Questions — AQA Further AS Paper 1 (126 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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AQA Further AS Paper 1 2019 June Q7
5 marks Standard +0.8
  1. Show that $$\frac{1}{r-1} - \frac{1}{r+1} = \frac{A}{r^2-1}$$ where \(A\) is a constant to be found. [1 mark]
  2. Hence use the method of differences to show that $$\sum_{r=2}^n \frac{1}{r^2-1} = \frac{an^2 + bn + c}{4n(n+1)}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4 marks]
AQA Further AS Paper 1 2019 June Q8
7 marks Standard +0.3
Given that \(z_1 = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) and \(z_2 = 2\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\)
  1. Find the value of \(|z_1z_2|\) [1 mark]
  2. Find the value of \(\arg\left(\frac{z_1}{z_2}\right)\) [1 mark]
  3. Sketch \(z_1\) and \(z_2\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively. [2 marks]
  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 \(PRQ\). Fully justify your answer. [3 marks]
AQA Further AS Paper 1 2019 June Q9
7 marks Challenging +1.2
  1. Saul is solving the equation $$2\cosh x + \sinh^2 x = 1$$ He writes his steps as follows: $$2\cosh x + \sinh^2 x = 1$$ $$2\cosh x + 1 - \cosh^2 x = 1$$ $$2\cosh x - \cosh^2 x = 0$$ $$\cosh x \neq 0 \therefore 2 - \cosh x = 0$$ $$\cosh x = 2$$ $$x = \pm \cosh^{-1}(2)$$ Identify and explain the error in Saul's method. [2 marks]
  2. Anna is solving the different equation $$\sinh^2(2x) - 2\cosh(2x) = 1$$ 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. [5 marks]
AQA Further AS Paper 1 2019 June Q10
6 marks Standard +0.3
  1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(e^x\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). [3 marks]
  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 \(\cosh(ix)\). [3 marks]
AQA Further AS Paper 1 2019 June Q11
8 marks Challenging +1.2
  1. Curve \(C\) has equation $$y = \frac{x^2 + px - q}{x^2 - r}$$ where \(p\), \(q\) and \(r\) are positive constants. Write down the equations of its asymptotes. [2 marks]
  2. Find the set of possible \(y\)-coordinates for the graph of $$y = \frac{x^2 + x - 6}{x^2 - 1}, \quad x \neq \pm 1$$ giving your answer in exact form. No credit will be given for solutions based on differentiation. [6 marks]
AQA Further AS Paper 1 2019 June Q12
12 marks Standard +0.8
The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$$
  1. Prove by induction that, for all integers \(n \geq 1\), $$\mathbf{A}^n = \begin{bmatrix} 1 & 3^n - 1 \\ 0 & 3^n \end{bmatrix}$$ [4 marks]
  2. Find all invariant lines under the transformation matrix \(\mathbf{A}\). Fully justify your answer. [6 marks]
  3. Find a line of invariant points under the transformation matrix \(\mathbf{A}\). [2 marks]
AQA Further AS Paper 1 2019 June Q13
10 marks Standard +0.3
Line \(l_1\) has Cartesian equation $$x - 3 = \frac{2y + 2}{3} = 2 - z$$
  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. [2 marks]
  2. Line \(l_2\) passes through the points \(P(3, 2, 0)\) and \(Q(n, 5, n)\), where \(n\) is a constant.
    1. Show that the lines \(l_1\) and \(l_2\) are not perpendicular. [3 marks]
    2. Explain briefly why lines \(l_1\) and \(l_2\) cannot be parallel. [2 marks]
    3. Given that \(\theta\) is the acute angle between lines \(l_1\) and \(l_2\), show that $$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$ where \(p\) and \(q\) are constants to be found. [3 marks]
AQA Further AS Paper 1 2019 June Q14
7 marks Standard +0.8
The graph of \(y = x^3 - 3x\) is shown below. \includegraphics{figure_14} The two stationary points have \(x\)-coordinates of \(-1\) and \(1\) The cubic equation $$x^3 - 3x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha\), \(\beta\) and \(\gamma\). The roots \(\alpha\) and \(\beta\) are not real.
