) and \(r\) are positive constants.
   Write down the equations of its asymptotes.

12 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 2
0 & 3 \end{array} \right]$$ 12

1. Prove by induction that, for all integers \(n \geq 1\), $$\mathbf { A } ^ { n } = \left[ \begin{array} { c c } 1 & 3 ^ { n } - 1
   0 & 3 ^ { n } \end{array} \right]$$ 12
2. Find all invariant lines under the transformation matrix \(A\). Fully justify your answer.
   12
3. Find a line of invariant points under the transformation matrix \(\mathbf { A }\).

13 Line \(l _ { 1 }\) has Cartesian equation $$x - 3 = \frac { 2 y + 2 } { 3 } = 2 - z$$ 13

1. Write the equation of line \(l _ { 1 }\) in the form $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$$ where \(\lambda\) is a parameter and \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.
   13
2. Line \(l _ { 2 }\) passes through the points \(P ( 3,2,0 )\) and \(Q ( n , 5 , n )\), where \(n\) is a constant.
   13
3. 1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
      

      13 (ii) Explain briefly why lines \(l _ { 1 }\) and \(l _ { 2 }\) cannot be parallel.

      13 (iii) Given that \(\theta\) is the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\), show that

      \(\cos \theta = \frac { p } { \sqrt { 34 n ^ { 2 } + q n + 306 } }\)

      where \(p\) and \(q\) are constants to be found.

14 The graph of \(y = x ^ { 3 } - 3 x\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-22_718_771_370_632} The two stationary points have \(x\)-coordinates of - 1 and 1
The cubic equation $$x ^ { 3 } - 3 x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha , \beta\) and \(\gamma\).
The roots \(\alpha\) and \(\beta\) are not real.
14

1. Explain why \(\alpha + \beta = - \gamma\)
   14
2. Find the set of possible values for the real constant \(p\).
   14
3. \(\quad \mathrm { f } ( x ) = 0\) is a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\)
   14
4. 1. Show that the constant term of \(\mathrm { f } ( x )\) is \(p + 2\)
      14
5. (ii) Write down the \(x\)-coordinates of the stationary points of \(y = \mathrm { f } ( x )\)
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-24_2488_1719_219_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin.