AQA Further AS Paper 1 (Further AS Paper 1) 2023 June

1 Which expression below is equivalent to \(\tanh x\) ? Circle your answer.
\(\sinh x \cosh x\)
\(\frac { \sinh x } { \cosh x }\)
\(\frac { \cosh x } { \sinh x }\)
\(\sinh x + \cosh x\)

2 The two vectors \(\mathbf { a }\) and \(\mathbf { b }\) are such that \(\mathbf { a } \cdot \mathbf { b } = 0\) State the angle between the vectors \(\mathbf { a }\) and \(\mathbf { b }\)
Circle your answer.
[0pt] [1 mark]
\(0 ^ { \circ } 45 ^ { \circ } 90 ^ { \circ } 180 ^ { \circ }\)

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 3 & 1
0 & 5 \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 0 & 4
7 & 1 \end{array} \right]$$ \section*{Calculate AB} Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l l } 3 & 5
7 & 6 \end{array} \right] \quad \left[ \begin{array} { c c } 0 & 20
21 & 12 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 4
0 & 5 \end{array} \right] \quad \left[ \begin{array} { c c } 7 & 13
35 & 5 \end{array} \right]$$

4 The roots of the equation $$5 x ^ { 3 } + 2 x ^ { 2 } - 3 x + p = 0$$ are \(\alpha , \beta\) and \(\gamma\) Given that \(p\) is a constant, state the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\) Circle your answer.
\(- \frac { 3 } { 5 }\)
\(- \frac { 2 } { 5 }\)
\(\frac { 2 } { 5 }\)
\(\frac { 3 } { 5 }\)

5 The function f is defined by $$f ( x ) = 3 x ^ { 2 } \quad 1 \leq x \leq 5$$ 5

1. Find the mean value of f
   5
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { f } ( x ) + c \quad 1 \leq x \leq 5$$ The mean value of \(g\) is 40
   Calculate the value of the constant \(c\)

6

1. Find and simplify the first five terms in the Maclaurin series for \(\mathrm { e } ^ { 2 x }\)
   6
2. Hence, or otherwise, write down the first five terms in the Maclaurin series for \(\mathrm { e } ^ { - 2 x }\)
   6
3. Hence, or otherwise, show that the Maclaurin series for \(\cosh ( 2 x )\) is $$a + b x ^ { 2 } + c x ^ { 4 } + \ldots$$ where \(a\), \(b\) and \(c\) are rational numbers to be determined.

7

1. Show that, for all integers \(r\), $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) }$$ 7
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } = \frac { a n } { b n + c }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
   7
3. Hence, or otherwise, evaluate $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { 99 \times 101 }$$

8 Abdoallah wants to write the complex number \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) Here is his method: $$\begin{array} { r l r l } r & = \sqrt { ( - 1 ) ^ { 2 } + ( \sqrt { 3 } ) ^ { 2 } } & & \tan \theta = \frac { \sqrt { 3 } } { - 1 }
& = \sqrt { 1 + 3 } & & \Rightarrow
& = \sqrt { 4 } & & \tan \theta = - \sqrt { 3 }
& = 2 & & \theta = \tan ^ { - 1 } ( - \sqrt { 3 } )
& & \theta = - \frac { \pi } { 3 }
& - 1 + i \sqrt { 3 } = 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) \end{array}$$ There is an error in Abdoallah's method. 8

1. Show that Abdoallah's answer is wrong by writing $$2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right)$$ in the form \(x + \mathrm { i } y\)
   Simplify your answer.
   8
2. Explain the error in Abdoallah's method.
   8
3. Express \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\)
   8
4. Write down the complex conjugate of \(- 1 + i \sqrt { 3 }\)

9 The matrix \(\mathbf { M }\) represents the transformation T and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 p + 1 & 12
p + 2 & p ^ { 2 } - 3 \end{array} \right]$$ 9

1. In the case when \(p = 0\) show that the image of the point \(( 4,5 )\) under T is the point \(( 64 , - 7 )\)
   9
2. In the case when \(p = - 2\) find the gradient of the line of invariant points under \(T\)
   9
3. Show that \(p = 3\) is the only real value of \(p\) for which \(\mathbf { M }\) is singular.
   The curve \(C\) has equation $$y = \frac { 3 x ^ { 2 } + m x + p } { x ^ { 2 } + p x + m }$$ where \(m\) and \(p\) are integers.
   The vertical asymptotes of \(C\) are \(x = - 4\) and \(x = - 1\)
   The curve \(C\) is shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-12_867_1102_733_463}

10

1. Write down the equation of the horizontal asymptote of \(C\) 10
2. Find the value of \(m\) and the value of \(p\)
   

   10 10	Hence, or otherwise, write down the coordinates of the \(y\)-intercept of \(C\)
   	Without using calculus, show that the line \(y = - 1\) does not intersect \(C\)

11 A point has Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) 11

1. Express \(r\) in terms of \(x\) and \(y\) 11
2. Express \(x\) in terms of \(r\) and \(\theta\) 11
3. The curve \(C _ { 1 }\) has the polar equation $$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$ 11
4. 1. Show that the Cartesian equation of \(C _ { 1 }\) can be written as $$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$ where \(a\) and \(b\) are integers to be determined.
      11
5. (ii) The curve \(C _ { 2 }\) has the Cartesian equation $$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$ where \(a\) and \(b\) take the same values as in part (c)(i). Describe fully a single transformation that maps the curve \(C _ { 1 }\) onto the curve \(C _ { 2 }\)

12

1. Show that \(( 1 + i ) ^ { 4 } = - 4\)
   12
2. The function f is defined by $$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$ 12
3. 1. Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
      12
4. (ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
   12
5. (iii) One of the linear factors of \(\mathrm { f } ( \mathrm { z } )\) is $$( z - ( 1 + i ) )$$ Write down another linear factor and hence, or otherwise, find a quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
   12
6. (iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
   12
7. (v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.

13

1. Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ [4 marks]
   13
2. Hence, or otherwise, write down a factorised expression for the sum of the first \(2 n\) squares $$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
3. Use the formula in part (a) to write down a factorised expression for the sum of the first \(n\) even squares $$2 ^ { 2 } + 4 ^ { 2 } + 6 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
4. Hence, or otherwise, show that the sum of the first \(n\) odd squares is $$a n ( b n - 1 ) ( b n + 1 )$$ where \(a\) and \(b\) are rational numbers to be determined.

14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers \(a\) and \(b\)
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