AQA Further Paper 1 (Further Paper 1) 2023 June

1 Find the number of solutions of the equation \(\tanh x = \cosh x\)
Circle your answer.
0
1

2
3 2 The diagram below shows a locus on an Argand diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-02_855_962_1085_539} Which of the equations below represents the locus shown above?
Circle your answer.
[0pt] [1 mark]
\(| z - 2 + 3 \mathrm { i } | = 2\)
\(| z + 2 - 3 \mathrm { i } | = 2\)
\(| z - 2 + 3 \mathrm { i } | = 4\)
\(| z + 2 - 3 \mathrm { i } | = 4\)

3 The matrix \(\mathbf { A } = \left[ \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right]\) represents a transformation.
Which one of the points below is an invariant point under this transformation?
Circle your answer.
\(( 1,1 )\)
\(( 0,2 )\)
\(( 3,0 )\)
\(( 2,1 )\)

4 The solution of a second order differential equation is \(\mathrm { f } ( t )\)
The differential equation models heavy damping.
Which one of the statements below could be true?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \mathrm { f } ( t ) = 2 \mathrm { e } ^ { - t } \cos ( 3 t ) + 5 \mathrm { e } ^ { - t } \sin ( 3 t )
& \mathrm { f } ( t ) = 3 \mathrm { e } ^ { - t } + 4 t \mathrm { e } ^ { - t }
& \mathrm { f } ( t ) = 7 \mathrm { e } ^ { - t } + 2 \mathrm { e } ^ { - 2 t }
& \mathrm { f } ( t ) = 8 \mathrm { e } ^ { - t } \cos ( 3 t - 0.1 ) \end{aligned}$$ □
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□
□

5 The function f is defined by $$f ( r ) = 2 ^ { r } ( r - 2 ) \quad ( r \in \mathbb { Z } )$$ 5

1. Show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r 2 ^ { r }$$ 5
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } r 2 ^ { r } = 2 ^ { n + 1 } ( n - 1 ) + 2$$

6 The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \frac { 1 } { 10 } \left[ \begin{array} { c c c } a & a & - 6
0 & 10 & 0
9 & 14 & - 13 \end{array} \right]$$ where \(a\) is a real number. The vectors \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\), and \(\mathbf { v } _ { 3 }\) are eigenvectors of \(\mathbf { M }\)
The corresponding eigenvalues are \(\lambda _ { 1 } , \lambda _ { 2 }\), and \(\lambda _ { 3 }\) respectively.
It is given that \(\lambda _ { 2 } = 1\) and \(\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 1
0
3 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 1
1
1 \end{array} \right]\) and \(\mathbf { v } _ { 3 } = \left[ \begin{array} { l } c
0
1 \end{array} \right]\),
where \(c\) is an integer. 6

1. 1. Find the value of \(\lambda _ { 1 }\)
      6
2. (ii) Find the value of \(a\)
   6
3. Find the integer \(c\) and the value of \(\lambda _ { 3 }\)
   6
4. Find matrices \(\mathbf { U } , \mathbf { D }\) and \(\mathbf { U } ^ { - 1 }\), such that \(\mathbf { D }\) is diagonal and \(\mathbf { M } = \mathbf { U D U } ^ { - 1 }\)

7 The function f is defined by $$f ( x ) = \left| \sin x + \frac { 1 } { 2 } \right| \quad ( 0 \leq x \leq 2 \pi )$$ Find the set of values of \(x\) for which $$f ( x ) \geq \frac { 1 } { 2 }$$ Give your answer in set notation.

8 The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { \sin x } \quad ( 0 \leq x \leq 2 \pi )$$ The diagram below shows the graph of \(y = \mathrm { g } ( x )\)
\includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-09_369_593_548_721} 8

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = \mathrm { g } ( x )\), giving your answers in exact form. 8
2. Use Simpson's rule with 3 ordinates to estimate $$\int _ { 0 } ^ { \pi } g ( x ) d x$$ giving your answer to two decimal places.
   8
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b).

