Questions Further AS Paper 1 (126 questions)

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AQA Further AS Paper 1 2023 June Q1
1 marks Easy -2.0
1 Which expression below is equivalent to \(\tanh x\) ? Circle your answer. \(\sinh x \cosh x\) \(\frac { \sinh x } { \cosh x }\) \(\frac { \cosh x } { \sinh x }\) \(\sinh x + \cosh x\)
AQA Further AS Paper 1 2023 June Q2
1 marks Easy -1.8
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] \(0 ^ { \circ } 45 ^ { \circ } 90 ^ { \circ } 180 ^ { \circ }\)
AQA Further AS Paper 1 2023 June Q3
1 marks Easy -1.8
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] $$\left[ \begin{array} { l l } 3 & 5 \\ 7 & 6 \end{array} \right] \quad \left[ \begin{array} { c c } 0 & 20 \\ 21 & 12 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 4 \\ 0 & 5 \end{array} \right] \quad \left[ \begin{array} { c c } 7 & 13 \\ 35 & 5 \end{array} \right]$$
AQA Further AS Paper 1 2023 June Q4
1 marks Easy -1.2
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. \(- \frac { 3 } { 5 }\) \(- \frac { 2 } { 5 }\) \(\frac { 2 } { 5 }\) \(\frac { 3 } { 5 }\)
AQA Further AS Paper 1 2023 June Q5
4 marks Moderate -0.8
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 \(c\)
AQA Further AS Paper 1 2023 June Q6
6 marks Moderate -0.3
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.
AQA Further AS Paper 1 2023 June Q7
7 marks Standard +0.3
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 $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { 99 \times 101 }$$
AQA Further AS Paper 1 2023 June Q8
4 marks Moderate -0.3
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 \(- 1 + i \sqrt { 3 }\)
AQA Further AS Paper 1 2023 June Q9
11 marks Standard +0.3
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}
AQA Further AS Paper 1 2023 June Q10
9 marks Standard +0.3
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
  3. 10
  		4. 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 \(C\)
AQA Further AS Paper 1 2023 June Q11
8 marks Moderate -0.5
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 (c) (i) 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 (c) (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 \(C _ { 2 }\)
AQA Further AS Paper 1 2023 June Q12
13 marks Standard +0.3
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 (b) (i) Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\) 12 (b) (ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\) 12 (b) (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 (b) (iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12 (b) (v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.
AQA Further AS Paper 1 2023 June Q13
10 marks Standard +0.3
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.
AQA Further AS Paper 1 2023 June Q14
4 marks Standard +0.8
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 \(b\) \includegraphics[max width=\textwidth, alt={}]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-21_2491_1755_173_123} number Additional page, if required. Write the question numbers in the left-hand margin. \(\_\_\_\_\) number \section*{Additional page, if required. Write the question numbers in the left-hand margin.
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Question numberAdditional page, if required. Write the question numbers in the left-hand margin.
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AQA Further AS Paper 1 2021 June Q1
1 marks Easy -1.8
1 The complex number \(\omega\) is shown below on the Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-02_597_650_632_689} Which of the following complex numbers could be \(\omega\) ?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \cos ( - 2 ) + i \sin ( - 2 ) \\ & \cos ( - 1 ) + i \sin ( - 1 ) \\ & \cos ( 1 ) + i \sin ( 1 ) \\ & \cos ( 2 ) + i \sin ( 2 ) \end{aligned}$$ □
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AQA Further AS Paper 1 2021 June Q2
1 marks Easy -1.8
2 Given that \(\mathrm { f } ( x ) = 3 x - 1\) find the mean value of \(\mathrm { f } ( x )\) over the interval \(4 \leq x \leq 8\) Circle your answer. 6111717
AQA Further AS Paper 1 2021 June Q3
1 marks Moderate -0.3
3 The matrix \(\mathbf { M }\) represents a rotation about the \(x\)-axis. $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a & \frac { \sqrt { 3 } } { 2 } \\ 0 & b & - \frac { 1 } { 2 } \end{array} \right]$$ Which of the following pairs of values is correct?
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } a = \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\ a = \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \square \\ a = - \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\ a = - \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \end{array}$$
AQA Further AS Paper 1 2021 June Q4
1 marks Moderate -0.8
4 The point \(( 2 , - 1 )\) is invariant under the transformation represented by the matrix \(\mathbf { N }\) Which of the following matrices could be \(\mathbf { N }\) ? Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l l } 4 & 6 \\ 2 & 5 \end{array} \right]\) \(\left[ \begin{array} { l l } 6 & 5 \\ 4 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 5 & 2 \\ 6 & 4 \end{array} \right]\) \(\left[ \begin{array} { l l } 2 & 4 \\ 5 & 6 \end{array} \right]\)
AQA Further AS Paper 1 2021 June Q5
2 marks Easy -1.8
5 Show that the vectors \(\left[ \begin{array} { c } 1 \\ - 3 \\ 5 \end{array} \right]\) and \(\left[ \begin{array} { l } 7 \\ 4 \\ 1 \end{array} \right]\) are perpendicular.
AQA Further AS Paper 1 2021 June Q6
2 marks Easy -1.2
6 Prove the identity $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
AQA Further AS Paper 1 2021 June Q7
3 marks Standard +0.3
7 Show that the Maclaurin series for \(\ln ( \mathrm { e } + 2 \mathrm { e } x )\) is $$1 + 2 x - 2 x ^ { 2 } + a x ^ { 3 } - \ldots$$ where \(a\) is to be determined.
AQA Further AS Paper 1 2021 June Q8
6 marks Standard +0.3
8 Stephen is correctly told that \(( 1 + \mathrm { i } )\) and - 1 are two roots of the polynomial equation $$z ^ { 3 } - 2 \mathrm { i } z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are complex numbers.
8
  1. Stephen states that ( \(1 - \mathrm { i }\) ) must also be a root of the equation because roots of polynomial equations occur in conjugate pairs. Explain why Stephen's reasoning is wrong. 8
  2. \(\quad\) Find \(p\) and \(q\)
AQA Further AS Paper 1 2021 June Q9
7 marks Challenging +1.2
9
  1. Use the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = a n ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are constants to be determined.
    [0pt] [4 marks]
    9
  2. Hence, or otherwise, find a fully factorised expression for $$\sum _ { r = n + 1 } ^ { 5 n } r ( r + 3 )$$ $$\mathbf { A } = \left[ \begin{array} { c c } 3 & i - 1 \\ i & 2 \end{array} \right]$$
AQA Further AS Paper 1 2021 June Q10
8 marks Standard +0.8
10
  1. Show that \(\operatorname { det } \mathbf { A } = a + \mathrm { i }\) where \(a\) is an integer to be determined. 10 Matrix A is given by 10
  2. Matrix B is given by $$\mathbf { B } = \left[ \begin{array} { c c } 14 - 2 \mathrm { i } & b \\ c & d \end{array} \right] \quad \text { and } \quad \mathbf { A B } = p$$ where \(b , c , d \in \mathbb { C }\) and \(p \in \mathbb { N }\) Find \(b , c , d\) and \(p\)
AQA Further AS Paper 1 2021 June Q11
4 marks Standard +0.8
11
  1. Show that, for all positive integers \(r\), $$\frac { 1 } { ( r - 1 ) ! } - \frac { 1 } { r ! } = \frac { r - 1 } { r ! }$$ ⟶
    11
  2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { r - 1 } { r ! } = a + \frac { b } { n ! }$$ where \(a\) and \(b\) are integers to be determined.