  1. Explain why \(\alpha + \beta = -\gamma\) [1 mark]
  2. Find the set of possible values for the real constant \(p\). [2 marks]
  3. \(f(x) = 0\) is a cubic equation with roots \(\alpha + 1\), \(\beta + 1\) and \(\gamma + 1\)
    1. Show that the constant term of \(f(x)\) is \(p + 2\) [3 marks]
    2. Write down the \(x\)-coordinates of the stationary points of \(y = f(x)\) [1 mark]
AQA Further AS Paper 1 2020 June Q1
1 marks Easy -1.2
Express the complex number \(1 - i\sqrt{3}\) in modulus-argument form. Tick \((\checkmark)\) one box. [1 mark] \(2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) \(2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\) \(2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)\) \(2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)\)
AQA Further AS Paper 1 2020 June Q2
1 marks Moderate -0.8
Given that \(1 - i\) is a root of the equation \(z^3 - 3z^2 + 4z - 2 = 0\), find the other two roots. Tick \((\checkmark)\) one box. [1 mark] \(-1 + i\) and \(-1\) \(1 + i\) and \(1\) \(-1 + i\) and \(1\) \(1 + i\) and \(-1\)
AQA Further AS Paper 1 2020 June Q3
1 marks Moderate -0.8
Given \((x - 1)(x - 2)(x - a) < 0\) and \(a > 2\) Find the set of possible values of \(x\). Tick \((\checkmark)\) one box. [1 mark] \(\{x : x < 1\} \cup \{x : 2 < x < a\}\) \(\{x : 1 < x < 2\} \cup \{x : x > a\}\) \(\{x : x < -a\} \cup \{x : -2 < x < -1\}\) \(\{x : -a < x < -2\} \cup \{x : x > -1\}\)
AQA Further AS Paper 1 2020 June Q4
5 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A} = \begin{bmatrix} 2 & a & 3 \\ 0 & -2 & 1 \end{bmatrix} \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 1 & -3 \\ -2 & 4a \\ 0 & 5 \end{bmatrix}$$
  1. Find the product \(\mathbf{AB}\) in terms of \(a\). [2 marks]
  2. Find the determinant of \(\mathbf{AB}\) in terms of \(a\). [1 mark]
  3. Show that \(\mathbf{AB}\) is singular when \(a = -1\) [2 marks]
AQA Further AS Paper 1 2020 June Q5
4 marks Standard +0.3
  1. Show that $$r^2(r + 1)^2 - (r - 1)^2r^2 = pr^3$$ where \(p\) is an integer to be found. [1 mark]
  2. Hence use the method of differences to show that $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [3 marks]
AQA Further AS Paper 1 2020 June Q6
2 marks Standard +0.3
Anna has been asked to describe the transformation given by the matrix $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$ She writes her answer as follows: The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where $$\sin \theta = \frac{1}{2} \quad \text{and} \quad -\sin \theta = -\frac{1}{2}$$ $$\theta = 30°$$ Identify and correct the error in Anna's work. [2 marks]
AQA Further AS Paper 1 2020 June Q7
4 marks Moderate -0.3
Prove by induction that, for all integers \(n \geq 1\), the expression \(7^n - 3^n\) is divisible by 4 [4 marks]
AQA Further AS Paper 1 2020 June Q8
8 marks Standard +0.3
  1. Prove that $$\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)$$ [5 marks]
  2. Prove that the graphs of $$y = \sinh x \quad \text{and} \quad y = \cosh x$$ do not intersect. [3 marks]
AQA Further AS Paper 1 2020 June Q9
8 marks Standard +0.3
The quadratic equation \(2x^2 + px + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\).
    1. Write down the value of \(\alpha\beta\). [1 mark]
    2. Express \(\alpha + \beta\) in terms of \(p\). [1 mark]
  2. Hence find \((\alpha - \beta)^2\) in terms of \(p\). [2 marks]
  3. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\) [4 marks]
AQA Further AS Paper 1 2020 June Q10
8 marks Standard +0.3
  1. Show that the equation $$y = \frac{3x - 5}{2x + 4}$$ can be written in the form $$(x + a)(y + b) = c$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
  2. Write down the equations of the asymptotes of the graph of $$y = \frac{3x - 5}{2x + 4}$$ [2 marks]
  3. Sketch, on the axes provided, the graph of $$y = \frac{3x - 5}{2x + 4}$$ \includegraphics{figure_10} [3 marks]
AQA Further AS Paper 1 2020 June Q11
3 marks Challenging +1.2
Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]
AQA Further AS Paper 1 2020 June Q12
2 marks Standard +0.8
The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\). The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\) [2 marks]
AQA Further AS Paper 1 2020 June Q13
9 marks Standard +0.8
Line \(l_1\) has equation $$\frac{x - 2}{3} = \frac{1 - 2y}{4} = -z$$ and line \(l_2\) has equation $$\mathbf{r} = \begin{bmatrix} -7 \\ 4 \\ -2 \end{bmatrix} + \mu \begin{bmatrix} 12 \\ a + 3 \\ 2b \end{bmatrix}$$
  1. In the case when \(l_1\) and \(l_2\) are parallel, show that \(a = -11\) and find the value of \(b\). [4 marks]
  2. In a different case, the lines \(l_1\) and \(l_2\) intersect at exactly one point, and the value of \(b\) is 3 Find the value of \(a\). [5 marks]
AQA Further AS Paper 1 2020 June Q14
7 marks Standard +0.8
  1. Given $$\frac{x + 7}{x + 1} \leq x + 1$$ show that $$\frac{(x + a)(x + b)}{x + c} \geq 0$$ where \(a\), \(b\), and \(c\) are integers to be found. [4 marks]
  2. Briefly explain why this statement is incorrect. $$\frac{(x + p)(x + q)}{x + r} \geq 0 \Leftrightarrow (x + p)(x + q)(x + r) \geq 0$$ [1 mark]
  3. Solve $$\frac{x + 7}{x + 1} \leq x + 1$$ [2 marks]
AQA Further AS Paper 1 2020 June Q15
4 marks Standard +0.8
A segment of the line \(y = kx\) is rotated about the \(x\)-axis to generate a cone with vertex \(O\). The distance of \(O\) from the centre of the base of the cone is \(h\). The radius of the base of the cone is \(r\). \includegraphics{figure_15}
  1. Find \(k\) in terms of \(r\) and \(h\). [1 mark]
  2. Use calculus to prove that the volume of the cone is $$\frac{1}{3}\pi r^2 h$$ [3 marks]
AQA Further AS Paper 1 2020 June Q16
4 marks Moderate -0.8
\(\mathbf{A}\) and \(\mathbf{B}\) are non-singular square matrices.
  1. Write down the product \(\mathbf{AA}^{-1}\) as a single matrix. [1 mark]
  2. \(\mathbf{M}\) is a matrix such that \(\mathbf{M} = \mathbf{AB}\). Prove that \(\mathbf{M}^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\) [3 marks]
AQA Further AS Paper 1 2020 June Q17
4 marks Standard +0.8
The polar equation of the circle \(C\) is $$r = a(\cos \theta + \sin \theta)$$ Find, in terms of \(a\), the radius of \(C\). Fully justify your answer. [4 marks]