9 The position vectors of the points \(A , B\) and \(C\) are $$\begin{aligned} & \mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }
& \mathbf { b } = - \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k }
& \mathbf { c } = - 2 \mathbf { j } \end{aligned}$$ respectively.
9

1. Find the area of the triangle \(A B C\)
   9
2. The points \(A , B\) and \(C\) all lie in the plane \(\Pi\) Find an equation of the plane \(\Pi\), in the form \(\mathbf { r } \cdot \mathbf { n } = d\)
   \(\mathbf { 9 ( c ) } \quad\) The point \(P\) has position vector \(\mathbf { p } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\)
   Find the exact distance of \(P\) from \(\Pi\)

10 The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 2 & - 1 & 1
- 1 & - 1 & - 2
1 & 2 & c \end{array} \right]$$ where \(c\) is a real number. 10

1. The linear transformation T is represented by the matrix \(\mathbf { M }\)
   Show that, for one particular value of \(c\), the image under \(T\) of every point lies in the plane $$x + 5 y + 3 z = 0$$ State the value of \(c\) for which this occurs.
   10
2. It is given that \(\mathbf { M }\) is a non-singular matrix.
   10
3. 1. State any restrictions on the value of \(c\)
      

      10 (iii) Using your answer from part (b)(ii), solve \(\begin{array} { r } 2 x - y + z = - 3
      - x - y - 2 z = - 6
      x + 2 y + 4 z = 13 \end{array}\)	\(\_\_\_\_\)

11 The function f is defined by $$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } - 51 x - 45 \quad ( x \in \mathbb { R } )$$ 11

1. 1. Fully factorise \(\mathrm { f } ( x )\)
      11
2. (ii) Hence, solve the inequality \(\mathrm { f } ( x ) < 0\)
   11
3. The graph of \(y = \mathrm { f } ( x )\) is translated by the vector \(\left[ \begin{array} { l } 7
   0 \end{array} \right]\)
   The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = \mathrm { g } ( x )\)
   Solve the inequality \(\mathrm { g } ( x ) \leq 0\)

12

1. Starting from the identities for \(\sinh 2 x\) and \(\cosh 2 x\), prove the identity $$\tanh 2 x = \frac { 2 \tanh x } { 1 + \tanh ^ { 2 } x }$$ 12
2. 1. The function f is defined by $$\mathrm { f } ( x ) = \tanh x \quad ( x > 0 )$$ State the range of f
      12
3. (ii) Use part (a) and part (b)(i) to prove that \(\tanh 2 x > \tanh x\) if \(x > 0\)

13 Use l'Hôpital's rule to prove that $$\lim _ { x \rightarrow \pi } \left( \frac { x \sin 2 x } { \cos \left( \frac { x } { 2 } \right) } \right) = - 4 \pi$$

14 The curve \(C\) has polar equation $$r = \frac { 4 } { 5 + 3 \cos \theta } \quad ( - \pi < \theta \leq \pi )$$ 14

1. Show that \(r\) takes values in the range \(\frac { 1 } { k } \leq r \leq k\), where \(k\) is an integer.
   [0pt] [2 marks] 14
2. Find the Cartesian equation of \(C\) in the form \(y ^ { 2 } = \mathrm { f } ( x )\) 14
3. The ellipse \(E\) has equation $$y ^ { 2 } + \frac { 16 x ^ { 2 } } { 25 } = 1$$ Find the transformation that maps the graph of \(E\) onto \(C\)
   [0pt] [4 marks]
   

   15	Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 4 y = \cos 2 x + 5 x\)

16

1. Show that $$\int _ { 0.5 } ^ { 4 } \frac { 1 } { t } \ln t \mathrm {~d} t = a ( \ln 2 ) ^ { 2 }$$ where \(a\) is a rational number to be found.
   16
2. A curve \(C\) is defined parametrically for \(t > 0\) by $$x = 2 t \quad y = \frac { 1 } { 2 } t ^ { 2 } - \ln t$$ The arc formed by the graph of \(C\) from \(t = 0.5\) to \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to generate a surface with area \(S\) Find the exact value of \(S\), giving your answer in the form $$S = \pi \left( b + c \ln 2 + d ( \ln 2 ) ^ { 2 } \right)$$ where \(b , c\) and \(d\) are rational numbers to be found.
   [0pt] [7 marks]
   
   \includegraphics[max width=\textwidth, alt={}]{a9f88195-e545-43f2-a13a-6459d14e1cda-25_2488_1719_219_150}
   Question number Additional page, if required.
   Write the question numbers in the left-hand margin.